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Small fixes

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Antti H S Laaksonen 2017-05-30 20:50:04 +03:00
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chapter27.tex
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@ -283,7 +283,7 @@ For example, consider the following operation:
First, we calculate the minimum distance First, we calculate the minimum distance
from the white cell marked with * to a black cell. from the white cell marked with * to a black cell.
The minimum distance is 2, because we can move The minimum distance is 2, because we can move
to steps left to a black cell. two steps left to a black cell.
Then, we paint the white cell black: Then, we paint the white cell black:
\begin{center} \begin{center}
@ -316,7 +316,8 @@ $O(\sqrt n)$ \emph{batches}, each of which consists
of $O(\sqrt n)$ operations. of $O(\sqrt n)$ operations.
At the beginning of each batch, At the beginning of each batch,
we perform Algorithm 1. we perform Algorithm 1.
Then, we use Algorithm 2 to process the batches. Then, we use Algorithm 2 to process the operations
in the batch.
We clear the list of Algorithm 2 between We clear the list of Algorithm 2 between
the batches. the batches.
At each operation, At each operation,
@ -341,13 +342,12 @@ if a positive integer $n$ is represented as
a sum of positive integers, a sum of positive integers,
such a sum always contains at most such a sum always contains at most
$O(\sqrt n)$ \emph{distinct} numbers. $O(\sqrt n)$ \emph{distinct} numbers.
The reason for this is that to construct The reason for this is that to construct
a sum that contains a maximum number of distinct a sum that contains a maximum number of distinct
numbers, we should choose \emph{small} numbers. numbers, we should choose \emph{small} numbers.
If we choose the numbers $1,2,\ldots,k$, If we choose the numbers $1,2,\ldots,k$,
the resulting sum is the resulting sum is
\[\frac{k(k+1)}{2} \le n.\] \[\frac{k(k+1)}{2}.\]
Thus, the maximum amount of distinct numbers is $k = O(\sqrt n)$. Thus, the maximum amount of distinct numbers is $k = O(\sqrt n)$.
Next we will discuss two problems that can be solved Next we will discuss two problems that can be solved
efficiently using this observation. efficiently using this observation.
@ -371,19 +371,20 @@ $\{1,3,3\}$, the possible sums are as follows:
Using the standard knapsack approach (see Chapter 7.4), Using the standard knapsack approach (see Chapter 7.4),
the problem can be solved as follows: the problem can be solved as follows:
we define a function $f(k,s)$ whose value is 1 we define a function $\texttt{possible}(x,k)$ whose value is 1
if the sum $s$ can be formed using the first $k$ weights, if the sum $x$ can be formed using the first $k$ weights,
and 0 otherwise. and 0 otherwise.
All values of this function can be calculated Since the sum of the weights is $n$,
there are at most $n$ weights and
all values of the function can be calculated
in $O(n^2)$ time using dynamic programming. in $O(n^2)$ time using dynamic programming.
However, we can make the algorithm more efficient However, we can make the algorithm more efficient
by using the fact that the sum of the weights is $n$, by using the fact that there are at most $O(\sqrt n)$
which means that there are at most $O(\sqrt n)$ \emph{distinct} weights.
distinct weights.
Thus, we can process the weights in groups Thus, we can process the weights in groups
such that all weights in each group are equal. that consists of similar weights.
It turns out that we can process each group We can process each group
in $O(n)$ time, which yields an $O(n \sqrt n)$ time algorithm. in $O(n)$ time, which yields an $O(n \sqrt n)$ time algorithm.
The idea is to use an array that records the sums of weights The idea is to use an array that records the sums of weights
@ -396,12 +397,13 @@ can be formed using this group and the previous groups.
\subsubsection{String construction} \subsubsection{String construction}
Given a string \texttt{s} and a dictionary $D$ of strings, Given a string \texttt{s} of length $n$
and a set of strings $D$ whose total length is $m$,
consider the problem of counting the number of ways consider the problem of counting the number of ways
\texttt{s} can be formed as a concatenation of strings in $D$. \texttt{s} can be formed as a concatenation of strings in $D$.
For example, For example,
if \texttt{s} is \texttt{ABAB} and $D$ is if $\texttt{s}=\texttt{ABAB}$ and
$\{\texttt{A},\texttt{B},\texttt{AB}\}$, $D=\{\texttt{A},\texttt{B},\texttt{AB}\}$,
there are 4 ways: there are 4 ways:
\begin{itemize}[noitemsep] \begin{itemize}[noitemsep]
@ -411,12 +413,10 @@ there are 4 ways:
\item $\texttt{AB}+\texttt{AB}$ \item $\texttt{AB}+\texttt{AB}$
\end{itemize} \end{itemize}
Assume that the length of \texttt{s} is $n$
and the total length of the strings in $D$ is $m$.
We can solve the problem using dynamic programming: We can solve the problem using dynamic programming:
Let $f(k)$ denote the number of ways to construct the prefix Let $\texttt{count}(k)$ denote the number of ways to construct the prefix
$\texttt{s}[0 \ldots k]$ using the strings in $D$. $\texttt{s}[0 \ldots k]$ using the strings in $D$.
Now $f(n-1)$ gives the answer to the problem, Now $\texttt{count}(n-1)$ gives the answer to the problem,
and we can solve the problem in $O(n^2)$ time and we can solve the problem in $O(n^2)$ time
using a trie structure. using a trie structure.
@ -425,12 +425,12 @@ by using string hashing and the fact that there
are at most $O(\sqrt m)$ distinct string lengths in $D$. are at most $O(\sqrt m)$ distinct string lengths in $D$.
First, we construct a set $H$ that contains all First, we construct a set $H$ that contains all
hash values of the strings in $D$. hash values of the strings in $D$.
Then, when calculating a value of $f(k)$, Then, when calculating a value of $\texttt{count}(k)$,
we consider each integer $p$ we go through all values of $p$
such that there is a string of length $p$ in $D$, such that there is a string of length $p$ in $D$,
calculate the hash value of $\texttt{s}[k-p+1 \ldots k]$ calculate the hash value of $\texttt{s}[k-p+1 \ldots k]$
and check if it belongs to $H$. and check if it belongs to $H$.
Since there are at most $O(\sqrt m)$ distinct word lengths, Since there are at most $O(\sqrt m)$ distinct string lengths,
this results in an algorithm whose running time is $O(n \sqrt m)$. this results in an algorithm whose running time is $O(n \sqrt m)$.
\section{Mo's algorithm} \section{Mo's algorithm}