Fixes and clean code
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Antti H S Laaksonen
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chapter07.tex
225
chapter07.tex
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@ -30,8 +30,9 @@ counting the solutions.
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Understanding dynamic programming is a milestone
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in every competitive programmer's career.
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While the basic idea of the technique is simple,
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the challenge is how to apply it to different problems.
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While the basic idea is simple,
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the challenge is how to apply
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dynamic programming to different problems.
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This chapter introduces a set of classic problems
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that are a good starting point.
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@ -73,11 +74,11 @@ subproblems.
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In the coin problem, a natural recursive
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problem is as follows:
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what is the smallest number of coins
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required for constructing a sum $x$?
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required to form a sum $x$?
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Let $f(x)$ be a function that gives the answer
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to the problem, i.e., $f(x)$ is the smallest
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number of coins required for constructing a sum $x$.
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number of coins required to form a sum $x$.
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The values of the function depend on the
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values of the coins.
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For example, if the coin values are $\{1,3,4\}$,
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@ -101,27 +102,27 @@ f(10) & = & 3 \\
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First, $f(0)=0$ because no coins are needed
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for the sum $0$.
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Moreover, $f(3)=1$ because the sum $3$
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Then, for example, $f(3)=1$ because the sum $3$
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can be formed using coin 3,
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and $f(5)=2$ because the sum 5 can
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be formed using coins 1 and 4.
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The essential property in the function is
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that each value of $f(x)$ can be calculated
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recursively from smaller values of the function.
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The essential property of $f$ is
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that its values can be
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recursively calculated from its smaller values.
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For example, if the coin set is $\{1,3,4\}$,
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there are three ways to select the first coin
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in a solution: we can choose coin 1, 3 or 4.
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If coin 1 is chosen, the remaining task is to
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form the sum $x-1$.
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Similarly, if coin 3 or 4 is chosen,
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we should form the sum $x-3$ or $x-4$.
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there are three ways how we can choose the
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first coin in a solution.
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If we choose coin 1, the remaining task
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is to form the sum $x-1$.
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Similarly, after choosing coins 3 and 4,
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the remaining sums are $x-3$ and $x-4$.
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Thus, the recursive formula is
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\[f(x) = \min(f(x-1),f(x-3),f(x-4))+1\]
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where the function $\min$ gives the smallest
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of its parameters.
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In the general case, for the coin set
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In the general case, for a coin set
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$\{c_1,c_2,\ldots,c_k\}$,
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the recursive formula is
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\[f(x) = \min(f(x-c_1),f(x-c_2),\ldots,f(x-c_k))+1.\]
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@ -130,9 +131,9 @@ The base case for the function is
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because no coins are needed for constructing
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the sum 0.
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In addition, it is convenient to define
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\[f(x)=\infty\hspace{8px}\textrm{if $x<0$}.\]
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This means that an infinite number of coins
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is needed for forming a negative sum of money.
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\[f(x)=\infty\hspace{8px}\textrm{if $x<0$},\]
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which means that to get a negative sum of money,
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an infinite number of coins is needed.
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This prevents the function from constructing
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a solution where the initial sum of money is negative.
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@ -144,16 +145,16 @@ we can directly implement a solution in C++:
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int f(int x) {
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if (x < 0) return INF;
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if (x == 0) return 0;
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int u = INF;
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for (int i = 1; i <= k; i++) {
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u = min(u, f(x-c[i])+1);
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int best = INF;
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for (auto c : coins) {
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best = min(best, f(x-c)+1);
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}
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return u;
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return best;
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}
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\end{lstlisting}
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The code assumes that the available coins are
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stored in an array $\texttt{c}$,
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stored in an array $\texttt{coins}$,
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and the constant \texttt{INF} denotes infinity.
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This function works but it is not efficient yet,
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because it goes through a large number
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@ -161,7 +162,7 @@ of ways to construct the sum.
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However, the function can be made efficient by
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using memoization.
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\subsubsection{Memoization}
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\subsubsection{Using memoization}
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\index{memoization}
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@ -175,16 +176,18 @@ For each parameter, the value of the function
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is calculated recursively only once, and after this,
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the value can be directly retrieved from the array.
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In this problem, we can use an array
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In this problem, we use arrays
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\begin{lstlisting}
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int d[N];
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bool ready[N];
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int value[N];
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\end{lstlisting}
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where $\texttt{d}[x]$ will contain
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the value of $f(x)$.
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The constant $N$ has to be chosen so
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that all required values of the function fit
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in the array.
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where $\texttt{ready}[x]$ indicates
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whether the value of $f(x)$ has been calculated,
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and if it is, $\texttt{value}[x]$
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contains this value.
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The constant $N$ has been chosen so
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that all required values fit in the arrays.
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After this, the function can be efficiently
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implemented as follows:
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@ -193,109 +196,103 @@ implemented as follows:
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int f(int x) {
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if (x < 0) return INF;
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if (x == 0) return 0;
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if (d[x]) return d[x];
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int u = INF;
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for (int i = 1; i <= k; i++) {
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u = min(u, f(x-c[i])+1);
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if (ready[x]) return value[x];
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int best = INF;
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for (auto c : coins) {
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best = min(best, f(x-c)+1);
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}
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d[x] = u;
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return d[x];
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ready[x] = true;
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value[x] = best;
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return best;
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}
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\end{lstlisting}
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The function handles the base cases
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$x<0$ and $x=0$ as previously.
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Then the function checks if
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Then the function checks from
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$\texttt{ready}[x]$ if
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$f(x)$ has already been stored
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in $\texttt{d}[x]$.
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If the value of $f(x)$ is found in the array,
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the function directly returns it.
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in $\texttt{value}[x]$,
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and if it is, the function directly returns it.
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Otherwise the function calculates the value
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recursively and stores it in $\texttt{d}[x]$.
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recursively and stores it in $\texttt{value}[x]$.
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Using memoization the function works
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efficiently, because the answer for each parameter $x$
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is calculated recursively only once.
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After a value of $f(x)$ has been stored in the array,
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After a value of $f(x)$ has been stored in $\texttt{value}[x]$,
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it can be efficiently retrieved whenever the
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function will be called again with the parameter $x$.
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The time complexity of the resulting algorithm
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is $O(xk)$ where the sum is $x$ and the number of
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coins is $k$.
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The resulting algorithm works in $O(xk)$ time,
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where the sum is $x$ and the number of coins is $k$.
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In practice, the algorithm can be used if
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$x$ is so small that it is possible to allocate
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an array for all possible function parameters.
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Note that the array can also be constructed using
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a loop that calculates all the values
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instead of a recursive function:
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Note that we can also construct the array \texttt{value}
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\emph{iteratively} using
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a loop that simply calculates all the values
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of $f$ for parameters $0 \ldots x$:
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\begin{lstlisting}
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d[0] = 0;
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value[0] = 0;
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for (int i = 1; i <= x; i++) {
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int u = INF;
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for (int j = 1; j <= k; j++) {
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if (i-c[j] < 0) continue;
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u = min(u, d[i-c[j]]+1);
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}
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d[i] = u;
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}
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\end{lstlisting}
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This implementation is shorter and somewhat
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more efficient than recursion,
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and experienced competitive programmers
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often prefer dynamic programming solutions
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that are implemented using loops.
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Still, the underlying idea is the same as
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in the recursive function.
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\subsubsection{Constructing a solution}
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Sometimes we are asked both to find the value
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of an optimal solution and also to give
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an example how such a solution can be constructed.
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In the coin problem, this means that the algorithm
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should show how to select the coins that produce
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the sum $x$ using as few coins as possible.
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We can construct the solution by adding another
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array to the code. The new array indicates for
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each sum of money the first coin that should be
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chosen in an optimal solution.
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In the following code, the array \texttt{e}
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is used for this:
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\begin{lstlisting}
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d[0] = 0;
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for (int i = 1; i <= x; i++) {
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d[i] = INF;
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for (int j = 1; j <= k; j++) {
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if (i-c[j] < 0) continue;
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int u = d[i-c[j]]+1;
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if (u < d[i]) {
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d[i] = u;
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e[i] = c[j];
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value[i] = INF;
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for (auto c : coins) {
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if (i-c >= 0) {
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value[i] = min(value[i], value[i-c]+1);
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}
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}
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}
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\end{lstlisting}
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After this, we can print the coins needed
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for the sum $x$ as follows:
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Since the iterative solution is shorter and a bit
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more efficient than recursion,
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competitive programmers often prefer this solution.
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\subsubsection{Constructing an example solution}
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Sometimes we are asked both to find the value
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of an optimal solution and to give
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an example how such a solution can be constructed.
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In the coin problem, for example,
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we might be asked both the minimum number of coins
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and an example how to choose the coins.
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We can do this by using an array \texttt{first}
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that indicates for
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each sum of money the first coin
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in an optimal solution:
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\begin{lstlisting}
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value[0] = 0;
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for (int i = 1; i <= x; i++) {
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value[i] = INF;
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for (auto c : coins) {
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if (i-c < 0) continue;
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int v = value[i-c]+1;
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if (v < value[i]) {
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value[i] = v;
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first[i] = c;
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}
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}
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}
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\end{lstlisting}
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After this, we can print the coins that
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form the sum $x$ as follows:
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\begin{lstlisting}
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while (x > 0) {
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cout << e[x] << "\n";
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x -= e[x];
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cout << first[x] << "\n";
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x -= first[x];
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}
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\end{lstlisting}
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\subsubsection{Counting the number of solutions}
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Let us now consider a variant of the problem
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Let us now consider a variant of the coin problem
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that is otherwise like the original problem,
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but we should count the total number of solutions instead
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but we are asked to count the total number of solutions instead
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of finding the optimal solution.
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For example, if the coins are $\{1,3,4\}$ and
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the target sum is $5$,
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@ -312,21 +309,19 @@ there are a total of 6 solutions:
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\end{itemize}
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\end{multicols}
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The number of the solutions can be calculated
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The number of solutions can be calculated
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using the same idea as finding the optimal solution.
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The difference is that when finding the optimal solution,
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we maximize or minimize something in the recursion,
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but now we will calculate sums of numbers of solutions.
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but now we calculate sums of numbers of solutions.
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To solve the problem, we can define a function $f(x)$
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To solve the problem, we define a function $f(x)$
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that gives the number of ways to construct
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a sum $x$ using the coins.
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For example, $f(5)=6$ when the coins are $\{1,3,4\}$.
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The value of $f(x)$ can be calculated recursively
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using the formula
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\[ f(x) = f(x-c_1)+f(x-c_2)+\cdots+f(x-c_{k}),\]
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because to form the sum $x$, we have to first
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choose some coin $c_i$ and then form the sum $x-c_i$.
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\[ f(x) = f(x-c_1)+f(x-c_2)+\cdots+f(x-c_{k}).\]
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The base cases are $f(0)=1$, because there is exactly
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one way to form the sum 0 using an empty set of coins,
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and $f(x)=0$, when $x<0$, because it is not possible
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@ -352,14 +347,15 @@ f(9) & = & 40 \\
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The following code calculates the value of $f(x)$
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using dynamic programming by filling the array
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\texttt{d} for parameters $0 \ldots x$:
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\texttt{count} for parameters $0 \ldots x$:
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\begin{lstlisting}
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d[0] = 1;
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count[0] = 1;
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for (int i = 1; i <= x; i++) {
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for (int j = 1; j <= k; j++) {
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if (i-c[j] < 0) continue;
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d[i] += d[i-c[j]];
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for (auto c : coins) {
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if (i-c >= 0) {
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count[i] += count[i-c];
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}
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}
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}
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\end{lstlisting}
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@ -372,20 +368,19 @@ This can be done by changing the code so that
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all calculations are done modulo $m$.
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In the above code, it suffices to add the line
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\begin{lstlisting}
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d[i] %= m;
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count[i] %= m;
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\end{lstlisting}
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after the line
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\begin{lstlisting}
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d[i] += d[i-c[j]];
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count[i] += count[i-c];
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\end{lstlisting}
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Now we have discussed all basic
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techniques related to
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dynamic programming.
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ideas of dynamic programming.
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Since dynamic programming can be used
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in many different situations,
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we will now go through a set of problems
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that show further examples about
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that show further examples about the
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possibilities of dynamic programming.
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\section{Longest increasing subsequence}
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