（转载）动态规划求最长公共子序列
通过这篇来加深对动态规划的理解。动态规划法
经常会遇到复杂问题不能简单地分解成几个子问题，而会分解出一系列的子问题。简单地采用把大问题分解成子问题，并综合子问题的解导出大问题的解的方法，问题求解耗时会按问题规模呈幂级数增加。
为了节约重复求相同子问题的时间，引入一个数组，不管它们是否对最终解有用，把所有子问题的解存于该数组中，这就是动态规划法所采用的基本方法。
【问题】 求两字符序列的最长公共字符子序列
问题描述：字符序列的子序列是指从给定字符序列中随意地（不一定连续）去掉若干个字符（可能一个也不去掉）后所形成的字符序列。令给定的字符序列X=“x0，x1，…，xm-1”，序列Y=“y0，y1，…，yk-1”是X的子序列，存在X的一个严格递增下标序列<i0，i1，…，ik-1>，使得对所有的j=0，1，…，k-1，有xij=yj。例如，X=“ABCBDAB”，Y=“BCDB”是X的一个子序列。
考虑最长公共子序列问题如何分解成子问题，设A=“a0，a1，…，am-1”，B=“b0，b1，…，bn-1”，并Z=“z0，z1，…，zk-1”为它们的最长公共子序列。不难证明有以下性质：
（1） 如果am-1=bn-1，则zk-1=am-1=bn-1，且“z0，z1，…，zk-2”是“a0，a1，…，am-2”和“b0，b1，…，bn-2”的一个最长公共子序列；
（2） 如果am-1!=bn-1，则若zk-1!=am-1，蕴涵“z0，z1，…，zk-1”是“a0，a1，…，am-2”和“b0，b1，…，bn-1”的一个最长公共子序列；
（3） 如果am-1!=bn-1，则若zk-1!=bn-1，蕴涵“z0，z1，…，zk-1”是“a0，a1，…，am-1”和“b0，b1，…，bn-2”的一个最长公共子序列。
这样，在找A和B的公共子序列时，如有am-1=bn-1，则进一步解决一个子问题，找“a0，a1，…，am-2”和“b0，b1，…，bm-2”的一个最长公共子序列；如果am-1!=bn-1，则要解决两个子问题，找出“a0，a1，…，am-2”和“b0，b1，…，bn-1”的一个最长公共子序列和找出“a0，a1，…，am-1”和“b0，b1，…，bn-2”的一个最长公共子序列，再取两者中较长者作为A和B的最长公共子序列。
求解：
引进一个二维数组c[][]，用c记录X与Y 的LCS 的长度，b记录c是通过哪一个子问题的值求得的，以决定搜索的方向。
我们是自底向上进行递推计算，那么在计算c之前，c，c与c均已计算出来。此时我们根据X = Y还是X != Y，就可以计算出c。
问题的递归式写成：
http://p.blog.csdn.net/images/p_blog_csdn_net/hhygcy/EntryImages/20090302/lcs_1.PNG
回溯输出最长公共子序列过程：
http://p.blog.csdn.net/images/p_blog_csdn_net/hhygcy/EntryImages/20090302/lcs_2.PNG
算法分析：
由于每次调用至少向上或向左（或向上向左同时）移动一步，故最多调用(m + n)次就会遇到i = 0或j = 0的情况，此时开始返回。返回时与递归调用时方向相反，步数相同，故算法时间复杂度为Θ(m + n)。
代码：
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/None.gif#include <stdio.h>
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/None.gif#include <string.h>
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/None.gif#define MAXLEN 100
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/None.gif
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/None.gifvoid LCSLength(char *x, char *y, int m, int n, int c[], int b[])
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedBlockStart.gif{
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif int i, j;
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif for(i = 0; i <= m; i++)
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif c[0] = 0;
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif for(j = 1; j <= n; j++)
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif c[0] = 0;
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif for(i = 1; i<= m; i++)
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockStart.gif {
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif for(j = 1; j <= n; j++)
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockStart.gif {
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif if(x-1] == y-1])
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockStart.gif {
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif c = c-1]-1] + 1;
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif b = 0;
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockEnd.gif }
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif else if(c-1] >= c-1])
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockStart.gif {
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif c = c-1];
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif b = 1;
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockEnd.gif }
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif else
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockStart.gif {
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif c = c-1];
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif b = -1;
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockEnd.gif }
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockEnd.gif }
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockEnd.gif }
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedBlockEnd.gif}
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/None.gif
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/None.gifvoid PrintLCS(int b[], char *x, int i, int j)
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedBlockStart.gif{
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif if(i == 0 || j == 0)
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif return;
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif if(b == 0)
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockStart.gif {
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif PrintLCS(b, x, i-1, j-1);
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif printf("%c ", x-1]);
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockEnd.gif }
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif else if(b == 1)
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif PrintLCS(b, x, i-1, j);
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif else
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif PrintLCS(b, x, i, j-1);
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedBlockEnd.gif}
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/None.gif
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/None.gifint main(int argc, char **argv)
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedBlockStart.gif{
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockStart.gif char x = {"ABCBDAB"};
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedSubBlockStart.gif char y = {"BDCABA"};
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif int b;
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif int c;
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif int m, n;
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif m = strlen(x);
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif n = strlen(y);
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif LCSLength(x, y, m, n, c, b);
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif PrintLCS(b, x, m, n);
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/InBlock.gif return 0;
http://images.csdn.net/syntaxhighlighting/OutliningIndicators/ExpandedBlockEnd.gif}
转载自：http://blog.csdn.net/yysdsyl/archive/2009/05/30/4226630.aspx
页:
[1]