References etc.

kirj.tex

@@ -20,12 +20,6 @@
   \emph{Programming Pearls}.
   Addison-Wesley, 1986.
 
-\bibitem{ber93}
-  O. Berkman and U. Vishkin.
-  Recursive star-tree parallel data structure.
-  \emph{SIAM Journal on Computing},
-  22(2):221--242, 1993.
-
 \bibitem{cod15}
   Codeforces: On ''Mo's algorithm'',
   \url{http://codeforces.com/blog/entry/20032}
@@ -55,6 +49,11 @@
   A new data structure for cumulative frequency tables.
   \emph{Software: Practice and Experience}, 24(3):327--336, 1994.
 
+\bibitem{fis06}
+  J. Fischer and V. Heun.
+  Theoretical and practical improvements on the RMQ-problem, with applications to LCA and LCE.
+  In \emph{Annual Symposium on Combinatorial Pattern Matching}, 36--48, 2006.
+
 \bibitem{fis11}
   J. Fischer and V. Heun.
   Space-efficient preprocessing schemes for range minimum queries on static arrays.
@@ -102,6 +101,11 @@
   A method for the construction of minimum-redundancy codes.
   \emph{Proceedings of the IRE}, 40(9):1098--1101, 1952.
 
+\bibitem{kar87}
+  R. M. Karp and M. O. Rabin.
+  Efficient randomized pattern-matching algorithms.
+  \emph{IBM Journal of Research and Development}, 31(2):249--260, 1987.
+
 \bibitem{kas61}
   P. W. Kasteleyn.  
   The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice.
@@ -117,6 +121,17 @@
   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{ben00}
+  M. A. Bender and M. Farach-Colton.
+  The LCA problem revisited. In
+  \emph{Latin American Symposium on Theoretical Informatics}, 88--94, 2000.
+
+\bibitem{mai84}
+  M. G. Main and R. J. Lorentz.
+  An $O(n \log n)$ algorithm for finding all repetitions in a string.
+  \emph{Journal of Algorithms}, 5(3):422--432, 1984.
+
+
 \bibitem{pac13}
   J. Pachocki and J. Radoszweski.
   Where to use and how not to use polynomial string hashing.
@@ -137,59 +152,9 @@
   Gaussian elimination is not optimal.
   \emph{Numerische Mathematik}, 13(4):354--356, 1969.
 
-\bibitem{tar84}
-  R. E. Tarjan and U. Vishkin.
-  Finding biconnected components and computing tree functions
-  in logarithmic parallel time.
-  \emph{25th Annual Symposium on Foundations of Computer Science}, 12--20, 1984.
-
 \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}

kkkk.tex

@@ -1,6 +1,6 @@
 \documentclass[twoside,12pt,a4paper,english]{book}
 
-\includeonly{luku27,kirj}
+\includeonly{luku18,kirj}
 
 \usepackage[english]{babel}
 \usepackage[utf8]{inputenc}

luku02.tex

@@ -353,10 +353,10 @@ time and even in $O(n)$ time.
 
 Given an array of $n$ integers $x_1,x_2,\ldots,x_n$,
 our task is to find the
-\key{maximum subarray sum}, i.e.,
+\key{maximum subarray sum}\footnote{Bentley's
+book \emph{Programming Pearls} \cite{ben86} made this problem popular.}, i.e.,
 the largest possible sum of numbers
-in a contiguous region in the array\footnote{Jon Bentley's
+in a contiguous region in the array.
-book \emph{Programming Pearls} \cite{ben86} made this problem popular.}.
 The problem is interesting when there may be
 negative numbers in the array.
 For example, in the array

luku09.tex

@@ -245,7 +245,7 @@ to the position of $X$.
 Next we will see how we can
 process range minimum queries in $O(1)$ time
 after an $O(n \log n)$ time preprocessing\footnote{There are also
-sophisticated techniques \cite{fis11} where the preprocessing time
+sophisticated techniques \cite{fis06} 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

luku18.tex

@@ -85,9 +85,9 @@ as a sum where each term is a power of two.
 
 \section{Subtrees and paths}
 
-\index{node array}
+\index{tree traversal array}
 
-A \key{node array}\footnote{A similar idea is sometimes called the \key{Euler tour technique} \cite{tar84}.} contains the nodes of a rooted tree
+A \key{tree traversal array} contains the nodes of a rooted tree
 in the order in which a depth-first search
 from the root node visits them.
 For example, in the tree
@@ -155,7 +155,7 @@ a depth-first search proceeds as follows:
 
 \end{tikzpicture}
 \end{center}
-Hence, the corresponding node array is as follows:
+Hence, the corresponding tree traversal array is as follows:
 \begin{center}
 \begin{tikzpicture}[scale=0.7]
 \draw (0,0) grid (9,1);
@@ -186,8 +186,8 @@ Hence, the corresponding node array is as follows:
 \subsubsection{Subtree queries}
 
 Each subtree of a tree corresponds to a subarray
-in the node array,
+in the tree traversal array such that
-where the first element is the root node.
+the first element in the subarray is the root node.
 For example, the following subarray contains the
 nodes in the subtree of node $4$:
 \begin{center}
@@ -265,7 +265,7 @@ is $3+4+3+1=11$.
 \end{tikzpicture}
 \end{center}
 
-The idea is to construct a node array that contains
+The idea is to construct a tree traversal array that contains
 three values for each node: (1) the identifier of the node,
 (2) the size of the subtree, and (3) the value of the node.
 For example, the array for the above tree is as follows:
@@ -381,7 +381,7 @@ and calculate the sum of values in $O(\log n)$ time.
 
 \subsubsection{Path queries}
 
-Using a node array, we can also efficiently
+Using a tree traversal array, we can also efficiently
 calculate sums of values on
 paths from the root node to any other
 node in the tree.
@@ -484,7 +484,7 @@ For example, the following array corresponds to the above tree:
 When the value of a node increases by $x$,
 the sums of all nodes in its subtree increase by $x$.
 For example, if the value of node 4 increases by 1,
-the node array changes as follows:
+the array changes as follows:
 
 \begin{center}
 \begin{tikzpicture}[scale=0.7]
@@ -650,7 +650,7 @@ performed in $O(\log n)$ time.
 \subsubsection{Method 2}
 
 Another way to solve the problem is based on
-a node array.
+a tree traversal array \cite{ben00}.
 Once again, the idea is to traverse the nodes
 using a depth-first search:
 
@@ -689,12 +689,13 @@ using a depth-first search:
 \end{tikzpicture}
 \end{center}
 
-However, in this problem,
+
-we add each node to the node array \emph{always}
+However, we use a bit different technique where
+we add each node to the tree traversal array \emph{always}
 when the depth-first search visits the node,
 and not only at the first visit.
 Hence, a node that has $k$ children appears $k+1$ times
-in the node array, and there are a total of $2n-1$
+in the array, and there are a total of $2n-1$
 nodes in the array.
 
 We store two values in the array:

luku26.tex

@@ -196,7 +196,9 @@ from node $s$ using character $c$.
 
 \key{String hashing} is a technique that
 allows us to efficiently check whether two
-substrings in a string are equal.
+substrings in a string are equal\footnote{The technique
+was popularized by the Karp–Rabin pattern matching
+algorithm \cite{kar87}.}.
 The idea is to compare the hash values of the
 substrings instead of their individual characters.
 
@@ -428,7 +430,10 @@ gives for each position $k$ in the string
 the length of the longest substring
 that begins at position $k$ and is a prefix of the string.
 Such an array can be efficiently constructed
-using the \key{Z-algorithm} \cite{gus97}.
+using the \key{Z-algorithm}\footnote{The Z-algorithm
+was presented in \cite{gus97} as the simplest known
+method for linear-time pattern matching, and the original idea
+was attributed to \cite{mai84}.}.
 
 For example, the Z-array for the string
 \texttt{ACBACDACBACBACDA} is as follows: