[poj]3187 Backward Digit Sums [dfs]

最新推荐文章于 2021-05-07 16:03:47 发布

原创最新推荐文章于 2021-05-07 16:03:47 发布 · 301 阅读

0 ·

CC 4.0 BY-SA版权

文章标签：

#poj #dfs

dfs 专栏收录该内容

16 篇文章

订阅专栏

本文介绍了一种算法，该算法能够通过给定的最终数值和数列长度，复原出符合特定加法规则的初始数列。采用深度优先搜索（DFS）策略并结合动态规划预先计算数列系数，确保了找到字典序最小的解决方案。

FJ and his cows enjoy playing a mental game. They write down the numbers from 1 to N (1 <= N <= 10) in a certain order and then sum adjacent numbers to produce a new list with one fewer number. They repeat this until only a single number is left. For example, one instance of the game (when N=4) might go like this:

3 1 2 4
4 3 6
7 9
16
Behind FJ’s back, the cows have started playing a more difficult game, in which they try to determine the starting sequence from only the final total and the number N. Unfortunately, the game is a bit above FJ’s mental arithmetic capabilities.

Write a program to help FJ play the game and keep up with the cows.
Input
Line 1: Two space-separated integers: N and the final sum.
Output
Line 1: An ordering of the integers 1..N that leads to the given sum. If there are multiple solutions, choose the one that is lexicographically least, i.e., that puts smaller numbers first.
Sample Input
4 16
Sample Output
3 1 2 4
Hint
Explanation of the sample:

There are other possible sequences, such as 3 2 1 4, but 3 1 2 4 is the lexicographically smallest.

link

题解

分析一下4个数的情况

times
1	A B C D
2	A+B B+C C+D
3	A+2B+C B+2C+D
4	A+3(B+C)+D

那么n=5呢，归纳得：A+3(B+C)+D+B+3(C+D)+E=A+4B+6C+4D+E
具体什么规律，不好言说，但是我们可以预处理得到n时的系数

    dp[1][1]=1;
    for(int i=2;i<=10;i++){
        for(int j=1;j<=i;j++)
            dp[i][j]=dp[i-1][j-1]+dp[i-1][j];
    }

然后dfs即可，复杂度 $o(n!)$

#include<stdio.h>
#include<stack>
using namespace std;
int dp[11][11];
int n,sum;

void init()
{
    dp[1][1]=1;
    for(int i=2;i<=10;i++){
        for(int j=1;j<=i;j++)
            dp[i][j]=dp[i-1][j-1]+dp[i-1][j];
    }
}

stack<int> sta;

int vis[11];
bool dfs(int cnt,int now){
    if(cnt==n&&now==sum) return true;
    if(now>=sum) return false;
    for(int i=1;i<=n;i++){
        if(!vis[i]){
            vis[i]=true;
            if(dfs(cnt+1,now+i*dp[n][cnt+1])){
                sta.push(i);
                return true;
            }
            vis[i]=false;
        }
    }
    return false;
}

int main()
{
    init();
    scanf("%d%d",&n,&sum);
    dfs(0,0);
    while(!sta.empty()){
        printf("%d",sta.top());
        sta.pop();
        if(!sta.empty()) putchar(' ');
    }
    putchar('\n');
    return 0;
}