UVa 10453 - Make Palindrome

#DP #字符串 #回文串

Published on 2016-02-10

描述

给定一个长度为 $n(n\le 1000)$ 的字符串，你需要在任意位置添加尽量少的字符，使新串是回文串。输出最少添加的字符个数以及新串。

样例输入

abcd
aaaa
abc
aab
abababaabababa
pqrsabcdpqrs

样例输出

3 abcdcba
0 aaaa
2 abcba
1 baab
0 abababaabababa
9 pqrsabcdpqrqpdcbasrqp

分析

乍一看我又是断了思路，看着这个题目 $n\le 1000$ ，应该是 $O(n^2)$ 的算法，于是强上区间 DP， $dp[i][j]$ 为将区间 $[i, j]$ 构成回文串需要添加多少字符。怎么决策呢？我又想，大回文串要么自己本身就是，要么是由小回文串添加得来的，于是写出方程：

决策一不用说，预处理出 $is[i][j]$ ，表示区间 $[i, j]$ 是否是回文串就行了，方法是枚举回文串中心，然后向左向右扩展（注意奇偶回文）。复杂度 $O(n^2)$ 。
决策二是在区间 $[i + 1, j - 1]$ 是回文串的基础上添加，如果 $str[i] == str[j]$ ，那么不需要添加，否则左右都要添加一个字符以便形成回文串。例如 $str[i] + str[j] + \text{str}_{i + 1, j - 1} + str[j] + str[i]$ 。
决策三是在区间 $[i, j - 1]$ 是回文串的基础上添加，只需在左边添加一个 $str[j]$ 即可形成回文串。
决策四是在区间 $[i + 1, j]$ 是回文串的基础上添加，只需在右边添加一个 $str[i]$ 即可形成回文串。
答案是 $dp[0][n - 1]$ ，于是这个 DP 就搞出来了。
打印解也好办，这里用时间换一下空间，递归的时候模拟一下决策，然后按照决策的内容添加字符并递归下去即可。
本题有多解，所以附加输出仅作参考，我给的是我的程序的输出。

代码

//  Created by Sengxian on 2/9/16.
//  Copyright (c) 2015年 Sengxian. All rights reserved.
//  UVa 10453 造回文串（区间 DP）
#include <algorithm>
#include <iostream>
#include <cassert>
#include <cctype>
#include <climits>
#include <cstring>
#include <cstdio>
#include <cmath>
#include <vector>
#include <stack>
#include <queue>
#define br putchar('\n');
using namespace std;
const int maxn = 1000 + 3, INF = 0x3f3f3f3f;
bool is[maxn][maxn];
char str[maxn];
int dp[maxn][maxn], n;

void process() {
    memset(is, 0, sizeof(is));
    for(int i = 0; i < n; ++i) {
        int p = i - 1, q = i + 1;
        is[i][i] = true;
        while(p >= 0 && q < n && str[p] == str[q]) is[p--][q++] = true;
        p = i, q = i + 1;
        while(p >= 0 && q < n && str[p] == str[q]) is[p--][q++] = true;
    }
}


int solve() {
    for(int i = 0; i < n; ++i) dp[i][i] = 0;
    for(int w = 2; w <= n; ++w)
        for(int i = 0; i + w <= n; ++i) {
            int j = i + w - 1;
            dp[i][j] = INF;
            if(is[i][j]) dp[i][j] = 0;
            else {
                if(w > 2) dp[i][j] = min(dp[i][j], dp[i + 1][j - 1] + (str[i] == str[j] ? 0 : 2));
                dp[i][j] = min(dp[i][j], dp[i + 1][j] + 1);
                dp[i][j] = min(dp[i][j], dp[i][j - 1] + 1);
            }
        }
    return dp[0][n - 1];
}

void print(int i, int j) {
    if(is[i][j]) {
        for(int k = i; k <= j; ++k) putchar(str[k]);
        return;
    }
    if((j - i + 1) > 2 && dp[i][j] == dp[i + 1][j - 1] + (str[i] == str[j] ? 0 : 2)) {
        putchar(str[i]);
        if(str[i] != str[j]) putchar(str[j]);
        print(i + 1, j - 1);
        putchar(str[j]);
        if(str[i] != str[j]) putchar(str[i]);
    }else if(dp[i][j] == dp[i + 1][j] + 1) {
        putchar(str[i]);
        print(i + 1, j);
        putchar(str[i]);
    }else if(dp[i][j] == dp[i][j - 1] + 1) {
        putchar(str[j]);
        print(i, j - 1);
        putchar(str[j]);
    }else assert(false);
}

int main() {
    while(~scanf("%s", str)) {
        n = strlen(str);
        process();
        printf("%d ", solve());
        print(0, n - 1); br
    }
    return 0;
}

附加输入

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mvtwlls
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附加输出

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Dynamic Programming Exercises Beginner 归档题解(2) UVa 1351 - String Compression

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