From 2283ee768f2a006fb8019dbc95edf4bb9b7567da Mon Sep 17 00:00:00 2001 From: Bibin Muttappillil Date: Sun, 31 Jan 2021 20:46:25 +0100 Subject: [PATCH 1/5] README.md -- SOI Edition Adapted original README.md to communicate our purpose --- README.md | 18 ++++++------------ 1 file changed, 6 insertions(+), 12 deletions(-) diff --git a/README.md b/README.md index 53d7c87..c6c7ccc 100644 --- a/README.md +++ b/README.md @@ -1,22 +1,16 @@ -# Competitive Programmer's Handbook +# Competitive Programmer's Handbook -- SOI Edition Competitive Programmer's Handbook is a modern introduction to competitive programming. The book discusses programming tricks and algorithm design techniques relevant in competitive programming. -## CSES Problem Set +We adapted it to fit our SOI training. -The CSES Problem Set contains a collection of competitive programming problems. -You can practice the techniques presented in the book by solving the problems. +## Credits -https://cses.fi/problemset/ +All credits belong to Antti Laaksonen. + +See original: https://github.com/pllk/cphb ## License The license of the book is Creative Commons BY-NC-SA. - -## Other books - -Guide to Competitive Programming is a printed book, published by Springer, based on Competitive Programmer's Handbook. -There is also a Russian edition Олимпиадное программирование (Olympiad Programming) and a Korean edition 알고리즘 트레이닝: 프로그래밍 대회 입문 가이드. - -https://cses.fi/book/ From 00f6d0b74b9da83eff875265020e9b801b011a6b Mon Sep 17 00:00:00 2001 From: bibin Date: Wed, 3 Feb 2021 14:09:17 +0100 Subject: [PATCH 2/5] added output folder --- output/book.aux | 58 + output/book.dvi | Bin 0 -> 4030180 bytes output/book.fdb_latexmk | 355 +++++++ output/book.fls | 819 ++++++++++++++ output/book.idx | 345 ++++++ output/book.ilg | 6 + output/book.ind | 401 +++++++ output/book.log | 2245 +++++++++++++++++++++++++++++++++++++++ output/book.out | 154 +++ output/book.pdf | Bin 0 -> 1114339 bytes output/book.toc | 375 +++++++ output/chapter01.aux | 75 ++ output/chapter02.aux | 54 + output/chapter03.aux | 60 ++ output/chapter04.aux | 61 ++ output/chapter05.aux | 59 + output/chapter06.aux | 55 + output/chapter07.aux | 56 + output/chapter08.aux | 49 + output/chapter09.aux | 62 ++ output/chapter10.aux | 63 ++ output/chapter11.aux | 54 + output/chapter12.aux | 49 + output/chapter13.aux | 61 ++ output/chapter14.aux | 50 + output/chapter15.aux | 57 + output/chapter16.aux | 55 + output/chapter17.aux | 51 + output/chapter18.aux | 57 + output/chapter19.aux | 56 + output/chapter20.aux | 64 ++ output/chapter21.aux | 64 ++ output/chapter22.aux | 59 + 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100644 index 0000000..0e34e52 --- /dev/null +++ b/output/book.fdb_latexmk @@ -0,0 +1,355 @@ +# Fdb version 3 +["makeindex output/book.idx"] 1612133277 "output/book.idx" "output/book.ind" "output/book" 1612133640 + "output/book.idx" 1612133640 14726 a4f79dc0584fe9e391a7573780c527ed "pdflatex" + (generated) + "output/book.ind" + "output/book.ilg" +["pdflatex"] 1612133616 "book.tex" "output/book.pdf" "book" 1612133640 + "/etc/texmf/web2c/texmf.cnf" 1604433920 475 c0e671620eb5563b2130f56340a5fde8 "" + "/usr/share/texlive/texmf-dist/fonts/enc/dvips/base/8r.enc" 1165713224 4850 80dc9bab7f31fb78a000ccfed0e27cab "" + "/usr/share/texlive/texmf-dist/fonts/enc/dvips/inconsolata/i4-t1-0.enc" 1558214095 7631 5169ff1acd97ff1c4ab83563c99f96f4 "" + "/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/adobe/ncntrsbk/pncb8r.tfm" 1136768653 3220 7b5d1a9569a08754f1bdfbc1c7ad8dbd "" + 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"/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecss1000.tfm" 1136768653 3584 ab55608ad377891510479d34d90acc46 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecsx1440.tfm" 1136768653 3584 9bd2baa3a21be3460ad52867ca0084f1 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1246382020 916 f87d7c45f9c908e672703b83b72241a3 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1246382020 928 2dc8d444221b7a635bb58038579b861a "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1246382020 908 2921f8a10601f252058503cc6570e581 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1246382020 940 228d6584342e91276bf566bcf9716b83 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr12.tfm" 1136768653 1288 655e228510b4c2a1abe905c368440826 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/fourier/fourier-bb.tfm" 1575236847 588 30b875a179a99eec0f6f0a2540c942c1 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/fourier/fourier-mcl.tfm" 1136768653 620 19c124db3cfab09e2a270e93c0b94c1c "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/fourier/fourier-mex.tfm" 1575236847 1572 233cc1d2fe9da0b3f64c907e5d58b292 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/fourier/fourier-ml.tfm" 1136768653 1120 281c7872a1f3fc686c053051e921a432 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/fourier/fourier-mlit.tfm" 1136768653 1076 2f8bbd0154a18a0fd665f046ebdfecb3 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/fourier/fourier-ms.tfm" 1136768653 1184 c045430d3c50e3ac06be4ea09e23cd04 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/fourier/futr8t.tfm" 1136768653 16984 6d4d92d3efaef5cab6a470d5d5548d56 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/fouriernc/fncmi.tfm" 1177591358 1368 99619f7bc8a4a375fcb21bf87f7e8b17 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/fouriernc/fncmii.tfm" 1177591358 1704 843e3f158d760e961a55e682635af76d "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/fouriernc/fncsy.tfm" 1177591358 1468 cf5c4cb2bf4b2c1807b5813f5820e2dd "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/inconsolata/t1-zi4r-0.tfm" 1558214095 1328 8f7396ab15954094ca9576957bd516dd "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/skaknew/SkakNew-Diagram.tfm" 1234978116 728 5bcda46594b54e6701133e5bcfdb07af "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/skaknew/SkakNew-Figurine.tfm" 1136768653 656 d5f1171a2115ce2d27a740d37272175a "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb" 1248133631 31764 459c573c03a4949a528c2cc7f557e217 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb" 1248133631 34694 ad62b13721ee8eda1dcc8993c8bd7041 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/fourier/fourier-bb.pfb" 1136849748 6232 c3fc7954848548c26c82678efc1400a6 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/fourier/fourier-mex.pfb" 1575236847 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query|hyperpage}{83} +\indexentry{minimum query|hyperpage}{83} +\indexentry{maximum query|hyperpage}{83} +\indexentry{prefix sum array|hyperpage}{84} +\indexentry{sparse table|hyperpage}{85} +\indexentry{binary indexed tree|hyperpage}{86} +\indexentry{Fenwick tree|hyperpage}{86} +\indexentry{segment tree|hyperpage}{89} +\indexentry{index compression|hyperpage}{93} +\indexentry{difference array|hyperpage}{93} +\indexentry{bit representation|hyperpage}{95} +\indexentry{and operation|hyperpage}{96} +\indexentry{or operation|hyperpage}{96} +\indexentry{xor operation|hyperpage}{97} +\indexentry{not operation|hyperpage}{97} +\indexentry{bit shift|hyperpage}{97} +\indexentry{Hamming distance|hyperpage}{100} +\indexentry{graph|hyperpage}{109} +\indexentry{node|hyperpage}{109} +\indexentry{edge|hyperpage}{109} +\indexentry{path|hyperpage}{109} +\indexentry{cycle|hyperpage}{109} +\indexentry{connected graph|hyperpage}{110} +\indexentry{component|hyperpage}{110} +\indexentry{tree|hyperpage}{110} +\indexentry{directed graph|hyperpage}{110} +\indexentry{weighted graph|hyperpage}{111} +\indexentry{neighbor|hyperpage}{111} +\indexentry{degree|hyperpage}{111} +\indexentry{regular graph|hyperpage}{111} +\indexentry{complete graph|hyperpage}{111} +\indexentry{indegree|hyperpage}{111} +\indexentry{outdegree|hyperpage}{111} +\indexentry{coloring|hyperpage}{112} +\indexentry{bipartite graph|hyperpage}{112} +\indexentry{simple graph|hyperpage}{112} +\indexentry{adjacency list|hyperpage}{113} +\indexentry{adjacency matrix|hyperpage}{114} +\indexentry{edge list|hyperpage}{115} +\indexentry{depth-first search|hyperpage}{117} +\indexentry{breadth-first search|hyperpage}{119} +\indexentry{connected graph|hyperpage}{121} +\indexentry{cycle|hyperpage}{121} +\indexentry{bipartite graph|hyperpage}{122} +\indexentry{shortest path|hyperpage}{123} +\indexentry{Bellman–Ford algorithm|hyperpage}{123} +\indexentry{negative cycle|hyperpage}{125} +\indexentry{SPFA algorithm|hyperpage}{126} +\indexentry{Dijkstra's algorithm|hyperpage}{126} +\indexentry{Floyd–Warshall algorithm|hyperpage}{129} +\indexentry{tree|hyperpage}{133} +\indexentry{leaf|hyperpage}{133} +\indexentry{root|hyperpage}{133} +\indexentry{rooted tree|hyperpage}{133} +\indexentry{child|hyperpage}{133} +\indexentry{parent|hyperpage}{133} +\indexentry{subtree|hyperpage}{133} +\indexentry{diameter|hyperpage}{135} +\indexentry{binary tree|hyperpage}{139} +\indexentry{pre-order|hyperpage}{139} +\indexentry{in-order|hyperpage}{139} +\indexentry{post-order|hyperpage}{139} +\indexentry{spanning tree|hyperpage}{141} +\indexentry{minimum spanning tree|hyperpage}{141} +\indexentry{maximum spanning tree|hyperpage}{142} +\indexentry{Kruskal's algorithm|hyperpage}{142} +\indexentry{union-find structure|hyperpage}{145} +\indexentry{Prim's algorithm|hyperpage}{147} +\indexentry{topological sorting|hyperpage}{149} +\indexentry{cycle|hyperpage}{149} +\indexentry{Dijkstra's algorithm|hyperpage}{153} +\indexentry{successor graph|hyperpage}{154} +\indexentry{functional graph|hyperpage}{154} +\indexentry{cycle|hyperpage}{155} +\indexentry{cycle detection|hyperpage}{155} +\indexentry{Floyd's algorithm|hyperpage}{156} +\indexentry{strongly connected graph|hyperpage}{157} +\indexentry{strongly connected component|hyperpage}{157} +\indexentry{component graph|hyperpage}{157} +\indexentry{Kosaraju's algorithm|hyperpage}{158} +\indexentry{2SAT problem|hyperpage}{160} +\indexentry{3SAT problem|hyperpage}{162} +\indexentry{tree query|hyperpage}{163} +\indexentry{ancestor|hyperpage}{163} +\indexentry{tree traversal array|hyperpage}{164} +\indexentry{lowest common ancestor|hyperpage}{167} +\indexentry{Euler tour technique|hyperpage}{168} +\indexentry{Eulerian path|hyperpage}{173} +\indexentry{Eulerian circuit|hyperpage}{174} +\indexentry{Hierholzer's algorithm|hyperpage}{175} +\indexentry{Hamiltonian path|hyperpage}{177} +\indexentry{Hamiltonian circuit|hyperpage}{177} +\indexentry{Dirac's theorem|hyperpage}{177} +\indexentry{Ore's theorem|hyperpage}{177} +\indexentry{De Bruijn sequence|hyperpage}{178} +\indexentry{knight's tour|hyperpage}{179} +\indexentry{heuristic|hyperpage}{179} +\indexentry{Warnsdorf's rule|hyperpage}{179} +\indexentry{flow|hyperpage}{181} +\indexentry{maximum flow|hyperpage}{181} +\indexentry{cut|hyperpage}{182} +\indexentry{minimum cut|hyperpage}{182} +\indexentry{Ford–Fulkerson algorithm|hyperpage}{182} +\indexentry{Edmonds–Karp algorithm|hyperpage}{184} +\indexentry{scaling algorithm|hyperpage}{185} +\indexentry{minimum cut|hyperpage}{185} +\indexentry{matching|hyperpage}{187} +\indexentry{maximum matching|hyperpage}{187} +\indexentry{Hall's theorem|hyperpage}{189} +\indexentry{perfect matching|hyperpage}{189} +\indexentry{Kőnig's theorem|hyperpage}{189} +\indexentry{node cover|hyperpage}{189} +\indexentry{minimum node cover|hyperpage}{189} +\indexentry{independent set|hyperpage}{190} +\indexentry{maximum independent set|hyperpage}{190} +\indexentry{path cover|hyperpage}{190} +\indexentry{Dilworth's theorem|hyperpage}{193} +\indexentry{antichain|hyperpage}{193} +\indexentry{number theory|hyperpage}{197} +\indexentry{divisibility|hyperpage}{197} +\indexentry{factor|hyperpage}{197} +\indexentry{divisor|hyperpage}{197} +\indexentry{prime|hyperpage}{197} +\indexentry{prime decomposition|hyperpage}{197} +\indexentry{perfect number|hyperpage}{198} +\indexentry{Goldbach's conjecture|hyperpage}{199} +\indexentry{twin prime|hyperpage}{199} +\indexentry{Legendre's conjecture|hyperpage}{199} +\indexentry{sieve of Eratosthenes|hyperpage}{200} +\indexentry{harmonic sum|hyperpage}{200} +\indexentry{greatest common divisor|hyperpage}{200} +\indexentry{least common multiple|hyperpage}{200} +\indexentry{Euclid's algorithm|hyperpage}{200} +\indexentry{coprime|hyperpage}{201} +\indexentry{Euler's totient function|hyperpage}{201} +\indexentry{modular arithmetic|hyperpage}{201} +\indexentry{Fermat's theorem|hyperpage}{202} +\indexentry{Euler's theorem|hyperpage}{202} +\indexentry{modular inverse|hyperpage}{202} +\indexentry{Diophantine equation|hyperpage}{204} +\indexentry{extended Euclid's algorithm|hyperpage}{204} +\indexentry{Chinese remainder theorem|hyperpage}{205} +\indexentry{Lagrange's theorem|hyperpage}{205} +\indexentry{Zeckendorf's theorem|hyperpage}{206} +\indexentry{Fibonacci number|hyperpage}{206} +\indexentry{Pythagorean triple|hyperpage}{206} +\indexentry{Euclid's formula|hyperpage}{206} +\indexentry{Wilson's theorem|hyperpage}{206} +\indexentry{combinatorics|hyperpage}{207} +\indexentry{binomial coefficient|hyperpage}{208} +\indexentry{Pascal's triangle|hyperpage}{209} +\indexentry{multinomial coefficient|hyperpage}{210} +\indexentry{Catalan number|hyperpage}{210} +\indexentry{parenthesis expression|hyperpage}{211} +\indexentry{inclusion-exclusion|hyperpage}{212} +\indexentry{derangement|hyperpage}{213} +\indexentry{Burnside's lemma|hyperpage}{214} +\indexentry{Cayley's formula|hyperpage}{215} +\indexentry{Prüfer code|hyperpage}{216} +\indexentry{matrix|hyperpage}{217} +\indexentry{vector|hyperpage}{217} +\indexentry{transpose|hyperpage}{217} +\indexentry{square matrix|hyperpage}{217} +\indexentry{matrix multiplication|hyperpage}{218} +\indexentry{identity matrix|hyperpage}{218} +\indexentry{matrix power|hyperpage}{219} +\indexentry{determinant|hyperpage}{219} +\indexentry{cofactor|hyperpage}{219} +\indexentry{inverse matrix|hyperpage}{220} +\indexentry{linear recurrence|hyperpage}{220} +\indexentry{Fibonacci number|hyperpage}{220} +\indexentry{Kirchhoff's theorem|hyperpage}{223} +\indexentry{spanning tree|hyperpage}{223} +\indexentry{Laplacean matrix|hyperpage}{224} +\indexentry{probability|hyperpage}{225} +\indexentry{conditional probability|hyperpage}{227} +\indexentry{independence|hyperpage}{228} +\indexentry{random variable|hyperpage}{228} +\indexentry{expected value|hyperpage}{229} +\indexentry{distribution|hyperpage}{229} +\indexentry{uniform distribution|hyperpage}{230} +\indexentry{binomial distribution|hyperpage}{230} +\indexentry{geometric distribution|hyperpage}{230} +\indexentry{Markov chain|hyperpage}{230} +\indexentry{randomized algorithm|hyperpage}{231} +\indexentry{Monte Carlo algorithm|hyperpage}{231} +\indexentry{Las Vegas algorithm|hyperpage}{231} +\indexentry{order statistic|hyperpage}{232} +\indexentry{matrix multiplication|hyperpage}{232} +\indexentry{quicksort|hyperpage}{232} +\indexentry{quickselect|hyperpage}{232} +\indexentry{Freivalds' algoritm|hyperpage}{232} +\indexentry{coloring|hyperpage}{233} +\indexentry{winning state|hyperpage}{235} +\indexentry{losing state|hyperpage}{235} +\indexentry{nim game|hyperpage}{237} +\indexentry{nim sum|hyperpage}{237} +\indexentry{misère game|hyperpage}{238} +\indexentry{Sprague–Grundy theorem|hyperpage}{238} +\indexentry{Grundy number|hyperpage}{238} +\indexentry{mex function|hyperpage}{238} +\indexentry{Grundy's game|hyperpage}{241} +\indexentry{pattern matching|hyperpage}{243} +\indexentry{string|hyperpage}{243} +\indexentry{alphabet|hyperpage}{243} +\indexentry{substring|hyperpage}{243} +\indexentry{subsequence|hyperpage}{243} +\indexentry{prefix|hyperpage}{243} +\indexentry{suffix|hyperpage}{243} +\indexentry{rotation|hyperpage}{243} +\indexentry{period|hyperpage}{243} +\indexentry{border|hyperpage}{244} +\indexentry{lexicographical order|hyperpage}{244} +\indexentry{trie|hyperpage}{244} +\indexentry{hashing|hyperpage}{245} +\indexentry{string hashing|hyperpage}{245} +\indexentry{hash value|hyperpage}{245} +\indexentry{polynomial hashing|hyperpage}{245} +\indexentry{collision|hyperpage}{246} +\indexentry{birthday paradox|hyperpage}{247} +\indexentry{Z-algorithm|hyperpage}{247} +\indexentry{Z-array|hyperpage}{247} +\indexentry{square root algorithm|hyperpage}{251} +\indexentry{Mo's algorithm|hyperpage}{255} +\indexentry{segment tree|hyperpage}{257} +\indexentry{lazy propagation|hyperpage}{258} +\indexentry{lazy segment tree|hyperpage}{258} +\indexentry{dynamic segment tree|hyperpage}{261} +\indexentry{sparse segment tree|hyperpage}{261} +\indexentry{persistent segment tree|hyperpage}{262} +\indexentry{two-dimensional segment tree|hyperpage}{264} +\indexentry{geometry|hyperpage}{265} +\indexentry{Heron's formula|hyperpage}{265} +\indexentry{complex number|hyperpage}{266} +\indexentry{point|hyperpage}{266} +\indexentry{vector|hyperpage}{266} +\indexentry{complex@\texttt{complex}|hyperpage}{266} +\indexentry{cross product|hyperpage}{268} +\indexentry{line segment intersection|hyperpage}{269} +\indexentry{shoelace formula|hyperpage}{271} +\indexentry{Pick's theorem|hyperpage}{272} +\indexentry{distance function|hyperpage}{272} +\indexentry{Euclidean distance|hyperpage}{272} +\indexentry{Manhattan distance|hyperpage}{272} +\indexentry{sweep line|hyperpage}{275} +\indexentry{intersection point|hyperpage}{276} +\indexentry{closest pair|hyperpage}{277} +\indexentry{Andrew's algorithm|hyperpage}{279} diff --git a/output/book.ilg b/output/book.ilg new file mode 100644 index 0000000..511b999 --- /dev/null +++ b/output/book.ilg @@ -0,0 +1,6 @@ +This is makeindex, version 2.15 [TeX Live 2020] (kpathsea + Thai support). +Scanning input file output/book.idx....done (345 entries accepted, 0 rejected). +Sorting entries......done (3076 comparisons). +Generating output file output/book.ind....done (401 lines written, 0 warnings). +Output written in output/book.ind. +Transcript written in output/book.ilg. diff --git a/output/book.ind b/output/book.ind new file mode 100644 index 0000000..f3c505d --- /dev/null +++ b/output/book.ind @@ -0,0 +1,401 @@ +\begin{theindex} + + \item 2SAT problem, \hyperpage{160} + \item 2SUM problem, \hyperpage{78} + \item 3SAT problem, \hyperpage{162} + \item 3SUM problem, \hyperpage{79} + + \indexspace + + \item adjacency list, \hyperpage{113} + \item adjacency matrix, \hyperpage{114} + \item alphabet, \hyperpage{243} + \item amortized analysis, \hyperpage{77} + \item ancestor, \hyperpage{163} + \item and operation, \hyperpage{96} + \item Andrew's algorithm, \hyperpage{279} + \item antichain, \hyperpage{193} + \item arithmetic progression, \hyperpage{10} + + \indexspace + + \item backtracking, \hyperpage{50} + \item Bellman–Ford algorithm, \hyperpage{123} + \item binary code, \hyperpage{62} + \item binary indexed tree, \hyperpage{86} + \item binary search, \hyperpage{31} + \item binary tree, \hyperpage{139} + \item Binet's formula, \hyperpage{14} + \item binomial coefficient, \hyperpage{208} + \item binomial distribution, \hyperpage{230} + \item bipartite graph, \hyperpage{112}, \hyperpage{122} + \item birthday paradox, \hyperpage{247} + \item bit representation, \hyperpage{95} + \item bit shift, \hyperpage{97} + \item bitset, \hyperpage{41} + \item border, \hyperpage{244} + \item breadth-first search, \hyperpage{119} + \item bubble sort, \hyperpage{25} + \item Burnside's lemma, \hyperpage{214} + + \indexspace + + \item Catalan number, \hyperpage{210} + \item Cayley's formula, \hyperpage{215} + \item child, \hyperpage{133} + \item Chinese remainder theorem, \hyperpage{205} + \item closest pair, \hyperpage{277} + \item codeword, \hyperpage{62} + \item cofactor, \hyperpage{219} + \item collision, \hyperpage{246} + \item coloring, \hyperpage{112}, \hyperpage{233} + \item combinatorics, \hyperpage{207} + \item comparison function, \hyperpage{31} + \item comparison operator, \hyperpage{30} + \item complement, \hyperpage{12} + \item complete graph, \hyperpage{111} + \item \texttt{complex}, \hyperpage{266} + \item complex number, \hyperpage{266} + \item complexity classes, \hyperpage{20} + \item component, \hyperpage{110} + \item component graph, \hyperpage{157} + \item conditional probability, \hyperpage{227} + \item conjuction, \hyperpage{13} + \item connected graph, \hyperpage{110}, \hyperpage{121} + \item constant factor, \hyperpage{21} + \item constant-time algorithm, \hyperpage{20} + \item coprime, \hyperpage{201} + \item counting sort, \hyperpage{28} + \item cross product, \hyperpage{268} + \item cubic algorithm, \hyperpage{20} + \item cut, \hyperpage{182} + \item cycle, \hyperpage{109}, \hyperpage{121}, \hyperpage{149}, + \hyperpage{155} + \item cycle detection, \hyperpage{155} + + \indexspace + + \item data compression, \hyperpage{62} + \item data structure, \hyperpage{35} + \item De Bruijn sequence, \hyperpage{178} + \item degree, \hyperpage{111} + \item depth-first search, \hyperpage{117} + \item deque, \hyperpage{42} + \item derangement, \hyperpage{213} + \item determinant, \hyperpage{219} + \item diameter, \hyperpage{135} + \item difference, \hyperpage{12} + \item difference array, \hyperpage{93} + \item Dijkstra's algorithm, \hyperpage{126}, \hyperpage{153} + \item Dilworth's theorem, \hyperpage{193} + \item Diophantine equation, \hyperpage{204} + \item Dirac's theorem, \hyperpage{177} + \item directed graph, \hyperpage{110} + \item disjunction, \hyperpage{13} + \item distance function, \hyperpage{272} + \item distribution, \hyperpage{229} + \item divisibility, \hyperpage{197} + \item divisor, \hyperpage{197} + \item dynamic array, \hyperpage{35} + \item dynamic programming, \hyperpage{65} + \item dynamic segment tree, \hyperpage{261} + + \indexspace + + \item edge, \hyperpage{109} + \item edge list, \hyperpage{115} + \item edit distance, \hyperpage{74} + \item Edmonds–Karp algorithm, \hyperpage{184} + \item equivalence, \hyperpage{13} + \item Euclid's algorithm, \hyperpage{200} + \item Euclid's formula, \hyperpage{206} + \item Euclidean distance, \hyperpage{272} + \item Euler tour technique, \hyperpage{168} + \item Euler's theorem, \hyperpage{202} + \item Euler's totient function, \hyperpage{201} + \item Eulerian circuit, \hyperpage{174} + \item Eulerian path, \hyperpage{173} + \item expected value, \hyperpage{229} + \item extended Euclid's algorithm, \hyperpage{204} + + \indexspace + + \item factor, \hyperpage{197} + \item factorial, \hyperpage{14} + \item Faulhaber's formula, \hyperpage{10} + \item Fenwick tree, \hyperpage{86} + \item Fermat's theorem, \hyperpage{202} + \item Fibonacci number, \hyperpage{14}, \hyperpage{206}, + \hyperpage{220} + \item floating point number, \hyperpage{7} + \item flow, \hyperpage{181} + \item Floyd's algorithm, \hyperpage{156} + \item Floyd–Warshall algorithm, \hyperpage{129} + \item Ford–Fulkerson algorithm, \hyperpage{182} + \item Freivalds' algoritm, \hyperpage{232} + \item functional graph, \hyperpage{154} + + \indexspace + + \item geometric distribution, \hyperpage{230} + \item geometric progression, \hyperpage{11} + \item geometry, \hyperpage{265} + \item Goldbach's conjecture, \hyperpage{199} + \item graph, \hyperpage{109} + \item greatest common divisor, \hyperpage{200} + \item greedy algorithm, \hyperpage{57} + \item Grundy number, \hyperpage{238} + \item Grundy's game, \hyperpage{241} + + \indexspace + + \item Hall's theorem, \hyperpage{189} + \item Hamiltonian circuit, \hyperpage{177} + \item Hamiltonian path, \hyperpage{177} + \item Hamming distance, \hyperpage{100} + \item harmonic sum, \hyperpage{11}, \hyperpage{200} + \item hash value, \hyperpage{245} + \item hashing, \hyperpage{245} + \item heap, \hyperpage{43} + \item Heron's formula, \hyperpage{265} + \item heuristic, \hyperpage{179} + \item Hierholzer's algorithm, \hyperpage{175} + \item Huffman coding, \hyperpage{63} + + \indexspace + + \item identity matrix, \hyperpage{218} + \item implication, \hyperpage{13} + \item in-order, \hyperpage{139} + \item inclusion-exclusion, \hyperpage{212} + \item indegree, \hyperpage{111} + \item independence, \hyperpage{228} + \item independent set, \hyperpage{190} + \item index compression, \hyperpage{93} + \item input and output, \hyperpage{4} + \item integer, \hyperpage{6} + \item intersection, \hyperpage{12} + \item intersection point, \hyperpage{276} + \item inverse matrix, \hyperpage{220} + \item inversion, \hyperpage{26} + \item iterator, \hyperpage{39} + + \indexspace + + \item Kadane's algorithm, \hyperpage{23} + \item Kirchhoff's theorem, \hyperpage{223} + \item knapsack, \hyperpage{72} + \item knight's tour, \hyperpage{179} + \item Kosaraju's algorithm, \hyperpage{158} + \item Kruskal's algorithm, \hyperpage{142} + \item Kőnig's theorem, \hyperpage{189} + + \indexspace + + \item Lagrange's theorem, \hyperpage{205} + \item Laplacean matrix, \hyperpage{224} + \item Las Vegas algorithm, \hyperpage{231} + \item lazy propagation, \hyperpage{258} + \item lazy segment tree, \hyperpage{258} + \item leaf, \hyperpage{133} + \item least common multiple, \hyperpage{200} + \item Legendre's conjecture, \hyperpage{199} + \item Levenshtein distance, \hyperpage{74} + \item lexicographical order, \hyperpage{244} + \item line segment intersection, \hyperpage{269} + \item linear algorithm, \hyperpage{20} + \item linear recurrence, \hyperpage{220} + \item logarithm, \hyperpage{14} + \item logarithmic algorithm, \hyperpage{20} + \item logic, \hyperpage{13} + \item longest increasing subsequence, \hyperpage{70} + \item losing state, \hyperpage{235} + \item lowest common ancestor, \hyperpage{167} + + \indexspace + + \item macro, \hyperpage{9} + \item Manhattan distance, \hyperpage{272} + \item map, \hyperpage{38} + \item Markov chain, \hyperpage{230} + \item matching, \hyperpage{187} + \item matrix, \hyperpage{217} + \item matrix multiplication, \hyperpage{218}, \hyperpage{232} + \item matrix power, \hyperpage{219} + \item maximum flow, \hyperpage{181} + \item maximum independent set, \hyperpage{190} + \item maximum matching, \hyperpage{187} + \item maximum query, \hyperpage{83} + \item maximum spanning tree, \hyperpage{142} + \item maximum subarray sum, \hyperpage{21} + \item meet in the middle, \hyperpage{54} + \item memoization, \hyperpage{67} + \item merge sort, \hyperpage{27} + \item mex function, \hyperpage{238} + \item minimum cut, \hyperpage{182}, \hyperpage{185} + \item minimum node cover, \hyperpage{189} + \item minimum query, \hyperpage{83} + \item minimum spanning tree, \hyperpage{141} + \item misère game, \hyperpage{238} + \item Mo's algorithm, \hyperpage{255} + \item modular arithmetic, \hyperpage{6}, \hyperpage{201} + \item modular inverse, \hyperpage{202} + \item Monte Carlo algorithm, \hyperpage{231} + \item multinomial coefficient, \hyperpage{210} + + \indexspace + + \item natural logarithm, \hyperpage{15} + \item nearest smaller elements, \hyperpage{79} + \item negation, \hyperpage{13} + \item negative cycle, \hyperpage{125} + \item neighbor, \hyperpage{111} + \item \texttt{next\_permutation}, \hyperpage{49} + \item nim game, \hyperpage{237} + \item nim sum, \hyperpage{237} + \item node, \hyperpage{109} + \item node cover, \hyperpage{189} + \item not operation, \hyperpage{97} + \item NP-hard problem, \hyperpage{20} + \item number theory, \hyperpage{197} + + \indexspace + + \item or operation, \hyperpage{96} + \item order statistic, \hyperpage{232} + \item Ore's theorem, \hyperpage{177} + \item outdegree, \hyperpage{111} + + \indexspace + + \item \texttt{pair}, \hyperpage{30} + \item parent, \hyperpage{133} + \item parenthesis expression, \hyperpage{211} + \item Pascal's triangle, \hyperpage{209} + \item path, \hyperpage{109} + \item path cover, \hyperpage{190} + \item pattern matching, \hyperpage{243} + \item perfect matching, \hyperpage{189} + \item perfect number, \hyperpage{198} + \item period, \hyperpage{243} + \item permutation, \hyperpage{49} + \item persistent segment tree, \hyperpage{262} + \item Pick's theorem, \hyperpage{272} + \item point, \hyperpage{266} + \item polynomial algorithm, \hyperpage{20} + \item polynomial hashing, \hyperpage{245} + \item post-order, \hyperpage{139} + \item pre-order, \hyperpage{139} + \item predicate, \hyperpage{13} + \item prefix, \hyperpage{243} + \item prefix sum array, \hyperpage{84} + \item Prim's algorithm, \hyperpage{147} + \item prime, \hyperpage{197} + \item prime decomposition, \hyperpage{197} + \item priority queue, \hyperpage{43} + \item probability, \hyperpage{225} + \item programming language, \hyperpage{3} + \item Prüfer code, \hyperpage{216} + \item Pythagorean triple, \hyperpage{206} + + \indexspace + + \item quadratic algorithm, \hyperpage{20} + \item quantifier, \hyperpage{13} + \item queen problem, \hyperpage{50} + \item queue, \hyperpage{43} + \item quickselect, \hyperpage{232} + \item quicksort, \hyperpage{232} + + \indexspace + + \item random variable, \hyperpage{228} + \item \texttt{random\_shuffle}, \hyperpage{39} + \item randomized algorithm, \hyperpage{231} + \item range query, \hyperpage{83} + \item regular graph, \hyperpage{111} + \item remainder, \hyperpage{6} + \item \texttt{reverse}, \hyperpage{39} + \item root, \hyperpage{133} + \item rooted tree, \hyperpage{133} + \item rotation, \hyperpage{243} + + \indexspace + + \item scaling algorithm, \hyperpage{185} + \item segment tree, \hyperpage{89}, \hyperpage{257} + \item set, \hyperpage{12}, \hyperpage{37} + \item set theory, \hyperpage{12} + \item shoelace formula, \hyperpage{271} + \item shortest path, \hyperpage{123} + \item sieve of Eratosthenes, \hyperpage{200} + \item simple graph, \hyperpage{112} + \item sliding window, \hyperpage{81} + \item sliding window minimum, \hyperpage{81} + \item \texttt{sort}, \hyperpage{29}, \hyperpage{39} + \item sorting, \hyperpage{25} + \item spanning tree, \hyperpage{141}, \hyperpage{223} + \item sparse segment tree, \hyperpage{261} + \item sparse table, \hyperpage{85} + \item SPFA algorithm, \hyperpage{126} + \item Sprague–Grundy theorem, \hyperpage{238} + \item square matrix, \hyperpage{217} + \item square root algorithm, \hyperpage{251} + \item stack, \hyperpage{42} + \item string, \hyperpage{36}, \hyperpage{243} + \item string hashing, \hyperpage{245} + \item strongly connected component, \hyperpage{157} + \item strongly connected graph, \hyperpage{157} + \item subsequence, \hyperpage{243} + \item subset, \hyperpage{12}, \hyperpage{47} + \item substring, \hyperpage{243} + \item subtree, \hyperpage{133} + \item successor graph, \hyperpage{154} + \item suffix, \hyperpage{243} + \item sum query, \hyperpage{83} + \item sweep line, \hyperpage{275} + + \indexspace + + \item time complexity, \hyperpage{17} + \item topological sorting, \hyperpage{149} + \item transpose, \hyperpage{217} + \item tree, \hyperpage{110}, \hyperpage{133} + \item tree query, \hyperpage{163} + \item tree traversal array, \hyperpage{164} + \item trie, \hyperpage{244} + \item \texttt{tuple}, \hyperpage{30} + \item \texttt{typedef}, \hyperpage{8} + \item twin prime, \hyperpage{199} + \item two pointers method, \hyperpage{77} + \item two-dimensional segment tree, \hyperpage{264} + + \indexspace + + \item uniform distribution, \hyperpage{230} + \item union, \hyperpage{12} + \item union-find structure, \hyperpage{145} + \item universal set, \hyperpage{12} + + \indexspace + + \item vector, \hyperpage{35}, \hyperpage{217}, \hyperpage{266} + + \indexspace + + \item Warnsdorf's rule, \hyperpage{179} + \item weighted graph, \hyperpage{111} + \item Wilson's theorem, \hyperpage{206} + \item winning state, \hyperpage{235} + + \indexspace + + \item xor operation, \hyperpage{97} + + \indexspace + + \item Z-algorithm, \hyperpage{247} + \item Z-array, \hyperpage{247} + \item Zeckendorf's theorem, \hyperpage{206} + +\end{theindex} diff --git a/output/book.log b/output/book.log new file mode 100644 index 0000000..0daf69a --- /dev/null +++ b/output/book.log @@ -0,0 +1,2245 @@ +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020/Debian) (preloaded format=pdflatex 2021.1.31) 31 JAN 2021 23:53 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**book.tex +(./book.tex +LaTeX2e <2020-02-02> patch level 5 +L3 programming layer <2020-07-17> +(/usr/share/texlive/texmf-dist/tex/latex/base/book.cls +Document Class: book 2019/12/20 v1.4l Standard LaTeX document class +(/usr/share/texlive/texmf-dist/tex/latex/base/bk12.clo +File: bk12.clo 2019/12/20 v1.4l Standard 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[1][-]{section.6.2}{Scheduling}{chapter.6}% 34 +\BOOKMARK [1][-]{section.6.3}{Tasks and deadlines}{chapter.6}% 35 +\BOOKMARK [1][-]{section.6.4}{Minimizing sums}{chapter.6}% 36 +\BOOKMARK [1][-]{section.6.5}{Data compression}{chapter.6}% 37 +\BOOKMARK [0][-]{chapter.7}{Dynamic programming}{part.1}% 38 +\BOOKMARK [1][-]{section.7.1}{Coin problem}{chapter.7}% 39 +\BOOKMARK [1][-]{section.7.2}{Longest increasing subsequence}{chapter.7}% 40 +\BOOKMARK [1][-]{section.7.3}{Paths in a grid}{chapter.7}% 41 +\BOOKMARK [1][-]{section.7.4}{Knapsack problems}{chapter.7}% 42 +\BOOKMARK [1][-]{section.7.5}{Edit distance}{chapter.7}% 43 +\BOOKMARK [1][-]{section.7.6}{Counting tilings}{chapter.7}% 44 +\BOOKMARK [0][-]{chapter.8}{Amortized analysis}{part.1}% 45 +\BOOKMARK [1][-]{section.8.1}{Two pointers method}{chapter.8}% 46 +\BOOKMARK [1][-]{section.8.2}{Nearest smaller elements}{chapter.8}% 47 +\BOOKMARK [1][-]{section.8.3}{Sliding window minimum}{chapter.8}% 48 +\BOOKMARK [0][-]{chapter.9}{Range queries}{part.1}% 49 +\BOOKMARK [1][-]{section.9.1}{Static array queries}{chapter.9}% 50 +\BOOKMARK [1][-]{section.9.2}{Binary indexed tree}{chapter.9}% 51 +\BOOKMARK [1][-]{section.9.3}{Segment tree}{chapter.9}% 52 +\BOOKMARK [1][-]{section.9.4}{Additional techniques}{chapter.9}% 53 +\BOOKMARK [0][-]{chapter.10}{Bit manipulation}{part.1}% 54 +\BOOKMARK [1][-]{section.10.1}{Bit representation}{chapter.10}% 55 +\BOOKMARK [1][-]{section.10.2}{Bit operations}{chapter.10}% 56 +\BOOKMARK [1][-]{section.10.3}{Representing sets}{chapter.10}% 57 +\BOOKMARK [1][-]{section.10.4}{Bit optimizations}{chapter.10}% 58 +\BOOKMARK [1][-]{section.10.5}{Dynamic programming}{chapter.10}% 59 +\BOOKMARK [-1][-]{part.2}{II Graph algorithms}{}% 60 +\BOOKMARK [0][-]{chapter.11}{Basics of graphs}{part.2}% 61 +\BOOKMARK [1][-]{section.11.1}{Graph terminology}{chapter.11}% 62 +\BOOKMARK [1][-]{section.11.2}{Graph representation}{chapter.11}% 63 +\BOOKMARK [0][-]{chapter.12}{Graph traversal}{part.2}% 64 +\BOOKMARK [1][-]{section.12.1}{Depth-first search}{chapter.12}% 65 +\BOOKMARK [1][-]{section.12.2}{Breadth-first search}{chapter.12}% 66 +\BOOKMARK [1][-]{section.12.3}{Applications}{chapter.12}% 67 +\BOOKMARK [0][-]{chapter.13}{Shortest paths}{part.2}% 68 +\BOOKMARK [1][-]{section.13.1}{Bellman\205Ford algorithm}{chapter.13}% 69 +\BOOKMARK [1][-]{section.13.2}{Dijkstra's algorithm}{chapter.13}% 70 +\BOOKMARK [1][-]{section.13.3}{Floyd\205Warshall algorithm}{chapter.13}% 71 +\BOOKMARK [0][-]{chapter.14}{Tree algorithms}{part.2}% 72 +\BOOKMARK [1][-]{section.14.1}{Tree traversal}{chapter.14}% 73 +\BOOKMARK [1][-]{section.14.2}{Diameter}{chapter.14}% 74 +\BOOKMARK [1][-]{section.14.3}{All longest paths}{chapter.14}% 75 +\BOOKMARK [1][-]{section.14.4}{Binary trees}{chapter.14}% 76 +\BOOKMARK [0][-]{chapter.15}{Spanning trees}{part.2}% 77 +\BOOKMARK [1][-]{section.15.1}{Kruskal's algorithm}{chapter.15}% 78 +\BOOKMARK [1][-]{section.15.2}{Union-find structure}{chapter.15}% 79 +\BOOKMARK [1][-]{section.15.3}{Prim's algorithm}{chapter.15}% 80 +\BOOKMARK [0][-]{chapter.16}{Directed graphs}{part.2}% 81 +\BOOKMARK [1][-]{section.16.1}{Topological sorting}{chapter.16}% 82 +\BOOKMARK [1][-]{section.16.2}{Dynamic programming}{chapter.16}% 83 +\BOOKMARK [1][-]{section.16.3}{Successor paths}{chapter.16}% 84 +\BOOKMARK [1][-]{section.16.4}{Cycle detection}{chapter.16}% 85 +\BOOKMARK [0][-]{chapter.17}{Strong connectivity}{part.2}% 86 +\BOOKMARK [1][-]{section.17.1}{Kosaraju's algorithm}{chapter.17}% 87 +\BOOKMARK [1][-]{section.17.2}{2SAT problem}{chapter.17}% 88 +\BOOKMARK [0][-]{chapter.18}{Tree queries}{part.2}% 89 +\BOOKMARK [1][-]{section.18.1}{Finding ancestors}{chapter.18}% 90 +\BOOKMARK [1][-]{section.18.2}{Subtrees and paths}{chapter.18}% 91 +\BOOKMARK [1][-]{section.18.3}{Lowest common ancestor}{chapter.18}% 92 +\BOOKMARK [1][-]{section.18.4}{Offline algorithms}{chapter.18}% 93 +\BOOKMARK [0][-]{chapter.19}{Paths and circuits}{part.2}% 94 +\BOOKMARK [1][-]{section.19.1}{Eulerian paths}{chapter.19}% 95 +\BOOKMARK [1][-]{section.19.2}{Hamiltonian paths}{chapter.19}% 96 +\BOOKMARK [1][-]{section.19.3}{De Bruijn sequences}{chapter.19}% 97 +\BOOKMARK [1][-]{section.19.4}{Knight's tours}{chapter.19}% 98 +\BOOKMARK [0][-]{chapter.20}{Flows and cuts}{part.2}% 99 +\BOOKMARK [1][-]{section.20.1}{Ford\205Fulkerson algorithm}{chapter.20}% 100 +\BOOKMARK [1][-]{section.20.2}{Disjoint paths}{chapter.20}% 101 +\BOOKMARK [1][-]{section.20.3}{Maximum matchings}{chapter.20}% 102 +\BOOKMARK [1][-]{section.20.4}{Path covers}{chapter.20}% 103 +\BOOKMARK [-1][-]{part.3}{III Advanced topics}{}% 104 +\BOOKMARK [0][-]{chapter.21}{Number theory}{part.3}% 105 +\BOOKMARK [1][-]{section.21.1}{Primes and factors}{chapter.21}% 106 +\BOOKMARK [1][-]{section.21.2}{Modular arithmetic}{chapter.21}% 107 +\BOOKMARK [1][-]{section.21.3}{Solving equations}{chapter.21}% 108 +\BOOKMARK [1][-]{section.21.4}{Other results}{chapter.21}% 109 +\BOOKMARK [0][-]{chapter.22}{Combinatorics}{part.3}% 110 +\BOOKMARK [1][-]{section.22.1}{Binomial coefficients}{chapter.22}% 111 +\BOOKMARK [1][-]{section.22.2}{Catalan numbers}{chapter.22}% 112 +\BOOKMARK [1][-]{section.22.3}{Inclusion-exclusion}{chapter.22}% 113 +\BOOKMARK [1][-]{section.22.4}{Burnside's lemma}{chapter.22}% 114 +\BOOKMARK [1][-]{section.22.5}{Cayley's formula}{chapter.22}% 115 +\BOOKMARK [0][-]{chapter.23}{Matrices}{part.3}% 116 +\BOOKMARK [1][-]{section.23.1}{Operations}{chapter.23}% 117 +\BOOKMARK [1][-]{section.23.2}{Linear recurrences}{chapter.23}% 118 +\BOOKMARK [1][-]{section.23.3}{Graphs and matrices}{chapter.23}% 119 +\BOOKMARK [0][-]{chapter.24}{Probability}{part.3}% 120 +\BOOKMARK [1][-]{section.24.1}{Calculation}{chapter.24}% 121 +\BOOKMARK [1][-]{section.24.2}{Events}{chapter.24}% 122 +\BOOKMARK [1][-]{section.24.3}{Random variables}{chapter.24}% 123 +\BOOKMARK [1][-]{section.24.4}{Markov chains}{chapter.24}% 124 +\BOOKMARK [1][-]{section.24.5}{Randomized algorithms}{chapter.24}% 125 +\BOOKMARK [0][-]{chapter.25}{Game theory}{part.3}% 126 +\BOOKMARK [1][-]{section.25.1}{Game states}{chapter.25}% 127 +\BOOKMARK [1][-]{section.25.2}{Nim game}{chapter.25}% 128 +\BOOKMARK [1][-]{section.25.3}{Sprague\205Grundy theorem}{chapter.25}% 129 +\BOOKMARK [0][-]{chapter.26}{String algorithms}{part.3}% 130 +\BOOKMARK [1][-]{section.26.1}{String terminology}{chapter.26}% 131 +\BOOKMARK [1][-]{section.26.2}{Trie structure}{chapter.26}% 132 +\BOOKMARK [1][-]{section.26.3}{String hashing}{chapter.26}% 133 +\BOOKMARK [1][-]{section.26.4}{Z-algorithm}{chapter.26}% 134 +\BOOKMARK [0][-]{chapter.27}{Square root algorithms}{part.3}% 135 +\BOOKMARK [1][-]{section.27.1}{Combining algorithms}{chapter.27}% 136 +\BOOKMARK [1][-]{section.27.2}{Integer partitions}{chapter.27}% 137 +\BOOKMARK [1][-]{section.27.3}{Mo's algorithm}{chapter.27}% 138 +\BOOKMARK [0][-]{chapter.28}{Segment trees revisited}{part.3}% 139 +\BOOKMARK [1][-]{section.28.1}{Lazy propagation}{chapter.28}% 140 +\BOOKMARK [1][-]{section.28.2}{Dynamic trees}{chapter.28}% 141 +\BOOKMARK [1][-]{section.28.3}{Data structures}{chapter.28}% 142 +\BOOKMARK [1][-]{section.28.4}{Two-dimensionality}{chapter.28}% 143 +\BOOKMARK [0][-]{chapter.29}{Geometry}{part.3}% 144 +\BOOKMARK [1][-]{section.29.1}{Complex numbers}{chapter.29}% 145 +\BOOKMARK [1][-]{section.29.2}{Points and lines}{chapter.29}% 146 +\BOOKMARK [1][-]{section.29.3}{Polygon area}{chapter.29}% 147 +\BOOKMARK [1][-]{section.29.4}{Distance functions}{chapter.29}% 148 +\BOOKMARK [0][-]{chapter.30}{Sweep line algorithms}{part.3}% 149 +\BOOKMARK [1][-]{section.30.1}{Intersection points}{chapter.30}% 150 +\BOOKMARK [1][-]{section.30.2}{Closest pair problem}{chapter.30}% 151 +\BOOKMARK [1][-]{section.30.3}{Convex hull problem}{chapter.30}% 152 +\BOOKMARK [0][-]{section*.3}{Bibliography}{part.3}% 153 +\BOOKMARK [0][-]{chapter*.5}{Index}{part.3}% 154 diff --git a/output/book.pdf b/output/book.pdf new file mode 100644 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{subsubsection}{Algorithm 2}{6}{section.1.2}% +\contentsline {section}{\numberline {1.3}All longest paths}{7}{section.1.3}% +\contentsline {section}{\numberline {1.4}Binary trees}{9}{section.1.4}% +\contentsline {chapter}{Bibliography}{11}{section*.2}% +\contentsline {chapter}{Index}{17}{chapter*.4}% diff --git a/test.tex b/test.tex new file mode 100644 index 0000000..e108ae9 --- /dev/null +++ b/test.tex @@ -0,0 +1,100 @@ +\documentclass[twoside,12pt,a4paper,english]{book} + +%\includeonly{chapter04,list} + +\usepackage[english]{babel} +\usepackage[utf8]{inputenc} +\usepackage{listings} +\usepackage[table]{xcolor} +\usepackage{tikz} +\usepackage{multicol} +\usepackage{hyperref} +\usepackage{array} +\usepackage{microtype} + +\usepackage{fouriernc} +\usepackage[T1]{fontenc} + +\usepackage{graphicx} +\usepackage{framed} +\usepackage{amssymb} +\usepackage{amsmath} + +\usepackage{pifont} +\usepackage{ifthen} +\usepackage{makeidx} +\usepackage{enumitem} + +\usepackage{titlesec} + +\usepackage{skak} +\usepackage[scaled=0.95]{inconsolata} + + +\usetikzlibrary{patterns,snakes} +\pagestyle{plain} + +\definecolor{keywords}{HTML}{44548A} +\definecolor{strings}{HTML}{00999A} +\definecolor{comments}{HTML}{990000} + +\lstset{language=C++,frame=single,basicstyle=\ttfamily \small,showstringspaces=false,columns=flexible} +\lstset{ + literate={ö}{{\"o}}1 + {ä}{{\"a}}1 + {ü}{{\"u}}1 +} +\lstset{xleftmargin=20pt,xrightmargin=5pt} +\lstset{aboveskip=12pt,belowskip=8pt} + +\lstset{ + commentstyle=\color{comments}, + keywordstyle=\color{keywords}, + stringstyle=\color{strings} +} + +\date{Draft \today} + +\usepackage[a4paper,vmargin=30mm,hmargin=33mm,footskip=15mm]{geometry} + +\title{\Huge Competitive Programmer's Handbook} +\author{\Large Antti Laaksonen} + +\makeindex +\usepackage[totoc]{idxlayout} + +\titleformat{\subsubsection} +{\normalfont\large\bfseries\sffamily}{\thesubsection}{1em}{} + +\begin{document} + +%\selectlanguage{finnish} + +%\setcounter{page}{1} +%\pagenumbering{roman} + +%\frontmatter +%\maketitle +%\setcounter{tocdepth}{1} +%\tableofcontents + +%\include{preface} + +\mainmatter +\pagenumbering{arabic} +\setcounter{page}{1} + +\newcommand{\key}[1] {\textbf{#1}} + +%\part{Graph algorithms} +\include{chapter14} + +%\cleardoublepage +%\phantomsection +%\addcontentsline{toc}{chapter}{Bibliography} +%\include{list} + +%\cleardoublepage +%\printindex + +\end{document} \ No newline at end of file From 55a61a0050eea0d3e5e9f469c4b97eb759ed3699 Mon Sep 17 00:00:00 2001 From: Johannes Kapfhammer Date: Mon, 8 Feb 2021 23:10:42 +0100 Subject: [PATCH 3/5] graph day version --- book.pdf | Bin 1118908 -> 315920 bytes book.tex | 59 +++--- chapter11.tex | 93 ++++---- chapter12.tex | 12 +- chapter13.tex | 578 +++----------------------------------------------- chapter14.tex | 52 +++-- chapter16.tex | 475 +---------------------------------------- chapter17.tex | 202 ------------------ chapter18.tex | 2 +- preface.tex | 34 +-- 10 files changed, 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-\author{\Large Antti Laaksonen} +\title{\Huge SOI Camp 2021 -- Graph Day} +\author{\Large Competitive Programmer’s Handbook by Antti Laaksonen} \makeindex \usepackage[totoc]{idxlayout} @@ -86,39 +86,42 @@ \newcommand{\key}[1] {\textbf{#1}} -\part{Basic techniques} -\include{chapter01} -\include{chapter02} -\include{chapter03} -\include{chapter04} -\include{chapter05} -\include{chapter06} -\include{chapter07} -\include{chapter08} -\include{chapter09} -\include{chapter10} -\part{Graph algorithms} +%\part{Basic techniques} +%\include{chapter01} +%\include{chapter02} +%\include{chapter03} +%\include{chapter04} +%\include{chapter05} +%\include{chapter06} +%\include{chapter07} +%\include{chapter08} +%\include{chapter09} +%\include{chapter10} +\part{Main Topics} \include{chapter11} \include{chapter12} \include{chapter13} \include{chapter14} -\include{chapter15} \include{chapter16} +%\include{chapter20} + +\part{Advanced topics} +\include{chapter15} \include{chapter17} \include{chapter18} \include{chapter19} -\include{chapter20} -\part{Advanced topics} -\include{chapter21} -\include{chapter22} -\include{chapter23} -\include{chapter24} -\include{chapter25} -\include{chapter26} -\include{chapter27} -\include{chapter28} -\include{chapter29} -\include{chapter30} + +% \part{Advanced topics} +%\include{chapter21} +%\include{chapter22} +%\include{chapter23} +%\include{chapter24} +%\include{chapter25} +%\include{chapter26} +%\include{chapter27} +%\include{chapter28} +%\include{chapter29} +%\include{chapter30} \cleardoublepage \phantomsection diff --git a/chapter11.tex b/chapter11.tex index c659125..efa605d 100644 --- a/chapter11.tex +++ b/chapter11.tex @@ -478,21 +478,19 @@ way to represent graphs, and most algorithms can be efficiently implemented using them. A convenient way to store the adjacency lists is to declare -an array of vectors as follows: +a vector of vectors as follows: \begin{lstlisting} -vector adj[N]; +vector> g; \end{lstlisting} -The constant $N$ is chosen so that all -adjacency lists can be stored. For example, the graph \begin{center} \begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (1,3) {$1$}; -\node[draw, circle] (2) at (3,3) {$2$}; -\node[draw, circle] (3) at (5,3) {$3$}; -\node[draw, circle] (4) at (3,1) {$4$}; +\node[draw, circle] (1) at (1,3) {$0$}; +\node[draw, circle] (2) at (3,3) {$1$}; +\node[draw, circle] (3) at (5,3) {$2$}; +\node[draw, circle] (4) at (3,1) {$3$}; \path[draw,thick,->,>=latex] (1) -- (2); \path[draw,thick,->,>=latex] (2) -- (3); @@ -503,11 +501,12 @@ For example, the graph \end{center} can be stored as follows: \begin{lstlisting} -adj[1].push_back(2); -adj[2].push_back(3); -adj[2].push_back(4); -adj[3].push_back(4); -adj[4].push_back(1); +g.assign(4, {}); // g now consists of 4 empty arrays +g[0].push_back(1); +g[1].push_back(2); +g[1].push_back(3); +g[2].push_back(3); +g[3].push_back(0); \end{lstlisting} If the graph is undirected, it can be stored in a similar way, @@ -517,7 +516,7 @@ For a weighted graph, the structure can be extended as follows: \begin{lstlisting} -vector> adj[N]; +vector>> g; \end{lstlisting} In this case, the adjacency list of node $a$ @@ -527,10 +526,10 @@ with weight $w$. For example, the graph \begin{center} \begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (1,3) {$1$}; -\node[draw, circle] (2) at (3,3) {$2$}; -\node[draw, circle] (3) at (5,3) {$3$}; -\node[draw, circle] (4) at (3,1) {$4$}; +\node[draw, circle] (1) at (1,3) {$0$}; +\node[draw, circle] (2) at (3,3) {$1$}; +\node[draw, circle] (3) at (5,3) {$2$}; +\node[draw, circle] (4) at (3,1) {$3$}; \path[draw,thick,->,>=latex] (1) -- node[font=\small,label=above:5] {} (2); \path[draw,thick,->,>=latex] (2) -- node[font=\small,label=above:7] {} (3); @@ -541,11 +540,12 @@ with weight $w$. For example, the graph \end{center} can be stored as follows: \begin{lstlisting} -adj[1].push_back({2,5}); -adj[2].push_back({3,7}); -adj[2].push_back({4,6}); -adj[3].push_back({4,5}); -adj[4].push_back({1,2}); +g.assign(4, {}); +g[0].push_back({1,5}); +g[1].push_back({2,7}); +g[1].push_back({3,6}); +g[2].push_back({3,5}); +g[3].push_back({0,2}); \end{lstlisting} The benefit of using adjacency lists is that @@ -555,7 +555,7 @@ For example, the following loop goes through all nodes to which we can move from node $s$: \begin{lstlisting} -for (auto u : adj[s]) { +for (auto u : g[s]) { // process node u } \end{lstlisting} @@ -570,7 +570,8 @@ We can efficiently check from an adjacency matrix if there is an edge between two nodes. The matrix can be stored as an array \begin{lstlisting} -int adj[N][N]; +vector> adj; +adj.assign(n, vector(n, 0)); \end{lstlisting} where each value $\texttt{adj}[a][b]$ indicates whether the graph contains an edge from @@ -581,10 +582,10 @@ and otherwise $\texttt{adj}[a][b]=0$. For example, the graph \begin{center} \begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (1,3) {$1$}; -\node[draw, circle] (2) at (3,3) {$2$}; -\node[draw, circle] (3) at (5,3) {$3$}; -\node[draw, circle] (4) at (3,1) {$4$}; +\node[draw, circle] (1) at (1,3) {$0$}; +\node[draw, circle] (2) at (3,3) {$1$}; +\node[draw, circle] (3) at (5,3) {$2$}; +\node[draw, circle] (4) at (3,1) {$3$}; \path[draw,thick,->,>=latex] (1) -- (2); \path[draw,thick,->,>=latex] (2) -- (3); @@ -704,10 +705,10 @@ Thus, the graph \begin{center} \begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (1,3) {$1$}; -\node[draw, circle] (2) at (3,3) {$2$}; -\node[draw, circle] (3) at (5,3) {$3$}; -\node[draw, circle] (4) at (3,1) {$4$}; +\node[draw, circle] (1) at (1,3) {$0$}; +\node[draw, circle] (2) at (3,3) {$1$}; +\node[draw, circle] (3) at (5,3) {$2$}; +\node[draw, circle] (4) at (3,1) {$3$}; \path[draw,thick,->,>=latex] (1) -- (2); \path[draw,thick,->,>=latex] (2) -- (3); @@ -718,11 +719,11 @@ Thus, the graph \end{center} can be represented as follows: \begin{lstlisting} -edges.push_back({1,2}); -edges.push_back({2,3}); +edges.push_back({0,2}); +edges.push_back({1,3}); +edges.push_back({1,4}); edges.push_back({2,4}); -edges.push_back({3,4}); -edges.push_back({4,1}); +edges.push_back({3,1}); \end{lstlisting} \noindent @@ -738,10 +739,10 @@ For example, the graph \begin{center} \begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (1,3) {$1$}; -\node[draw, circle] (2) at (3,3) {$2$}; -\node[draw, circle] (3) at (5,3) {$3$}; -\node[draw, circle] (4) at (3,1) {$4$}; +\node[draw, circle] (1) at (1,3) {$0$}; +\node[draw, circle] (2) at (3,3) {$1$}; +\node[draw, circle] (3) at (5,3) {$2$}; +\node[draw, circle] (4) at (3,1) {$3$}; \path[draw,thick,->,>=latex] (1) -- node[font=\small,label=above:5] {} (2); \path[draw,thick,->,>=latex] (2) -- node[font=\small,label=above:7] {} (3); @@ -755,10 +756,10 @@ can be represented as follows\footnote{In some older compilers, the function \texttt{make\_tuple} must be used instead of the braces (for example, \texttt{make\_tuple(1,2,5)} instead of \texttt{\{1,2,5\}}).}: \begin{lstlisting} -edges.push_back({1,2,5}); -edges.push_back({2,3,7}); -edges.push_back({2,4,6}); -edges.push_back({3,4,5}); -edges.push_back({4,1,2}); +edges.push_back({0,2,5}); +edges.push_back({1,3,7}); +edges.push_back({1,4,6}); +edges.push_back({2,4,5}); +edges.push_back({3,1,2}); \end{lstlisting} \end{samepage} diff --git a/chapter12.tex b/chapter12.tex index fec597c..2bea634 100644 --- a/chapter12.tex +++ b/chapter12.tex @@ -129,11 +129,11 @@ a depth-first search at a given node. The function assumes that the graph is stored as adjacency lists in an array \begin{lstlisting} -vector adj[N]; +vector> g; \end{lstlisting} and also maintains an array \begin{lstlisting} -bool visited[N]; +vector visited; \end{lstlisting} that keeps track of the visited nodes. Initially, each array value is \texttt{false}, @@ -145,7 +145,7 @@ void dfs(int s) { if (visited[s]) return; visited[s] = true; // process node s - for (auto u: adj[s]) { + for (auto u : g[s]) { dfs(u); } } @@ -312,8 +312,8 @@ as adjacency lists and maintains the following data structures: \begin{lstlisting} queue q; -bool visited[N]; -int distance[N]; +vector visited(n); +vector distance(N); \end{lstlisting} The queue \texttt{q} @@ -336,7 +336,7 @@ q.push(x); while (!q.empty()) { int s = q.front(); q.pop(); // process node s - for (auto u : adj[s]) { + for (auto u : g[s]) { if (visited[u]) continue; visited[u] = true; distance[u] = distance[s]+1; diff --git a/chapter13.tex b/chapter13.tex index f318a98..910189d 100644 --- a/chapter13.tex +++ b/chapter13.tex @@ -20,313 +20,6 @@ where more sophisticated algorithms are needed for finding shortest paths. -\section{Bellman–Ford algorithm} - -\index{Bellman–Ford algorithm} - -The \key{Bellman–Ford algorithm}\footnote{The algorithm is named after -R. E. Bellman and L. R. Ford who published it independently -in 1958 and 1956, respectively \cite{bel58,for56a}.} finds -shortest paths from a starting node to all -nodes of the graph. -The algorithm can process all kinds of graphs, -provided that the graph does not contain a -cycle with negative length. -If the graph contains a negative cycle, -the algorithm can detect this. - -The algorithm keeps track of distances -from the starting node to all nodes of the graph. -Initially, the distance to the starting node is 0 -and the distance to all other nodes in infinite. -The algorithm reduces the distances by finding -edges that shorten the paths until it is not -possible to reduce any distance. - -\subsubsection{Example} - -Let us consider how the Bellman–Ford algorithm -works in the following graph: -\begin{center} -\begin{tikzpicture} -\node[draw, circle] (1) at (1,3) {1}; -\node[draw, circle] (2) at (4,3) {2}; -\node[draw, circle] (3) at (1,1) {3}; -\node[draw, circle] (4) at (4,1) {4}; -\node[draw, circle] (5) at (6,2) {6}; -\node[color=red] at (1,3+0.55) {$0$}; -\node[color=red] at (4,3+0.55) {$\infty$}; -\node[color=red] at (1,1-0.55) {$\infty$}; -\node[color=red] at (4,1-0.55) {$\infty$}; -\node[color=red] at (6,2-0.55) {$\infty$}; -\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2); -\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3); -\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5); -\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5); -\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4); -\end{tikzpicture} -\end{center} -Each node of the graph is assigned a distance. -Initially, the distance to the starting node is 0, -and the distance to all other nodes is infinite. - -The algorithm searches for edges that reduce distances. -First, all edges from node 1 reduce distances: -\begin{center} -\begin{tikzpicture} -\node[draw, circle] (1) at (1,3) {1}; -\node[draw, circle] (2) at (4,3) {2}; -\node[draw, circle] (3) at (1,1) {3}; -\node[draw, circle] (4) at (4,1) {4}; -\node[draw, circle] (5) at (6,2) {5}; -\node[color=red] at (1,3+0.55) {$0$}; -\node[color=red] at (4,3+0.55) {$5$}; -\node[color=red] at (1,1-0.55) {$3$}; -\node[color=red] at (4,1-0.55) {$7$}; -\node[color=red] at (6,2-0.55) {$\infty$}; -\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2); -\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3); -\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5); -\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5); -\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4); - -\path[draw=red,thick,->,line width=2pt] (1) -- (2); -\path[draw=red,thick,->,line width=2pt] (1) -- (3); -\path[draw=red,thick,->,line width=2pt] (1) -- (4); -\end{tikzpicture} -\end{center} -After this, edges -$2 \rightarrow 5$ and $3 \rightarrow 4$ -reduce distances: -\begin{center} -\begin{tikzpicture} -\node[draw, circle] (1) at (1,3) {1}; -\node[draw, circle] (2) at (4,3) {2}; -\node[draw, circle] (3) at (1,1) {3}; -\node[draw, circle] (4) at (4,1) {4}; -\node[draw, circle] (5) at (6,2) {5}; -\node[color=red] at (1,3+0.55) {$0$}; -\node[color=red] at (4,3+0.55) {$5$}; -\node[color=red] at (1,1-0.55) {$3$}; -\node[color=red] at (4,1-0.55) {$4$}; -\node[color=red] at (6,2-0.55) {$7$}; -\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2); -\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3); -\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5); -\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5); -\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4); - -\path[draw=red,thick,->,line width=2pt] (2) -- (5); -\path[draw=red,thick,->,line width=2pt] (3) -- (4); -\end{tikzpicture} -\end{center} -Finally, there is one more change: -\begin{center} -\begin{tikzpicture} -\node[draw, circle] (1) at (1,3) {1}; -\node[draw, circle] (2) at (4,3) {2}; -\node[draw, circle] (3) at (1,1) {3}; -\node[draw, circle] (4) at (4,1) {4}; -\node[draw, circle] (5) at (6,2) {5}; -\node[color=red] at (1,3+0.55) {$0$}; -\node[color=red] at (4,3+0.55) {$5$}; -\node[color=red] at (1,1-0.55) {$3$}; -\node[color=red] at (4,1-0.55) {$4$}; -\node[color=red] at (6,2-0.55) {$6$}; -\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2); -\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3); -\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5); -\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5); -\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4); - -\path[draw=red,thick,->,line width=2pt] (4) -- (5); -\end{tikzpicture} -\end{center} - -After this, no edge can reduce any distance. -This means that the distances are final, -and we have successfully -calculated the shortest distances -from the starting node to all nodes of the graph. - -For example, the shortest distance 3 -from node 1 to node 5 corresponds to -the following path: - -\begin{center} -\begin{tikzpicture} -\node[draw, circle] (1) at (1,3) {1}; -\node[draw, circle] (2) at (4,3) {2}; -\node[draw, circle] (3) at (1,1) {3}; -\node[draw, circle] (4) at (4,1) {4}; -\node[draw, circle] (5) at (6,2) {5}; -\node[color=red] at (1,3+0.55) {$0$}; -\node[color=red] at (4,3+0.55) {$5$}; -\node[color=red] at (1,1-0.55) {$3$}; -\node[color=red] at (4,1-0.55) {$4$}; -\node[color=red] at (6,2-0.55) {$6$}; -\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2); -\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3); -\path[draw,thick,-] (3) -- node[font=\small,label=below:1] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5); -\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5); -\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4); - -\path[draw=red,thick,->,line width=2pt] (1) -- (3); -\path[draw=red,thick,->,line width=2pt] (3) -- (4); -\path[draw=red,thick,->,line width=2pt] (4) -- (5); -\end{tikzpicture} -\end{center} - -\subsubsection{Implementation} - -The following implementation of the -Bellman–Ford algorithm determines the shortest distances -from a node $x$ to all nodes of the graph. -The code assumes that the graph is stored -as an edge list \texttt{edges} -that consists of tuples of the form $(a,b,w)$, -meaning that there is an edge from node $a$ to node $b$ -with weight $w$. - -The algorithm consists of $n-1$ rounds, -and on each round the algorithm goes through -all edges of the graph and tries to -reduce the distances. -The algorithm constructs an array \texttt{distance} -that will contain the distances from $x$ -to all nodes of the graph. -The constant \texttt{INF} denotes an infinite distance. - -\begin{lstlisting} -for (int i = 1; i <= n; i++) distance[i] = INF; -distance[x] = 0; -for (int i = 1; i <= n-1; i++) { - for (auto e : edges) { - int a, b, w; - tie(a, b, w) = e; - distance[b] = min(distance[b], distance[a]+w); - } -} -\end{lstlisting} - -The time complexity of the algorithm is $O(nm)$, -because the algorithm consists of $n-1$ rounds and -iterates through all $m$ edges during a round. -If there are no negative cycles in the graph, -all distances are final after $n-1$ rounds, -because each shortest path can contain at most $n-1$ edges. - -In practice, the final distances can usually -be found faster than in $n-1$ rounds. -Thus, a possible way to make the algorithm more efficient -is to stop the algorithm if no distance -can be reduced during a round. - -\subsubsection{Negative cycles} - -\index{negative cycle} - -The Bellman–Ford algorithm can also be used to -check if the graph contains a cycle with negative length. -For example, the graph - -\begin{center} -\begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (0,0) {$1$}; -\node[draw, circle] (2) at (2,1) {$2$}; -\node[draw, circle] (3) at (2,-1) {$3$}; -\node[draw, circle] (4) at (4,0) {$4$}; - -\path[draw,thick,-] (1) -- node[font=\small,label=above:$3$] {} (2); -\path[draw,thick,-] (2) -- node[font=\small,label=above:$1$] {} (4); -\path[draw,thick,-] (1) -- node[font=\small,label=below:$5$] {} (3); -\path[draw,thick,-] (3) -- node[font=\small,label=below:$-7$] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=right:$2$] {} (3); -\end{tikzpicture} -\end{center} -\noindent -contains a negative cycle -$2 \rightarrow 3 \rightarrow 4 \rightarrow 2$ -with length $-4$. - -If the graph contains a negative cycle, -we can shorten infinitely many times -any path that contains the cycle by repeating the cycle -again and again. -Thus, the concept of a shortest path -is not meaningful in this situation. - -A negative cycle can be detected -using the Bellman–Ford algorithm by -running the algorithm for $n$ rounds. -If the last round reduces any distance, -the graph contains a negative cycle. -Note that this algorithm can be used to -search for -a negative cycle in the whole graph -regardless of the starting node. - -\subsubsection{SPFA algorithm} - -\index{SPFA algorithm} - -The \key{SPFA algorithm} (''Shortest Path Faster Algorithm'') \cite{fan94} -is a variant of the Bellman–Ford algorithm, -that is often more efficient than the original algorithm. -The SPFA algorithm does not go through all the edges on each round, -but instead, it chooses the edges to be examined -in a more intelligent way. - -The algorithm maintains a queue of nodes that might -be used for reducing the distances. -First, the algorithm adds the starting node $x$ -to the queue. -Then, the algorithm always processes the -first node in the queue, and when an edge -$a \rightarrow b$ reduces a distance, -node $b$ is added to the queue. -% -% The following implementation uses a -% \texttt{queue} \texttt{q}. -% In addition, an array \texttt{inqueue} indicates -% if a node is already in the queue, -% in which case the algorithm does not add -% the node to the queue again. -% -% \begin{lstlisting} -% for (int i = 1; i <= n; i++) distance[i] = INF; -% distance[x] = 0; -% q.push(x); -% while (!q.empty()) { -% int a = q.front(); q.pop(); -% inqueue[a] = false; -% for (auto b : v[a]) { -% if (distance[a]+b.second < distance[b.first]) { -% distance[b.first] = distance[a]+b.second; -% if (!inqueue[b]) {q.push(b); inqueue[b] = true;} -% } -% } -% } -% \end{lstlisting} - -The efficiency of the SPFA algorithm depends -on the structure of the graph: -the algorithm is often efficient, -but its worst case time complexity is still -$O(nm)$ and it is possible to create inputs -that make the algorithm as slow as the -original Bellman–Ford algorithm. - \section{Dijkstra's algorithm} \index{Dijkstra's algorithm} @@ -334,15 +27,12 @@ original Bellman–Ford algorithm. \key{Dijkstra's algorithm}\footnote{E. W. Dijkstra published the algorithm in 1959 \cite{dij59}; however, his original paper does not mention how to implement the algorithm efficiently.} finds shortest -paths from the starting node to all nodes of the graph, -like the Bellman–Ford algorithm. -The benefit of Dijsktra's algorithm is that -it is more efficient and can be used for +paths from the starting node to all nodes of the graph. +Dijkstra's algorithm is very efficient and can be used for processing large graphs. However, the algorithm requires that there are no negative weight edges in the graph. -Like the Bellman–Ford algorithm, Dijkstra's algorithm maintains distances to the nodes and reduces them during the search. Dijkstra's algorithm is efficient, because @@ -543,9 +233,9 @@ The following implementation of Dijkstra's algorithm calculates the minimum distances from a node $x$ to other nodes of the graph. The graph is stored as adjacency lists -so that \texttt{adj[$a$]} contains a pair $(b,w)$ -always when there is an edge from node $a$ to node $b$ -with weight $w$. +so that \texttt{g[$v$]} contains a pair $(w,\text{cost})$ +always when there is an edge from node $v$ to node $w$ +with weight $\text{cost}$. An efficient implementation of Dijkstra's algorithm requires that it is possible to efficiently find the @@ -556,40 +246,42 @@ Using a priority queue, the next node to be processed can be retrieved in logarithmic time. In the following code, the priority queue -\texttt{q} contains pairs of the form $(-d,x)$, +\texttt{pq} contains pairs of the form $(d,x)$, meaning that the current distance to node $x$ is $d$. + The array $\texttt{distance}$ contains the distance to -each node, and the array $\texttt{processed}$ indicates -whether a node has been processed. -Initially the distance is $0$ to $x$ and $\infty$ to all other nodes. +each node. Initially, the distance is $0$ to $\text{start}$ and $-1$ to all +other nodes. We use $-1$ as invalid value to denote that the node +has not been reached yet. \begin{lstlisting} -for (int i = 1; i <= n; i++) distance[i] = INF; -distance[x] = 0; -q.push({0,x}); -while (!q.empty()) { - int a = q.top().second; q.pop(); - if (processed[a]) continue; - processed[a] = true; - for (auto u : adj[a]) { - int b = u.first, w = u.second; - if (distance[a]+w < distance[b]) { - distance[b] = distance[a]+w; - q.push({-distance[b],b}); - } - } +vector distance(n, -1); +priority_queue, + vector>, + greater>> pq; +distance[start] = 0; +pq.emplace(0, start); +while (!pq.empty()) { + auto [d, v] = q.top(); + q.pop(); + if (distance[v] != -1) continue; + distance[v] = d; + for (auto [w, cost] : g[v]) + q.emplace(d + cost, w); } \end{lstlisting} -Note that the priority queue contains \emph{negative} -distances to nodes. -The reason for this is that the -default version of the C++ priority queue finds maximum -elements, while we want to find minimum elements. -By using negative distances, -we can directly use the default priority queue\footnote{Of -course, we could also declare the priority queue as in Chapter 4.5 -and use positive distances, but the implementation would be a bit longer.}. +Note that the type of the priority queue is not +\verb|priority_queue| but instead +\verb|priority_queue,vector>,greater>>|. +This is because in C++, a priority queue by default puts the +\emph{largest} element on top, so we reverse the ordering by changing +the comparison operator from \verb|less| (the default) to +\verb|greater| (which does the opposite). + +In case you forget, you can look up the syntax for the priority queue +in the C++ cheatsheet linked on the camp page. + Also note that there may be several instances of the same node in the priority queue; however, only the instance with the minimum distance will be processed. @@ -598,205 +290,3 @@ The time complexity of the above implementation is $O(n+m \log m)$, because the algorithm goes through all nodes of the graph and adds for each edge at most one distance to the priority queue. - -\section{Floyd–Warshall algorithm} - -\index{Floyd–Warshall algorithm} - -The \key{Floyd–Warshall algorithm}\footnote{The algorithm -is named after R. W. Floyd and S. Warshall -who published it independently in 1962 \cite{flo62,war62}.} -provides an alternative way to approach the problem -of finding shortest paths. -Unlike the other algorithms of this chapter, -it finds all shortest paths between the nodes -in a single run. - -The algorithm maintains a two-dimensional array -that contains distances between the nodes. -First, distances are calculated only using -direct edges between the nodes, -and after this, the algorithm reduces distances -by using intermediate nodes in paths. - -\subsubsection{Example} - -Let us consider how the Floyd–Warshall algorithm -works in the following graph: - -\begin{center} -\begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (1,3) {$3$}; -\node[draw, circle] (2) at (4,3) {$4$}; -\node[draw, circle] (3) at (1,1) {$2$}; -\node[draw, circle] (4) at (4,1) {$1$}; -\node[draw, circle] (5) at (6,2) {$5$}; - -\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (2); -\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3); -\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5); -\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5); -\end{tikzpicture} -\end{center} - -Initially, the distance from each node to itself is $0$, -and the distance between nodes $a$ and $b$ is $x$ -if there is an edge between nodes $a$ and $b$ with weight $x$. -All other distances are infinite. - -In this graph, the initial array is as follows: -\begin{center} -\begin{tabular}{r|rrrrr} - & 1 & 2 & 3 & 4 & 5 \\ -\hline -1 & 0 & 5 & $\infty$ & 9 & 1 \\ -2 & 5 & 0 & 2 & $\infty$ & $\infty$ \\ -3 & $\infty$ & 2 & 0 & 7 & $\infty$ \\ -4 & 9 & $\infty$ & 7 & 0 & 2 \\ -5 & 1 & $\infty$ & $\infty$ & 2 & 0 \\ -\end{tabular} -\end{center} -\vspace{10pt} -The algorithm consists of consecutive rounds. -On each round, the algorithm selects a new node -that can act as an intermediate node in paths from now on, -and distances are reduced using this node. - -On the first round, node 1 is the new intermediate node. -There is a new path between nodes 2 and 4 -with length 14, because node 1 connects them. -There is also a new path -between nodes 2 and 5 with length 6. - -\begin{center} -\begin{tabular}{r|rrrrr} - & 1 & 2 & 3 & 4 & 5 \\ -\hline -1 & 0 & 5 & $\infty$ & 9 & 1 \\ -2 & 5 & 0 & 2 & \textbf{14} & \textbf{6} \\ -3 & $\infty$ & 2 & 0 & 7 & $\infty$ \\ -4 & 9 & \textbf{14} & 7 & 0 & 2 \\ -5 & 1 & \textbf{6} & $\infty$ & 2 & 0 \\ -\end{tabular} -\end{center} -\vspace{10pt} - -On the second round, node 2 is the new intermediate node. -This creates new paths between nodes 1 and 3 -and between nodes 3 and 5: - -\begin{center} -\begin{tabular}{r|rrrrr} - & 1 & 2 & 3 & 4 & 5 \\ -\hline -1 & 0 & 5 & \textbf{7} & 9 & 1 \\ -2 & 5 & 0 & 2 & 14 & 6 \\ -3 & \textbf{7} & 2 & 0 & 7 & \textbf{8} \\ -4 & 9 & 14 & 7 & 0 & 2 \\ -5 & 1 & 6 & \textbf{8} & 2 & 0 \\ -\end{tabular} -\end{center} -\vspace{10pt} - -On the third round, node 3 is the new intermediate round. -There is a new path between nodes 2 and 4: - -\begin{center} -\begin{tabular}{r|rrrrr} - & 1 & 2 & 3 & 4 & 5 \\ -\hline -1 & 0 & 5 & 7 & 9 & 1 \\ -2 & 5 & 0 & 2 & \textbf{9} & 6 \\ -3 & 7 & 2 & 0 & 7 & 8 \\ -4 & 9 & \textbf{9} & 7 & 0 & 2 \\ -5 & 1 & 6 & 8 & 2 & 0 \\ -\end{tabular} -\end{center} -\vspace{10pt} - -The algorithm continues like this, -until all nodes have been appointed intermediate nodes. -After the algorithm has finished, the array contains -the minimum distances between any two nodes: - -\begin{center} -\begin{tabular}{r|rrrrr} - & 1 & 2 & 3 & 4 & 5 \\ -\hline -1 & 0 & 5 & 7 & 3 & 1 \\ -2 & 5 & 0 & 2 & 8 & 6 \\ -3 & 7 & 2 & 0 & 7 & 8 \\ -4 & 3 & 8 & 7 & 0 & 2 \\ -5 & 1 & 6 & 8 & 2 & 0 \\ -\end{tabular} -\end{center} - -For example, the array tells us that the -shortest distance between nodes 2 and 4 is 8. -This corresponds to the following path: - -\begin{center} -\begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (1,3) {$3$}; -\node[draw, circle] (2) at (4,3) {$4$}; -\node[draw, circle] (3) at (1,1) {$2$}; -\node[draw, circle] (4) at (4,1) {$1$}; -\node[draw, circle] (5) at (6,2) {$5$}; - -\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (2); -\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3); -\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5); -\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5); - -\path[draw=red,thick,->,line width=2pt] (3) -- (4); -\path[draw=red,thick,->,line width=2pt] (4) -- (5); -\path[draw=red,thick,->,line width=2pt] (5) -- (2); -\end{tikzpicture} -\end{center} - -\subsubsection{Implementation} - -The advantage of the -Floyd–Warshall algorithm that it is -easy to implement. -The following code constructs a -distance matrix where $\texttt{distance}[a][b]$ -is the shortest distance between nodes $a$ and $b$. -First, the algorithm initializes \texttt{distance} -using the adjacency matrix \texttt{adj} of the graph: - -\begin{lstlisting} -for (int i = 1; i <= n; i++) { - for (int j = 1; j <= n; j++) { - if (i == j) distance[i][j] = 0; - else if (adj[i][j]) distance[i][j] = adj[i][j]; - else distance[i][j] = INF; - } -} -\end{lstlisting} -After this, the shortest distances can be found as follows: -\begin{lstlisting} -for (int k = 1; k <= n; k++) { - for (int i = 1; i <= n; i++) { - for (int j = 1; j <= n; j++) { - distance[i][j] = min(distance[i][j], - distance[i][k]+distance[k][j]); - } - } -} -\end{lstlisting} - -The time complexity of the algorithm is $O(n^3)$, -because it contains three nested loops -that go through the nodes of the graph. - -Since the implementation of the Floyd–Warshall -algorithm is simple, the algorithm can be -a good choice even if it is only needed to find a -single shortest path in the graph. -However, the algorithm can only be used when the graph -is so small that a cubic time complexity is fast enough. diff --git a/chapter14.tex b/chapter14.tex index 56db94f..5d69b2c 100644 --- a/chapter14.tex +++ b/chapter14.tex @@ -113,17 +113,16 @@ a depth-first search at an arbitrary node. The following recursive function can be used: \begin{lstlisting} -void dfs(int s, int e) { - // process node s - for (auto u : adj[s]) { - if (u != e) dfs(u, s); - } +void dfs(int v, int p) { + for (auto w : g[v]) + if (w != p) + dfs(w, v); } \end{lstlisting} -The function is given two parameters: the current node $s$ -and the previous node $e$. -The purpose of the parameter $e$ is to make sure +The function is given two parameters: the current node $v$ +and the previous node $p$. +The purpose of the parameter $p$ is to make sure that the search only moves to nodes that have not been visited yet. @@ -131,37 +130,36 @@ The following function call starts the search at node $x$: \begin{lstlisting} -dfs(x, 0); +dfs(x, -1); \end{lstlisting} -In the first call $e=0$, because there is no +In the first call $p=-1$, because there is no previous node, and it is allowed to proceed to any direction in the tree. -\subsubsection{Dynamic programming} - -Dynamic programming can be used to calculate -some information during a tree traversal. -Using dynamic programming, we can, for example, +\subsubsection{Storing Information} +We can calculate +some information during a tree traversal and store that for later use. +We can, for example, calculate in $O(n)$ time for each node of a rooted tree the number of nodes in its subtree or the length of the longest path from the node to a leaf. -As an example, let us calculate for each node $s$ -a value $\texttt{count}[s]$: the number of nodes in its subtree. +As an example, let us calculate for each node $v$ +a value $\texttt{subtreesize}[v]$: the number of nodes in its subtree. The subtree contains the node itself and all nodes in the subtrees of its children, so we can calculate the number of nodes recursively using the following code: \begin{lstlisting} -void dfs(int s, int e) { - count[s] = 1; - for (auto u : adj[s]) { - if (u == e) continue; - dfs(u, s); - count[s] += count[u]; +void dfs(int v, int p) { + subtreesize[s] = 1; + for (auto w : g[v]) { + if (w == p) continue; + dfs(w, v); + subtreesize[s] += subtreesize[u]; } } \end{lstlisting} @@ -220,7 +218,7 @@ to obtain another path with length 4. Next we will discuss two $O(n)$ time algorithms for calculating the diameter of a tree. -The first algorithm is based on dynamic programming, +The first algorithm is based on the previous idea of storing information, and the second algorithm uses two depth-first searches. \subsubsection{Algorithm 1} @@ -279,7 +277,7 @@ because there is a path $6 \rightarrow 2 \rightarrow 1 \rightarrow 4 \rightarrow 7$. In this case, $\texttt{maxLength}(1)$ equals the diameter. -Dynamic programming can be used to calculate the above +We can calculate the above values for all nodes in $O(n)$ time. First, to calculate $\texttt{toLeaf}(x)$, we go through the children of $x$, @@ -448,8 +446,8 @@ goes through its child 2: \path[draw,thick,->,color=red,line width=2pt] (3) -- (6); \end{tikzpicture} \end{center} -This part is easy to solve in $O(n)$ time, because we can use -dynamic programming as we have done previously. +This part is easy to solve in $O(n)$ time, because we can use a +similar technique to what we have done previously. Then, the second part of the problem is to calculate for every node $x$ the maximum length of a path diff --git a/chapter16.tex b/chapter16.tex index cf612b0..336e33a 100644 --- a/chapter16.tex +++ b/chapter16.tex @@ -1,20 +1,4 @@ -\chapter{Directed graphs} - -In this chapter, we focus on two classes of directed graphs: -\begin{itemize} -\item \key{Acyclic graphs}: -There are no cycles in the graph, -so there is no path from any node to itself\footnote{Directed acyclic -graphs are sometimes called DAGs.}. -\item \key{Successor graphs}: -The outdegree of each node is 1, -so each node has a unique successor. -\end{itemize} -It turns out that in both cases, -we can design efficient algorithms that are based -on the special properties of the graphs. - -\section{Topological sorting} +\chapter{Topological sorting} \index{topological sorting} \index{cycle} @@ -249,460 +233,3 @@ The search reaches node 2 whose state is 1, which means that the graph contains a cycle. In this example, there is a cycle $2 \rightarrow 3 \rightarrow 5 \rightarrow 2$. - -\section{Dynamic programming} - -If a directed graph is acyclic, -dynamic programming can be applied to it. -For example, we can efficiently solve the following -problems concerning paths from a starting node -to an ending node: - -\begin{itemize} -\item how many different paths are there? -\item what is the shortest/longest path? -\item what is the minimum/maximum number of edges in a path? -\item which nodes certainly appear in any path? -\end{itemize} - -\subsubsection{Counting the number of paths} - -As an example, let us calculate the number of paths -from node 1 to node 6 in the following graph: - -\begin{center} -\begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (1,5) {$1$}; -\node[draw, circle] (2) at (3,5) {$2$}; -\node[draw, circle] (3) at (5,5) {$3$}; -\node[draw, circle] (4) at (1,3) {$4$}; -\node[draw, circle] (5) at (3,3) {$5$}; -\node[draw, circle] (6) at (5,3) {$6$}; - -\path[draw,thick,->,>=latex] (1) -- (2); -\path[draw,thick,->,>=latex] (2) -- (3); -\path[draw,thick,->,>=latex] (1) -- (4); -\path[draw,thick,->,>=latex] (4) -- (5); -\path[draw,thick,->,>=latex] (5) -- (2); -\path[draw,thick,->,>=latex] (5) -- (3); -\path[draw,thick,->,>=latex] (3) -- (6); -\end{tikzpicture} -\end{center} -There are a total of three such paths: -\begin{itemize} -\item $1 \rightarrow 2 \rightarrow 3 \rightarrow 6$ -\item $1 \rightarrow 4 \rightarrow 5 \rightarrow 2 \rightarrow 3 \rightarrow 6$ -\item $1 \rightarrow 4 \rightarrow 5 \rightarrow 3 \rightarrow 6$ -\end{itemize} - -Let $\texttt{paths}(x)$ denote the number of paths from -node 1 to node $x$. -As a base case, $\texttt{paths}(1)=1$. -Then, to calculate other values of $\texttt{paths}(x)$, -we may use the recursion -\[\texttt{paths}(x) = \texttt{paths}(a_1)+\texttt{paths}(a_2)+\cdots+\texttt{paths}(a_k)\] -where $a_1,a_2,\ldots,a_k$ are the nodes from which there -is an edge to $x$. -Since the graph is acyclic, the values of $\texttt{paths}(x)$ -can be calculated in the order of a topological sort. -A topological sort for the above graph is as follows: -\begin{center} -\begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (0,0) {$1$}; -\node[draw, circle] (2) at (4.5,0) {$2$}; -\node[draw, circle] (3) at (6,0) {$3$}; -\node[draw, circle] (4) at (1.5,0) {$4$}; -\node[draw, circle] (5) at (3,0) {$5$}; -\node[draw, circle] (6) at (7.5,0) {$6$}; - -\path[draw,thick,->,>=latex] (1) edge [bend left=30] (2); -\path[draw,thick,->,>=latex] (2) -- (3); -\path[draw,thick,->,>=latex] (1) -- (4); -\path[draw,thick,->,>=latex] (4) -- (5); -\path[draw,thick,->,>=latex] (5) -- (2); -\path[draw,thick,->,>=latex] (5) edge [bend right=30] (3); -\path[draw,thick,->,>=latex] (3) -- (6); -\end{tikzpicture} -\end{center} -Hence, the numbers of paths are as follows: -\begin{center} -\begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (1,5) {$1$}; -\node[draw, circle] (2) at (3,5) {$2$}; -\node[draw, circle] (3) at (5,5) {$3$}; -\node[draw, circle] (4) at (1,3) {$4$}; -\node[draw, circle] (5) at (3,3) {$5$}; -\node[draw, circle] (6) at (5,3) {$6$}; - -\path[draw,thick,->,>=latex] (1) -- (2); -\path[draw,thick,->,>=latex] (2) -- (3); -\path[draw,thick,->,>=latex] (1) -- (4); -\path[draw,thick,->,>=latex] (4) -- (5); -\path[draw,thick,->,>=latex] (5) -- (2); -\path[draw,thick,->,>=latex] (5) -- (3); -\path[draw,thick,->,>=latex] (3) -- (6); - -\node[color=red] at (1,2.3) {$1$}; -\node[color=red] at (3,2.3) {$1$}; -\node[color=red] at (5,2.3) {$3$}; -\node[color=red] at (1,5.7) {$1$}; -\node[color=red] at (3,5.7) {$2$}; -\node[color=red] at (5,5.7) {$3$}; -\end{tikzpicture} -\end{center} - -For example, to calculate the value of $\texttt{paths}(3)$, -we can use the formula $\texttt{paths}(2)+\texttt{paths}(5)$, -because there are edges from nodes 2 and 5 -to node 3. -Since $\texttt{paths}(2)=2$ and $\texttt{paths}(5)=1$, we conclude that $\texttt{paths}(3)=3$. - -\subsubsection{Extending Dijkstra's algorithm} - -\index{Dijkstra's algorithm} - -A by-product of Dijkstra's algorithm is a directed, acyclic -graph that indicates for each node of the original graph -the possible ways to reach the node using a shortest path -from the starting node. -Dynamic programming can be applied to that graph. -For example, in the graph -\begin{center} -\begin{tikzpicture} -\node[draw, circle] (1) at (0,0) {$1$}; -\node[draw, circle] (2) at (2,0) {$2$}; -\node[draw, circle] (3) at (0,-2) {$3$}; -\node[draw, circle] (4) at (2,-2) {$4$}; -\node[draw, circle] (5) at (4,-1) {$5$}; - -\path[draw,thick,-] (1) -- node[font=\small,label=above:3] {} (2); -\path[draw,thick,-] (1) -- node[font=\small,label=left:5] {} (3); -\path[draw,thick,-] (2) -- node[font=\small,label=right:4] {} (4); -\path[draw,thick,-] (2) -- node[font=\small,label=above:8] {} (5); -\path[draw,thick,-] (3) -- node[font=\small,label=below:2] {} (4); -\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5); -\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (3); -\end{tikzpicture} -\end{center} -the shortest paths from node 1 may use the following edges: -\begin{center} -\begin{tikzpicture} -\node[draw, circle] (1) at (0,0) {$1$}; -\node[draw, circle] (2) at (2,0) {$2$}; -\node[draw, circle] (3) at (0,-2) {$3$}; -\node[draw, circle] (4) at (2,-2) {$4$}; -\node[draw, circle] (5) at (4,-1) {$5$}; - -\path[draw,thick,->] (1) -- node[font=\small,label=above:3] {} (2); -\path[draw,thick,->] (1) -- node[font=\small,label=left:5] {} (3); -\path[draw,thick,->] (2) -- node[font=\small,label=right:4] {} (4); -\path[draw,thick,->] (3) -- node[font=\small,label=below:2] {} (4); -\path[draw,thick,->] (4) -- node[font=\small,label=below:1] {} (5); -\path[draw,thick,->] (2) -- node[font=\small,label=above:2] {} (3); -\end{tikzpicture} -\end{center} - -Now we can, for example, calculate the number of -shortest paths from node 1 to node 5 -using dynamic programming: -\begin{center} -\begin{tikzpicture} -\node[draw, circle] (1) at (0,0) {$1$}; -\node[draw, circle] (2) at (2,0) {$2$}; -\node[draw, circle] (3) at (0,-2) {$3$}; -\node[draw, circle] (4) at (2,-2) {$4$}; -\node[draw, circle] (5) at (4,-1) {$5$}; - -\path[draw,thick,->] (1) -- node[font=\small,label=above:3] {} (2); -\path[draw,thick,->] (1) -- node[font=\small,label=left:5] {} (3); -\path[draw,thick,->] (2) -- node[font=\small,label=right:4] {} (4); -\path[draw,thick,->] (3) -- node[font=\small,label=below:2] {} (4); -\path[draw,thick,->] (4) -- node[font=\small,label=below:1] {} (5); -\path[draw,thick,->] (2) -- node[font=\small,label=above:2] {} (3); - -\node[color=red] at (0,0.7) {$1$}; -\node[color=red] at (2,0.7) {$1$}; -\node[color=red] at (0,-2.7) {$2$}; -\node[color=red] at (2,-2.7) {$3$}; -\node[color=red] at (4,-1.7) {$3$}; -\end{tikzpicture} -\end{center} - -\subsubsection{Representing problems as graphs} - -Actually, any dynamic programming problem -can be represented as a directed, acyclic graph. -In such a graph, each node corresponds to a dynamic programming state -and the edges indicate how the states depend on each other. - -As an example, consider the problem -of forming a sum of money $n$ -using coins -$\{c_1,c_2,\ldots,c_k\}$. -In this problem, we can construct a graph where -each node corresponds to a sum of money, -and the edges show how the coins can be chosen. -For example, for coins $\{1,3,4\}$ and $n=6$, -the graph is as follows: -\begin{center} -\begin{tikzpicture}[scale=0.9] -\node[draw, circle] (0) at (0,0) {$0$}; -\node[draw, circle] (1) at (2,0) {$1$}; -\node[draw, circle] (2) at (4,0) {$2$}; -\node[draw, circle] (3) at (6,0) {$3$}; -\node[draw, circle] (4) at (8,0) {$4$}; -\node[draw, circle] (5) at (10,0) {$5$}; -\node[draw, circle] (6) at (12,0) {$6$}; - -\path[draw,thick,->] (0) -- (1); -\path[draw,thick,->] (1) -- (2); -\path[draw,thick,->] (2) -- (3); -\path[draw,thick,->] (3) -- (4); -\path[draw,thick,->] (4) -- (5); -\path[draw,thick,->] (5) -- (6); - -\path[draw,thick,->] (0) edge [bend right=30] (3); -\path[draw,thick,->] (1) edge [bend right=30] (4); -\path[draw,thick,->] (2) edge [bend right=30] (5); -\path[draw,thick,->] (3) edge [bend right=30] (6); - -\path[draw,thick,->] (0) edge [bend left=30] (4); -\path[draw,thick,->] (1) edge [bend left=30] (5); -\path[draw,thick,->] (2) edge [bend left=30] (6); -\end{tikzpicture} -\end{center} - -Using this representation, -the shortest path from node 0 to node $n$ -corresponds to a solution with the minimum number of coins, -and the total number of paths from node 0 to node $n$ -equals the total number of solutions. - -\section{Successor paths} - -\index{successor graph} -\index{functional graph} - -For the rest of the chapter, -we will focus on \key{successor graphs}. -In those graphs, -the outdegree of each node is 1, i.e., -exactly one edge starts at each node. -A successor graph consists of one or more -components, each of which contains -one cycle and some paths that lead to it. - -Successor graphs are sometimes called -\key{functional graphs}. -The reason for this is that any successor graph -corresponds to a function that defines -the edges of the graph. -The parameter for the function is a node of the graph, -and the function gives the successor of that node. - -For example, the function -\begin{center} -\begin{tabular}{r|rrrrrrrrr} -$x$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ -\hline -$\texttt{succ}(x)$ & 3 & 5 & 7 & 6 & 2 & 2 & 1 & 6 & 3 \\ -\end{tabular} -\end{center} -defines the following graph: -\begin{center} -\begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (0,0) {$1$}; -\node[draw, circle] (2) at (2,0) {$2$}; -\node[draw, circle] (3) at (-2,0) {$3$}; -\node[draw, circle] (4) at (1,-3) {$4$}; -\node[draw, circle] (5) at (4,0) {$5$}; -\node[draw, circle] (6) at (2,-1.5) {$6$}; -\node[draw, circle] (7) at (-2,-1.5) {$7$}; -\node[draw, circle] (8) at (3,-3) {$8$}; -\node[draw, circle] (9) at (-4,0) {$9$}; - -\path[draw,thick,->] (1) -- (3); -\path[draw,thick,->] (2) edge [bend left=40] (5); -\path[draw,thick,->] (3) -- (7); -\path[draw,thick,->] (4) -- (6); -\path[draw,thick,->] (5) edge [bend left=40] (2); -\path[draw,thick,->] (6) -- (2); -\path[draw,thick,->] (7) -- (1); -\path[draw,thick,->] (8) -- (6); -\path[draw,thick,->] (9) -- (3); -\end{tikzpicture} -\end{center} - -Since each node of a successor graph has a -unique successor, we can also define a function $\texttt{succ}(x,k)$ -that gives the node that we will reach if -we begin at node $x$ and walk $k$ steps forward. -For example, in the above graph $\texttt{succ}(4,6)=2$, -because we will reach node 2 by walking 6 steps from node 4: - -\begin{center} -\begin{tikzpicture}[scale=0.9] -\node[draw, circle] (1) at (0,0) {$4$}; -\node[draw, circle] (2) at (1.5,0) {$6$}; -\node[draw, circle] (3) at (3,0) {$2$}; -\node[draw, circle] (4) at (4.5,0) {$5$}; -\node[draw, circle] (5) at (6,0) {$2$}; -\node[draw, circle] (6) at (7.5,0) {$5$}; -\node[draw, circle] (7) at (9,0) {$2$}; - -\path[draw,thick,->] (1) -- (2); -\path[draw,thick,->] (2) -- (3); -\path[draw,thick,->] (3) -- (4); -\path[draw,thick,->] (4) -- (5); -\path[draw,thick,->] (5) -- (6); -\path[draw,thick,->] (6) -- (7); -\end{tikzpicture} -\end{center} - -A straightforward way to calculate a value of $\texttt{succ}(x,k)$ -is to start at node $x$ and walk $k$ steps forward, which takes $O(k)$ time. -However, using preprocessing, any value of $\texttt{succ}(x,k)$ -can be calculated in only $O(\log k)$ time. - -The idea is to precalculate all values of $\texttt{succ}(x,k)$ where -$k$ is a power of two and at most $u$, where $u$ is -the maximum number of steps we will ever walk. -This can be efficiently done, because -we can use the following recursion: - -\begin{equation*} - \texttt{succ}(x,k) = \begin{cases} - \texttt{succ}(x) & k = 1\\ - \texttt{succ}(\texttt{succ}(x,k/2),k/2) & k > 1\\ - \end{cases} -\end{equation*} - -Precalculating the values takes $O(n \log u)$ time, -because $O(\log u)$ values are calculated for each node. -In the above graph, the first values are as follows: - -\begin{center} -\begin{tabular}{r|rrrrrrrrr} -$x$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ -\hline -$\texttt{succ}(x,1)$ & 3 & 5 & 7 & 6 & 2 & 2 & 1 & 6 & 3 \\ -$\texttt{succ}(x,2)$ & 7 & 2 & 1 & 2 & 5 & 5 & 3 & 2 & 7 \\ -$\texttt{succ}(x,4)$ & 3 & 2 & 7 & 2 & 5 & 5 & 1 & 2 & 3 \\ -$\texttt{succ}(x,8)$ & 7 & 2 & 1 & 2 & 5 & 5 & 3 & 2 & 7 \\ -$\cdots$ \\ -\end{tabular} -\end{center} - -After this, any value of $\texttt{succ}(x,k)$ can be calculated -by presenting the number of steps $k$ as a sum of powers of two. -For example, if we want to calculate the value of $\texttt{succ}(x,11)$, -we first form the representation $11=8+2+1$. -Using that, -\[\texttt{succ}(x,11)=\texttt{succ}(\texttt{succ}(\texttt{succ}(x,8),2),1).\] -For example, in the previous graph -\[\texttt{succ}(4,11)=\texttt{succ}(\texttt{succ}(\texttt{succ}(4,8),2),1)=5.\] - -Such a representation always consists of -$O(\log k)$ parts, so calculating a value of $\texttt{succ}(x,k)$ -takes $O(\log k)$ time. - -\section{Cycle detection} - -\index{cycle} -\index{cycle detection} - -Consider a successor graph that only contains -a path that ends in a cycle. -We may ask the following questions: -if we begin our walk at the starting node, -what is the first node in the cycle -and how many nodes does the cycle contain? - -For example, in the graph - -\begin{center} -\begin{tikzpicture}[scale=0.9] -\node[draw, circle] (5) at (0,0) {$5$}; -\node[draw, circle] (4) at (-2,0) {$4$}; -\node[draw, circle] (6) at (-1,1.5) {$6$}; -\node[draw, circle] (3) at (-4,0) {$3$}; -\node[draw, circle] (2) at (-6,0) {$2$}; -\node[draw, circle] (1) at (-8,0) {$1$}; - -\path[draw,thick,->] (1) -- (2); -\path[draw,thick,->] (2) -- (3); -\path[draw,thick,->] (3) -- (4); -\path[draw,thick,->] (4) -- (5); -\path[draw,thick,->] (5) -- (6); -\path[draw,thick,->] (6) -- (4); -\end{tikzpicture} -\end{center} -we begin our walk at node 1, -the first node that belongs to the cycle is node 4, and the cycle consists -of three nodes (4, 5 and 6). - -A simple way to detect the cycle is to walk in the -graph and keep track of -all nodes that have been visited. Once a node is visited -for the second time, we can conclude -that the node is the first node in the cycle. -This method works in $O(n)$ time and also uses -$O(n)$ memory. - -However, there are better algorithms for cycle detection. -The time complexity of such algorithms is still $O(n)$, -but they only use $O(1)$ memory. -This is an important improvement if $n$ is large. -Next we will discuss Floyd's algorithm that -achieves these properties. - -\subsubsection{Floyd's algorithm} - -\index{Floyd's algorithm} - -\key{Floyd's algorithm}\footnote{The idea of the algorithm is mentioned in \cite{knu982} -and attributed to R. W. Floyd; however, it is not known if Floyd actually -discovered the algorithm.} walks forward -in the graph using two pointers $a$ and $b$. -Both pointers begin at a node $x$ that -is the starting node of the graph. -Then, on each turn, the pointer $a$ walks -one step forward and the pointer $b$ -walks two steps forward. -The process continues until -the pointers meet each other: -\begin{lstlisting} -a = succ(x); -b = succ(succ(x)); -while (a != b) { - a = succ(a); - b = succ(succ(b)); -} -\end{lstlisting} - -At this point, the pointer $a$ has walked $k$ steps -and the pointer $b$ has walked $2k$ steps, -so the length of the cycle divides $k$. -Thus, the first node that belongs to the cycle -can be found by moving the pointer $a$ to node $x$ -and advancing the pointers -step by step until they meet again. -\begin{lstlisting} -a = x; -while (a != b) { - a = succ(a); - b = succ(b); -} -first = a; -\end{lstlisting} - -After this, the length of the cycle -can be calculated as follows: -\begin{lstlisting} -b = succ(a); -length = 1; -while (a != b) { - b = succ(b); - length++; -} -\end{lstlisting} diff --git a/chapter17.tex b/chapter17.tex index ca36a62..160e84c 100644 --- a/chapter17.tex +++ b/chapter17.tex @@ -359,205 +359,3 @@ that create the remaining strongly connected components: The time complexity of the algorithm is $O(n+m)$, because the algorithm performs two depth-first searches. - -\section{2SAT problem} - -\index{2SAT problem} - -Strong connectivity is also linked with the -\key{2SAT problem}\footnote{The algorithm presented here was -introduced in \cite{asp79}. -There is also another well-known linear-time algorithm \cite{eve75} -that is based on backtracking.}. -In this problem, we are given a logical formula -\[ -(a_1 \lor b_1) \land (a_2 \lor b_2) \land \cdots \land (a_m \lor b_m), -\] -where each $a_i$ and $b_i$ is either a logical variable -($x_1,x_2,\ldots,x_n$) -or a negation of a logical variable -($\lnot x_1, \lnot x_2, \ldots, \lnot x_n$). -The symbols ''$\land$'' and ''$\lor$'' denote -logical operators ''and'' and ''or''. -Our task is to assign each variable a value -so that the formula is true, or state -that this is not possible. - -For example, the formula -\[ -L_1 = (x_2 \lor \lnot x_1) \land - (\lnot x_1 \lor \lnot x_2) \land - (x_1 \lor x_3) \land - (\lnot x_2 \lor \lnot x_3) \land - (x_1 \lor x_4) -\] -is true when the variables are assigned as follows: - -\[ -\begin{cases} -x_1 = \textrm{false} \\ -x_2 = \textrm{false} \\ -x_3 = \textrm{true} \\ -x_4 = \textrm{true} \\ -\end{cases} -\] - -However, the formula -\[ -L_2 = (x_1 \lor x_2) \land - (x_1 \lor \lnot x_2) \land - (\lnot x_1 \lor x_3) \land - (\lnot x_1 \lor \lnot x_3) -\] -is always false, regardless of how we -assign the values. -The reason for this is that we cannot -choose a value for $x_1$ -without creating a contradiction. -If $x_1$ is false, both $x_2$ and $\lnot x_2$ -should be true which is impossible, -and if $x_1$ is true, both $x_3$ and $\lnot x_3$ -should be true which is also impossible. - -The 2SAT problem can be represented as a graph -whose nodes correspond to -variables $x_i$ and negations $\lnot x_i$, -and edges determine the connections -between the variables. -Each pair $(a_i \lor b_i)$ generates two edges: -$\lnot a_i \to b_i$ and $\lnot b_i \to a_i$. -This means that if $a_i$ does not hold, -$b_i$ must hold, and vice versa. - -The graph for the formula $L_1$ is: -\\ -\begin{center} -\begin{tikzpicture}[scale=1.0,minimum size=2pt] -\node[draw, circle, inner sep=1.3pt] (1) at (1,2) {$\lnot x_3$}; -\node[draw, circle] (2) at (3,2) {$x_2$}; -\node[draw, circle, inner sep=1.3pt] (3) at (1,0) {$\lnot x_4$}; -\node[draw, circle] (4) at (3,0) {$x_1$}; -\node[draw, circle, inner sep=1.3pt] (5) at (5,2) {$\lnot x_1$}; -\node[draw, circle] (6) at (7,2) {$x_4$}; -\node[draw, circle, inner sep=1.3pt] (7) at (5,0) {$\lnot x_2$}; -\node[draw, circle] (8) at (7,0) {$x_3$}; - -\path[draw,thick,->] (1) -- (4); -\path[draw,thick,->] (4) -- (2); -\path[draw,thick,->] (2) -- (1); -\path[draw,thick,->] (3) -- (4); -\path[draw,thick,->] (2) -- (5); -\path[draw,thick,->] (4) -- (7); -\path[draw,thick,->] (5) -- (6); -\path[draw,thick,->] (5) -- (8); -\path[draw,thick,->] (8) -- (7); -\path[draw,thick,->] (7) -- (5); -\end{tikzpicture} -\end{center} -And the graph for the formula $L_2$ is: -\\ -\begin{center} -\begin{tikzpicture}[scale=1.0,minimum size=2pt] -\node[draw, circle] (1) at (1,2) {$x_3$}; -\node[draw, circle] (2) at (3,2) {$x_2$}; -\node[draw, circle, inner sep=1.3pt] (3) at (5,2) {$\lnot x_2$}; -\node[draw, circle, inner sep=1.3pt] (4) at (7,2) {$\lnot x_3$}; -\node[draw, circle, inner sep=1.3pt] (5) at (4,3.5) {$\lnot x_1$}; -\node[draw, circle] (6) at (4,0.5) {$x_1$}; - -\path[draw,thick,->] (1) -- (5); -\path[draw,thick,->] (4) -- (5); -\path[draw,thick,->] (6) -- (1); -\path[draw,thick,->] (6) -- (4); -\path[draw,thick,->] (5) -- (2); -\path[draw,thick,->] (5) -- (3); -\path[draw,thick,->] (2) -- (6); -\path[draw,thick,->] (3) -- (6); -\end{tikzpicture} -\end{center} - -The structure of the graph tells us whether -it is possible to assign the values -of the variables so -that the formula is true. -It turns out that this can be done -exactly when there are no nodes -$x_i$ and $\lnot x_i$ such that -both nodes belong to the -same strongly connected component. -If there are such nodes, -the graph contains -a path from $x_i$ to $\lnot x_i$ -and also a path from $\lnot x_i$ to $x_i$, -so both $x_i$ and $\lnot x_i$ should be true -which is not possible. - -In the graph of the formula $L_1$ -there are no nodes $x_i$ and $\lnot x_i$ -such that both nodes -belong to the same strongly connected component, -so a solution exists. -In the graph of the formula $L_2$ -all nodes belong to the same strongly connected component, -so a solution does not exist. - -If a solution exists, the values for the variables -can be found by going through the nodes of the -component graph in a reverse topological sort order. -At each step, we process a component -that does not contain edges that lead to an -unprocessed component. -If the variables in the component -have not been assigned values, -their values will be determined -according to the values in the component, -and if they already have values, -they remain unchanged. -The process continues until each variable -has been assigned a value. - -The component graph for the formula $L_1$ is as follows: -\begin{center} -\begin{tikzpicture}[scale=1.0] -\node[draw, circle] (1) at (0,0) {$A$}; -\node[draw, circle] (2) at (2,0) {$B$}; -\node[draw, circle] (3) at (4,0) {$C$}; -\node[draw, circle] (4) at (6,0) {$D$}; - -\path[draw,thick,->] (1) -- (2); -\path[draw,thick,->] (2) -- (3); -\path[draw,thick,->] (3) -- (4); -\end{tikzpicture} -\end{center} - -The components are -$A = \{\lnot x_4\}$, -$B = \{x_1, x_2, \lnot x_3\}$, -$C = \{\lnot x_1, \lnot x_2, x_3\}$ and -$D = \{x_4\}$. -When constructing the solution, -we first process the component $D$ -where $x_4$ becomes true. -After this, we process the component $C$ -where $x_1$ and $x_2$ become false -and $x_3$ becomes true. -All variables have been assigned values, -so the remaining components $A$ and $B$ -do not change the variables. - -Note that this method works, because the -graph has a special structure: -if there are paths from node $x_i$ to node $x_j$ -and from node $x_j$ to node $\lnot x_j$, -then node $x_i$ never becomes true. -The reason for this is that there is also -a path from node $\lnot x_j$ to node $\lnot x_i$, -and both $x_i$ and $x_j$ become false. - -\index{3SAT problem} - -A more difficult problem is the \key{3SAT problem}, -where each part of the formula is of the form -$(a_i \lor b_i \lor c_i)$. -This problem is NP-hard, so no efficient algorithm -for solving the problem is known. diff --git a/chapter18.tex b/chapter18.tex index 6c933df..17ff5db 100644 --- a/chapter18.tex +++ b/chapter18.tex @@ -55,7 +55,7 @@ is $O(k)$, which may be slow, because a tree of $n$ nodes may have a chain of $n$ nodes. Fortunately, using a technique similar to that -used in Chapter 16.3, any value of $\texttt{ancestor}(x,k)$ +used in Chapter 16.3 (of the full book), any value of $\texttt{ancestor}(x,k)$ can be efficiently calculated in $O(\log k)$ time after preprocessing. The idea is to precalculate all values $\texttt{ancestor}(x,k)$ diff --git a/preface.tex b/preface.tex index 887127b..6d5fa7d 100644 --- a/preface.tex +++ b/preface.tex @@ -2,32 +2,10 @@ \markboth{\MakeUppercase{Preface}}{} \addcontentsline{toc}{chapter}{Preface} -The purpose of this book is to give you -a thorough introduction to competitive programming. -It is assumed that you already -know the basics of programming, but no previous -background in competitive programming is needed. +This script is based on the Competitive Programmer's Handbook +by Antti Laaksonen. -The book is especially intended for -students who want to learn algorithms and -possibly participate in -the International Olympiad in Informatics (IOI) or -in the International Collegiate Programming Contest (ICPC). -Of course, the book is also suitable for -anybody else interested in competitive programming. - -It takes a long time to become a good competitive -programmer, but it is also an opportunity to learn a lot. -You can be sure that you will get -a good general understanding of algorithms -if you spend time reading the book, -solving problems and taking part in contests. - -The book is under continuous development. -You can always send feedback on the book to -\texttt{ahslaaks@cs.helsinki.fi}. - -\begin{flushright} -Helsinki, August 2019 \\ -Antti Laaksonen -\end{flushright} +It contains the topics relevant for the graph day of the +SOI Camp 2021. +Most of the code was modified slightly, and also some minor +adjustments were made on the text. 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The nodes are numbered using integers $1,2,\ldots,n$. +Note: at SOI we usually say \key{vertex} (plural \key{vertices}) instead of \key{node}. +Vertex and node can be used interchangeably. +We also like to number the vertices 0-based as $0,1,\ldots,n-1$. + For example, the following graph consists of 5 nodes and 7 edges: \begin{center} @@ -51,9 +55,11 @@ For example, the following graph consists of 5 nodes and 7 edges: \index{path} -A \key{path} leads from node $a$ to node $b$ +A \key{walk} leads from node $a$ to node $b$ through edges of the graph. -The \key{length} of a path is the number of +A \key{path} is a walk where each node appears +at most once in the path. +The \key{length} of a path (or a walk) is the number of edges in it. For example, the above graph contains a path $1 \rightarrow 3 \rightarrow 4 \rightarrow 5$ @@ -84,12 +90,10 @@ from node 1 to node 5: \index{cycle} -A path is a \key{cycle} if the first and last -node is the same. +A \key{cycle} is a walk where the first and last +node is the same, and every other vertex appears at most once. For example, the above graph contains a cycle $1 \rightarrow 3 \rightarrow 4 \rightarrow 1$. -A path is \key{simple} if each node appears -at most once in the path. % From d0f6df5d24037704e8cbe9178218548931539a5f Mon Sep 17 00:00:00 2001 From: Johannes Kapfhammer Date: Wed, 10 Feb 2021 20:28:36 +0100 Subject: [PATCH 5/5] replace push_back with emplace_back and enable colorlinks --- book.pdf | Bin 322196 -> 324122 bytes book.tex | 7 +++---- chapter11.tex | 36 ++++++++++++++++++------------------ 3 files changed, 21 insertions(+), 22 deletions(-) diff --git a/book.pdf b/book.pdf index d33718bcae4b8f159337874b386ff1a8d9d144bd..8031c4ed6e9b56be9236e4c76cdb6b104f1a30f0 100644 GIT binary patch delta 89304 zcmXVXV|bWN({^kpjn&vzW7{?x+s+l+wr$&P+{U&W+h6bJ{r;ZE?#v#$Gw00i%qB0Q zXC9!}gMxu`v9cyPf>HyAnnLzjOc=e#8V*@M`v01nWSMl?{pOIX$~DJu|G4i=N;Vi$cWzZBgW><{~jBDpQyuFYBXFv z_5&WrT%YB}Zmz4b+*L{AbM|09e5GWCRi*CZ#&zRDNK!I9*bEu?3z;3JEP8QkU1YCT zxm=}ZaYd0Y=%2rs$64~-^df3i4hPA^-)%n5PmpQ!Bx(Xq0EAj`4?c84ELFWL0fd+$ 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