diff --git a/chapter24.tex b/chapter24.tex index a4bd1aa..6f91c2c 100644 --- a/chapter24.tex +++ b/chapter24.tex @@ -359,7 +359,8 @@ The expected value for $X$ in a geometric distribution is \index{Markov chain} -A \key{Markov chain} is a random process +A \key{Markov chain}\footnote{A. A. Markov (1856--1922) +was a Russian mathematician.} is a random process that consists of states and transitions between them. For each state, we know the probabilities for moving to other states. @@ -516,7 +517,9 @@ It turns out that we can find order statistics using a randomized algorithm without sorting the array. The algorithm is a Las Vegas algorithm: its running time is usually $O(n)$ -but $O(n^2)$ in the worst case. +but $O(n^2)$ in the worst case\footnote{C. A. R. Hoare +discovered both this algorithm, known as \key{quickselect} \cite{hoa61b}, +and a similar sorting algorithm, known as \key{quicksort} \cite{hoa61a}.}. The algorithm chooses a random element $x$ in the array, and moves elements smaller than $x$ @@ -561,7 +564,8 @@ answer would by easier than to calculate it from scratch. It turns out that we can solve the problem using a Monte Carlo algorithm whose -time complexity is only $O(n^2)$. +time complexity is only $O(n^2)$\footnote{This algorithm is sometimes +called \index{Freivalds' algoritm} \key{Freivalds' algorithm} \cite{fre77}.}. The idea is simple: we choose a random vector $X$ of $n$ elements, and calculate the matrices $ABX$ and $CX$. If $ABX=CX$, we report that $AB=C$, diff --git a/chapter25.tex b/chapter25.tex index 5c3b9dc..9c5f565 100644 --- a/chapter25.tex +++ b/chapter25.tex @@ -248,7 +248,8 @@ and this is always the final state. It turns out that we can easily classify any nim state by calculating the \key{nim sum} $x_1 \oplus x_2 \oplus \cdots \oplus x_n$, -where $\oplus$ is the xor operation. +where $\oplus$ is the xor operation\footnote{The optimal strategy +for nim was published in 1901 by C. L. Bouton \cite{bou01}}. The states whose nim sum is 0 are losing states, and all other states are winning states. For example, the nim sum for @@ -367,7 +368,8 @@ so the nim sum is not 0. \index{Sprague–Grundy theorem} -The \key{Sprague–Grundy theorem} generalizes the +The \key{Sprague–Grundy theorem}\footnote{The theorem was discovered +independently by R. Sprague \cite{spr35} and P. M. Grundy \cite{gru39}} generalizes the strategy used in nim to all games that fulfil the following requirements: diff --git a/list.tex b/list.tex index 3aa597c..1131cfa 100644 --- a/list.tex +++ b/list.tex @@ -35,6 +35,11 @@ \emph{Programming Pearls}. Addison-Wesley, 1986. +\bibitem{bou01} + C. L. Bouton. + Nim, a game with a complete mathematical theory. + \emph{Annals of Mathematics}, 3(1/4):35--39, 1901. + \bibitem{cod15} Codeforces: On ''Mo's algorithm'', \url{http://codeforces.com/blog/entry/20032} @@ -94,6 +99,11 @@ Maximal flow through a network. \emph{Canadian Journal of Mathematics}, 8(3):399--404, 1956. +\bibitem{fre77} + R. Freivalds. + Probabilistic machines can use less running time. + In \emph{IFIP congress}, 839--842, 1977. + \bibitem{gal14} F. Le Gall. Powers of tensors and fast matrix multiplication. @@ -116,6 +126,11 @@ \emph{2014 IEEE 55th Annual Symposium on Foundations of Computer Science}, 621--630, 2014. +\bibitem{gru39} + P. M. Grundy. + Mathematics and games. + \emph{Eureka}, 2(5):6--8, 1939. + \bibitem{gus97} D. Gusfield. \emph{Algorithms on Strings, Trees and Sequences: @@ -139,6 +154,16 @@ Über die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. \emph{Mathematische Annalen}, 6(1), 30--32, 1873. +\bibitem{hoa61a} + C. A. R. Hoare. + Algorithm 64: Quicksort. + \emph{Communications of the ACM}, 4(7):321, 1961. + +\bibitem{hoa61b} + C. A. R. Hoare. + Algorithm 65: Find. + \emph{Communications of the ACM}, 4(7):321--322, 1961. + \bibitem{hop71} J. E. Hopcroft and J. D. Ullman. A linear list merging algorithm. @@ -231,6 +256,11 @@ A strong-connectivity algorithm and its applications in data flow analysis. \emph{Computers \& Mathematics with Applications}, 7(1):67--72, 1981. +\bibitem{spr35} + R. Sprague. + Über mathematische Kampfspiele. + \emph{Tohoku Mathematical Journal}, 41:438--444, 1935. + \bibitem{sta06} P. Stańczyk. \emph{Algorytmika praktyczna w konkursach Informatycznych},