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\begin{thebibliography}{9}
\bibitem{aho83}
A. V. Aho, J. E. Hopcroft and J. Ullman.
\emph{Data Structures and Algorithms},
Addison-Wesley, 1983.
\bibitem{asp79}
B. Aspvall, M. F. Plass and R. E. Tarjan.
A linear-time algorithm for testing the truth of certain quantified boolean formulas.
\emph{Information Processing Letters}, 8(3):121--123, 1979.
\bibitem{bel58}
R. Bellman.
On a routing problem.
\emph{Quarterly of Applied Mathematics}, 16(1):87--90, 1958.
\bibitem{ben86}
J. Bentley.
\emph{Programming Pearls}.
Addison-Wesley, 1986.
\bibitem{cod15}
Codeforces: On ''Mo's algorithm'',
\url{http://codeforces.com/blog/entry/20032}
\bibitem{dij59}
E. W. Dijkstra.
A note on two problems in connexion with graphs.
\emph{Numerische Mathematik}, 1(1):269--271, 1959.
\bibitem{edm65}
J. Edmonds.
Paths, trees, and flowers.
\emph{Canadian Journal of Mathematics}, 17(3):449--467, 1965.
\bibitem{edm72}
J. Edmonds and R. M. Karp.
Theoretical improvements in algorithmic efficiency for network flow problems.
\emph{Journal of the ACM}, 19(2):248--264, 1972.
\bibitem{fan94}
D. Fanding.
A faster algorithm for shortest-path -- SPFA.
\emph{Journal of Southwest Jiaotong University}, 2, 1994.
\bibitem{fen94}
P. M. Fenwick.
A new data structure for cumulative frequency tables.
\emph{Software: Practice and Experience}, 24(3):327--336, 1994.
\bibitem{fis11}
J. Fischer and V. Heun.
Space-efficient preprocessing schemes for range minimum queries on static arrays.
\emph{SIAM Journal on Computing}, 40(2):465--492, 2011.
\bibitem{flo62}
R. W. Floyd
Algorithm 97: shortest path.
\emph{Communications of the ACM}, 5(6):345, 1962.
\bibitem{for56}
L. R. Ford and D. R. Fulkerson.
Maximal flow through a network.
\emph{Canadian Journal of Mathematics}, 8(3):399--404, 1956.
\bibitem{gal14}
F. Le Gall.
Powers of tensors and fast matrix multiplication.
In \emph{Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation},
296--303.
\bibitem{gar79}
M. R. Garey and D. S. Johnson.
\emph{Computers and Intractability:
A Guide to the Theory of NP-Completeness},
W. H. Freeman and Company, 1979.
\bibitem{goo16}
Google Code Jam Statistics (2016),
\url{https://www.go-hero.net/jam/16}
\bibitem{gro14}
A. Grønlund and S. Pettie.
Threesomes, degenerates, and love triangles.
\emph{2014 IEEE 55th Annual Symposium on Foundations of Computer Science},
621--630, 2014.
\bibitem{gus97}
D. Gusfield.
\emph{Algorithms on Strings, Trees and Sequences:
Computer Science and Computational Biology},
Cambridge University Press, 1997.
\bibitem{huf52}
A method for the construction of minimum-redundancy codes.
\emph{Proceedings of the IRE}, 40(9):1098--1101, 1952.
\bibitem{kas61}
P. W. Kasteleyn.
The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice.
\emph{Physica}, 27(12):1209--1225, 1961.
\bibitem{ken06}
C. Kent, G. m. Landau and M. Ziv-Ukelson.
On the complexity of sparse exon assembly.
\emph{Journal of Computational Biology}, 13(5):1013--1027, 2006.
\bibitem{kru56}
J. B. Kruskal.
On the shortest spanning subtree of a graph and the traveling salesman problem.
\emph{Proceedings of the American Mathematical Society}, 7(1):48--50, 1956.
\bibitem{pac13}
J. Pachocki and J. Radoszweski.
Where to use and how not to use polynomial string hashing.
\emph{Olympiads in Informatics}, 2013.
\bibitem{pri57}
R. C. Prim.
Shortest connection networks and some generalizations.
\emph{Bell System Technical Journal}, 36(6):1389--1401, 1957.
\bibitem{sha81}
M. Sharir.
A strong-connectivity algorithm and its applications in data flow analysis.
\emph{Computers \& Mathematics with Applications}, 7(1):67--72, 1981.
\bibitem{str69}
V. Strassen.
Gaussian elimination is not optimal.
\emph{Numerische Mathematik}, 13(4):354--356, 1969.
\bibitem{tem61}
H. N. V. Temperley and M. E. Fisher.
Dimer problem in statistical mechanics -- an exact result.
\emph{Philosophical Magazine}, 6(68):1061--1063, 1961.
\end{thebibliography}
%
%
% \chapter*{Literature}
% \markboth{\MakeUppercase{Literature}}{}
% \addcontentsline{toc}{chapter}{Literature}
%
% \subsubsection{Textbooks on algorithms}
%
% \begin{itemize}
% \item T. H. Cormen, C. E. Leiserson,
% R. L Rivest and C. Stein:
% \emph{Introduction to Algorithms},
% MIT Press, 2009 (3rd edition)
% \item J. Kleinberg and É. Tardos:
% \emph{Algorithm Design},
% Pearson, 2005
% \item S. S. Skiena:
% \emph{The Algorithm Design Manual},
% Springer, 2008 (2nd edition)
% \end{itemize}
%
% \subsubsection{Textbooks on competitive programming}
%
% \begin{itemize}
% \item K. Diks, T. Idziaszek,
% J. Łącki and J. Radoszewski:
% \emph{Looking for a Challenge?},
% University of Warsaw, 2012
% \item S. Halim and F. Halim:
% \emph{Competitive Programming},
% 2013 (3rd edition)
% \item S. S. Skiena and M. A. Revilla:
% \emph{Programming Challenges: The Programming
% Contest Training Manual},
% Springer, 2003
% \end{itemize}
%
% \subsubsection{Other books}
%
% \begin{itemize}
% \item J. Bentley:
% \emph{Programming Pearls},
% Addison-Wesley, 1999 (2nd edition)
% \end{itemize}

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@ -1,6 +1,6 @@
\documentclass[twoside,12pt,a4paper,english]{book} \documentclass[twoside,12pt,a4paper,english]{book}
%\includeonly{johdanto} \includeonly{luku27,kirj}
\usepackage[english]{babel} \usepackage[english]{babel}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -105,9 +105,9 @@
\include{luku28} \include{luku28}
\include{luku29} \include{luku29}
\include{luku30} \include{luku30}
%\include{kirj} \include{kirj}
\cleardoublepage \cleardoublepage
\printindex \printindex
\end{document} \end{document}la

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luku01.tex
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@ -49,7 +49,7 @@ For example, in Google Code Jam 2016,
among the best 3,000 participants, among the best 3,000 participants,
73 \% used C++, 73 \% used C++,
15 \% used Python and 15 \% used Python and
10 \% used Java. %\footnote{\url{https://www.go-hero.net/jam/16}} 10 \% used Java \cite{goo16}.
Some participants also used several languages. Some participants also used several languages.
Many people think that C++ is the best choice Many people think that C++ is the best choice

5
luku02.tex
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@ -281,7 +281,7 @@ Still, there are many important problems for which
no polynomial algorithm is known, i.e., no polynomial algorithm is known, i.e.,
nobody knows how to solve them efficiently. nobody knows how to solve them efficiently.
\key{NP-hard} problems are an important set \key{NP-hard} problems are an important set
of problems for which no polynomial algorithm is known. of problems for which no polynomial algorithm is known \cite{gar79}.
\section{Estimating efficiency} \section{Estimating efficiency}
@ -355,7 +355,8 @@ Given an array of $n$ integers $x_1,x_2,\ldots,x_n$,
our task is to find the our task is to find the
\key{maximum subarray sum}, i.e., \key{maximum subarray sum}, i.e.,
the largest possible sum of numbers the largest possible sum of numbers
in a contiguous region in the array. in a contiguous region in the array\footnote{Jon Bentley's
book \emph{Programming Pearls} \cite{ben86} made this problem popular.}.
The problem is interesting when there may be The problem is interesting when there may be
negative numbers in the array. negative numbers in the array.
For example, in the array For example, in the array

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luku06.tex
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@ -530,7 +530,7 @@ the string \texttt{AB} or the string \texttt{C}.
\subsubsection{Huffman coding} \subsubsection{Huffman coding}
\key{Huffman coding} is a greedy algorithm \key{Huffman coding} \cite{huf52} is a greedy algorithm
that constructs an optimal code for that constructs an optimal code for
compressing a given string. compressing a given string.
The algorithm builds a binary tree The algorithm builds a binary tree

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@ -983,7 +983,7 @@ $2^m$ distinct rows and the time complexity is
$O(n 2^{2m})$. $O(n 2^{2m})$.
As a final note, there is also a surprising direct formula As a final note, there is also a surprising direct formula
for calculating the number of tilings: for calculating the number of tilings \cite{kas61,tem61}:
\[ \prod_{a=1}^{\lceil n/2 \rceil} \prod_{b=1}^{\lceil m/2 \rceil} 4 \cdot (\cos^2 \frac{\pi a}{n + 1} + \cos^2 \frac{\pi b}{m+1})\] \[ \prod_{a=1}^{\lceil n/2 \rceil} \prod_{b=1}^{\lceil m/2 \rceil} 4 \cdot (\cos^2 \frac{\pi a}{n + 1} + \cos^2 \frac{\pi b}{m+1})\]
This formula is very efficient, because it calculates This formula is very efficient, because it calculates
the number of tilings in $O(nm)$ time, the number of tilings in $O(nm)$ time,

8
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@ -419,7 +419,13 @@ A more difficult problem is
the \key{3SUM problem} that asks to the \key{3SUM problem} that asks to
find \emph{three} numbers in the array find \emph{three} numbers in the array
such that their sum is $x$. such that their sum is $x$.
This problem can be solved in $O(n^2)$ time. Using the idea of the above algorithm,
this problem can be solved in $O(n^2)$ time\footnote{For a long time,
it was thought that solving
the 3SUM problem more efficiently than in $O(n^2)$ time
would not be possible.
However, in 2014, it turned out \cite{gro14}
that this is not the case.}.
Can you see how? Can you see how?
\section{Nearest smaller elements} \section{Nearest smaller elements}

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@ -242,9 +242,12 @@ to the position of $X$.
\subsubsection{Minimum queries} \subsubsection{Minimum queries}
It is also possible to process minimum queries Next we will see how we can
in $O(1)$ time, though it is more difficult than process range minimum queries in $O(1)$ time
to process sum queries. after an $O(n \log n)$ time preprocessing\footnote{There are also
sophisticated techniques \cite{fis11} where the preprocessing time
is only $O(n)$, but such algorithms are not needed in
competitive programming.}.
Note that minimum and maximum queries can always Note that minimum and maximum queries can always
be processed using similar techniques, be processed using similar techniques,
so it suffices to focus on minimum queries. so it suffices to focus on minimum queries.
@ -437,7 +440,7 @@ we can conclude that $\textrm{rmq}(2,7)=1$.
\index{binary indexed tree} \index{binary indexed tree}
\index{Fenwick tree} \index{Fenwick tree}
A \key{binary indexed tree} or \key{Fenwick tree} A \key{binary indexed tree} or \key{Fenwick tree} \cite{fen94}
can be seen as a dynamic variant of a sum array. can be seen as a dynamic variant of a sum array.
This data structure supports two $O(\log n)$ time operations: This data structure supports two $O(\log n)$ time operations:
calculating the sum of elements in a range calculating the sum of elements in a range

2
luku12.tex
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@ -546,4 +546,4 @@ it is difficult to find out if the nodes
in a graph can be colored using $k$ colors in a graph can be colored using $k$ colors
so that no adjacent nodes have the same color. so that no adjacent nodes have the same color.
Even when $k=3$, no efficient algorithm is known Even when $k=3$, no efficient algorithm is known
but the problem is NP-hard. but the problem is NP-hard \cite{gar79}.

8
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@ -24,7 +24,7 @@ for finding shortest paths.
\index{Bellman–Ford algorithm} \index{Bellman–Ford algorithm}
The \key{Bellman–Ford algorithm} finds the The \key{Bellman–Ford algorithm} \cite{bel58} finds the
shortest paths from a starting node to all shortest paths from a starting node to all
other nodes in the graph. other nodes in the graph.
The algorithm can process all kinds of graphs, The algorithm can process all kinds of graphs,
@ -280,7 +280,7 @@ regardless of the starting node.
\index{SPFA algorithm} \index{SPFA algorithm}
The \key{SPFA algorithm} (''Shortest Path Faster Algorithm'') The \key{SPFA algorithm} (''Shortest Path Faster Algorithm'') \cite{fan94}
is a variant of the Bellman–Ford algorithm, is a variant of the Bellman–Ford algorithm,
that is often more efficient than the original algorithm. that is often more efficient than the original algorithm.
The SPFA algorithm does not go through all the edges on each round, The SPFA algorithm does not go through all the edges on each round,
@ -331,7 +331,7 @@ original Bellman–Ford algorithm.
\index{Dijkstra's algorithm} \index{Dijkstra's algorithm}
\key{Dijkstra's algorithm} finds the shortest \key{Dijkstra's algorithm} \cite{dij59} finds the shortest
paths from the starting node to all other nodes, paths from the starting node to all other nodes,
like the Bellman–Ford algorithm. like the Bellman–Ford algorithm.
The benefit in Dijsktra's algorithm is that The benefit in Dijsktra's algorithm is that
@ -594,7 +594,7 @@ at most one distance to the priority queue.
\index{Floyd–Warshall algorithm} \index{Floyd–Warshall algorithm}
The \key{Floyd–Warshall algorithm} The \key{Floyd–Warshall algorithm} \cite{flo62}
is an alternative way to approach the problem is an alternative way to approach the problem
of finding shortest paths. of finding shortest paths.
Unlike the other algorihms in this chapter, Unlike the other algorihms in this chapter,

4
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@ -123,7 +123,7 @@ maximum spanning trees by processing the edges in reverse order.
\index{Kruskal's algorithm} \index{Kruskal's algorithm}
In \key{Kruskal's algorithm}, the initial spanning tree In \key{Kruskal's algorithm} \cite{kru56}, the initial spanning tree
only contains the nodes of the graph only contains the nodes of the graph
and does not contain any edges. and does not contain any edges.
Then the algorithm goes through the edges Then the algorithm goes through the edges
@ -567,7 +567,7 @@ the smaller set to the larger set.
\index{Prim's algorithm} \index{Prim's algorithm}
\key{Prim's algorithm} is an alternative method \key{Prim's algorithm} \cite{pri57} is an alternative method
for finding a minimum spanning tree. for finding a minimum spanning tree.
The algorithm first adds an arbitrary node The algorithm first adds an arbitrary node
to the tree. to the tree.

9
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@ -135,7 +135,10 @@ presented in Chapter 16.
\index{Kosaraju's algorithm} \index{Kosaraju's algorithm}
\key{Kosaraju's algorithm} is an efficient \key{Kosaraju's algorithm}\footnote{According to \cite{aho83},
S. R. Kosaraju invented this algorithm in 1978
but did not publish it. In 1981, the same algorithm was rediscovered
and published by M. Sharir \cite{sha81}.} is an efficient
method for finding the strongly connected components method for finding the strongly connected components
of a directed graph. of a directed graph.
The algorithm performs two depth-first searches: The algorithm performs two depth-first searches:
@ -365,7 +368,7 @@ performs two depth-first searches.
\index{2SAT problem} \index{2SAT problem}
Strongly connectivity is also linked with the Strongly connectivity is also linked with the
\key{2SAT problem}. \key{2SAT problem} \cite{asp79}.
In this problem, we are given a logical formula 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), (a_1 \lor b_1) \land (a_2 \lor b_2) \land \cdots \land (a_m \lor b_m),
@ -556,7 +559,7 @@ and both $x_i$ and $x_j$ become false.
A more difficult problem is the \key{3SAT problem} A more difficult problem is the \key{3SAT problem}
where each part of the formula is of the form where each part of the formula is of the form
$(a_i \lor b_i \lor c_i)$. $(a_i \lor b_i \lor c_i)$.
This problem is NP-hard, so no efficient algorithm This problem is NP-hard \cite{gar79}, so no efficient algorithm
for solving the problem is known. for solving the problem is known.

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luku20.tex
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@ -89,7 +89,7 @@ It is easy see that this flow is maximum,
because the total capacity of the edges because the total capacity of the edges
leading to the sink is $7$. leading to the sink is $7$.
\subsubsection{Minimum cuts} \subsubsection{Minimum cut}
\index{cut} \index{cut}
\index{minimum cut} \index{minimum cut}
@ -157,7 +157,7 @@ The algorithm also helps us to understand
\index{Ford–Fulkerson algorithm} \index{Ford–Fulkerson algorithm}
The \key{Ford–Fulkerson algorithm} finds The \key{Ford–Fulkerson algorithm} \cite{for56} finds
the maximum flow in a graph. the maximum flow in a graph.
The algorithm begins with an empty flow, The algorithm begins with an empty flow,
and at each step finds a path in the graph and at each step finds a path in the graph
@ -430,7 +430,7 @@ by using one of the following techniques:
\index{Edmonds–Karp algorithm} \index{Edmonds–Karp algorithm}
The \key{Edmonds–Karp algorithm} The \key{Edmonds–Karp algorithm} \cite{edm72}
is a variant of the is a variant of the
Ford–Fulkerson algorithm that Ford–Fulkerson algorithm that
chooses each path so that the number of edges chooses each path so that the number of edges
@ -459,7 +459,7 @@ because depth-first search can be used for finding paths.
Both algorithms are efficient enough for problems Both algorithms are efficient enough for problems
that typically appear in programming contests. that typically appear in programming contests.
\subsubsection{Minimum cut} \subsubsection{Minimum cuts}
\index{minimum cut} \index{minimum cut}
@ -779,7 +779,7 @@ such that each pair is connected with an edge and
each node belongs to at most one pair. each node belongs to at most one pair.
There are polynomial algorithms for finding There are polynomial algorithms for finding
maximum matchings in general graphs, maximum matchings in general graphs \cite{edm65},
but such algorithms are complex and do but such algorithms are complex and do
not appear in programming contests. not appear in programming contests.
However, in bipartite graphs, However, in bipartite graphs,

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@ -244,12 +244,16 @@ we can calculate the product of
two $n \times n$ matrices two $n \times n$ matrices
in $O(n^3)$ time. in $O(n^3)$ time.
There are also more efficient algorithms There are also more efficient algorithms
for matrix multiplication: for matrix multiplication\footnote{The first such
at the moment, the best known time complexity algorithm, with time complexity $O(n^{2.80735})$,
is $O(n^{2.37})$. was published in 1969 \cite{str69}, and
However, such special algorithms are not needed the best current algorithm
works in $O(n^{2.37286})$ time \cite{gal14}.},
but they are mostly of theoretical interest
and such special algorithms are not needed
in competitive programming. in competitive programming.
\subsubsection{Matrix power} \subsubsection{Matrix power}
\index{matrix power} \index{matrix power}

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luku26.tex
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@ -416,12 +416,7 @@ which is convenient, because operations with 32 and 64
bit integers are calculated modulo $2^{32}$ and $2^{64}$. bit integers are calculated modulo $2^{32}$ and $2^{64}$.
However, this is not a good choice, because it is possible However, this is not a good choice, because it is possible
to construct inputs that always generate collisions when to construct inputs that always generate collisions when
constants of the form $2^x$ are used. constants of the form $2^x$ are used \cite{pac13}.
% \footnote{
% J. Pachocki and Jakub Radoszweski:
% ''Where to use and how not to use polynomial string hashing''.
% \textit{Olympiads in Informatics}, 2013.
% }.
\section{Z-algorithm} \section{Z-algorithm}
@ -433,7 +428,7 @@ gives for each position $k$ in the string
the length of the longest substring the length of the longest substring
that begins at position $k$ and is a prefix of the string. that begins at position $k$ and is a prefix of the string.
Such an array can be efficiently constructed Such an array can be efficiently constructed
using the \key{Z-algorithm}. using the \key{Z-algorithm} \cite{gus97}.
For example, the Z-array for the string For example, the Z-array for the string
\texttt{ACBACDACBACBACDA} is as follows: \texttt{ACBACDACBACBACDA} is as follows:

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luku27.tex
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@ -314,7 +314,9 @@ because both cases take a total of $O(n \sqrt n)$ time.
\index{Mo's algorithm} \index{Mo's algorithm}
\key{Mo's algorithm} can be used in many problems \key{Mo's algorithm} \footnote{According to \cite{cod15}, this algorithm
is named after Mo Tao, a Chinese competitive programmer. However,
the technique has been appeared earlier in the literature \cite{ken06}.} can be used in many problems
that require processing range queries in that require processing range queries in
a \emph{static} array. a \emph{static} array.
Before processing the queries, the algorithm Before processing the queries, the algorithm