From 8a936ed2467c7673bdb238679558bab7de9528e3 Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Sun, 26 Feb 2017 11:57:14 +0200 Subject: [PATCH] Small fixes --- chapter23.tex | 10 ++++++---- chapter24.tex | 23 ++++++++++++++--------- chapter25.tex | 4 ++-- 3 files changed, 22 insertions(+), 15 deletions(-) diff --git a/chapter23.tex b/chapter23.tex index 5b5d0e1..b4d6c25 100644 --- a/chapter23.tex +++ b/chapter23.tex @@ -245,8 +245,9 @@ two $n \times n$ matrices in $O(n^3)$ time. There are also more efficient algorithms for matrix multiplication\footnote{The first such -algorithm, with time complexity $O(n^{2.80735})$, -was published in 1969 \cite{str69}, and +algorithm was Strassen's algorithm, +published in 1969 \cite{str69}, +whose time complexity is $O(n^{2.80735})$; the best current algorithm works in $O(n^{2.37286})$ time \cite{gal14}.}, but they are mostly of theoretical interest @@ -749,8 +750,9 @@ $2 \rightarrow 1 \rightarrow 4 \rightarrow 2 \rightarrow 5$. \index{Kirchhoff's theorem} \index{spanning tree} -\key{Kirchhoff's theorem}\footnote{G. R. Kirchhoff (1824--1887) was -a German physicist.} provides a way +\key{Kirchhoff's theorem} +%\footnote{G. R. Kirchhoff (1824--1887) was a German physicist.} +provides a way to calculate the number of spanning trees of a graph as a determinant of a special matrix. For example, the graph diff --git a/chapter24.tex b/chapter24.tex index 6f91c2c..a739613 100644 --- a/chapter24.tex +++ b/chapter24.tex @@ -359,8 +359,10 @@ The expected value for $X$ in a geometric distribution is \index{Markov chain} -A \key{Markov chain}\footnote{A. A. Markov (1856--1922) -was a Russian mathematician.} 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. @@ -515,11 +517,13 @@ just to find one element? It turns out that we can find order statistics using a randomized algorithm without sorting the array. -The algorithm is a Las Vegas algorithm: +The algorithm, called \key{quickselect}\footnote{In 1961, +C. A. R. Hoare published two algorithms that +are efficient on average: \index{quicksort} \index{quickselect} +\key{quicksort} \cite{hoa61a} for sorting arrays and +\key{quickselect} \cite{hoa61b} for finding order statistics.}, is a Las Vegas algorithm: its running time is usually $O(n)$ -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}.}. +but $O(n^2)$ in the worst case. The algorithm chooses a random element $x$ in the array, and moves elements smaller than $x$ @@ -563,9 +567,10 @@ but one could hope that verifying the 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 $O(n^2)$\footnote{This algorithm is sometimes -called \index{Freivalds' algoritm} \key{Freivalds' algorithm} \cite{fre77}.}. +using a Monte Carlo algorithm\footnote{R. M. Freivalds published +this algorithm in 1977 \cite{fre77}, and it is sometimes +called \index{Freivalds' algoritm} \key{Freivalds' algorithm}.} whose +time complexity is only $O(n^2)$. 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 9c5f565..a34227f 100644 --- a/chapter25.tex +++ b/chapter25.tex @@ -249,7 +249,7 @@ 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\footnote{The optimal strategy -for nim was published in 1901 by C. L. Bouton \cite{bou01}}. +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 @@ -369,7 +369,7 @@ so the nim sum is not 0. \index{Sprague–Grundy theorem} The \key{Sprague–Grundy theorem}\footnote{The theorem was discovered -independently by R. Sprague \cite{spr35} and P. M. Grundy \cite{gru39}} generalizes the +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: