UVA 116 Unidirectional TSP（动态规划）

116 - Unidirectional TSP

Background

Problems that require minimum paths through some domain appear in many different areas of computer science. For example, one of the constraints in VLSI routing problems is minimizing wire length. The Traveling Salesperson Problem (TSP) – finding whether all the cities in a salesperson’s route can be visited exactly once with a specified limit on travel time – is one of the canonical examples of an NP-complete problem; solutions appear to require an inordinate amount of time to generate, but are simple to check.

This problem deals with finding a minimal path through a grid of points while traveling only from left to right.

The Problem

Given an m×n matrix of integers, you are to write a program that computes a path of minimal weight. A path starts anywhere in column 1 (the first column) and consists of a sequence of steps terminating in column n (the last column). A step consists of traveling from column i to column i+1 in an adjacent (horizontal or diagonal) row. The first and last rows (rows 1 and m) of a matrix are considered adjacent, i.e., the matrix “wraps” so that it represents a horizontal cylinder. Legal steps are illustrated below.

The weight of a path is the sum of the integers in each of the n cells of the matrix that are visited.

For example, two slightly different 5×6 matrices are shown below (the only difference is the numbers in the bottom row).

The minimal path is illustrated for each matrix. Note that the path for the matrix on the right takes advantage of the adjacency property of the first and last rows.

The Input

The input consists of a sequence of matrix specifications. Each matrix specification consists of the row and column dimensions in that order on a line followed by m⋅n integers where m is the row dimension and n is the column dimension. The integers appear in the input in row major order, i.e., the first n integers constitute the first row of the matrix, the second n integers constitute the second row and so on. The integers on a line will be separated from other integers by one or more spaces. Note: integers are not restricted to being positive. There will be one or more matrix specifications in an input file. Input is terminated by end-of-file.

For each specification the number of rows will be between 1 and 10 inclusive; the number of columns will be between 1 and 100 inclusive. No path’s weight will exceed integer values representable using 30 bits.

The Output

Two lines should be output for each matrix specification in the input file, the first line represents a minimal-weight path, and the second line is the cost of a minimal path. The path consists of a sequence of n integers (separated by one or more spaces) representing the rows that constitute the minimal path. If there is more than one path of minimal weight the path that is lexicographically smallest should be output.

Sample Input

5 6 
3 4 1 2 8 6 
6 1 8 2 7 4 
5 9 3 9 9 5 
8 4 1 3 2 6 
3 7 2 8 6 4 
5 6 
3 4 1 2 8 6 
6 1 8 2 7 4 
5 9 3 9 9 5 
8 4 1 3 2 6 
3 7 2 1 2 3 
2 2 
9 10 9 10

Sample Output

1 2 3 4 4 5 
16 
1 2 1 5 4 5 
11 
1 1 
19

 【题解】 参考博客：http://http://blog.csdn.net/q547550831/article/details/51899460

 题意就是有一个m*n的方格，起点在第一列任何位置，目标位置是最后一列任何位置，问从起点到终点的最小值（即走过的每个点的数值和）。

 因为有右上，右和右下三个方向，所以要把这三个方向标记为一个状态，每次访问不同方向，由于行是循环的，所以要处理第一行和最后一行，其他的就常规做法。

 【AC代码】

#include<iostream>#include<cstdio>#include<algorithm>#include<cstring>using namespace std;const int N=105;int matrix[N][N];int m,n;int dp[N][N];const int INF=1e9+7;int path[N][N];//记录路径void solve(){    int ans=INF;//初始化为无穷远    int s=0;    for(int j=n-1;j>=0;--j)    {        for(int i=0;i<m;++i)        {            if(j == n-1)//最后一行特判            {                dp[i][j]=matrix[i][j];            }            else            {                int dx[3]={i-1,i,i+1}//处理循环行问题                if(dx[0]<0)                    dx[0]=m-1;                if(dx[2]>=m)                    dx[2]=0;                sort(dx,dx+3);                dp[i][j]=INF;                for(int k=0;k<3;++k)                {                   int sum= dp[dx[k]][j+1]+matrix[i][j];//状压                   if(sum<dp[i][j]){                      dp[i][j]=sum;                       path[i][j]=dx[k];//保存路径                    }                }            }            if(j==0 && dp[i][j]<ans)//到达第一列特判            {                ans = dp[i][j];                s=i;            }         }    }    printf("%d",s+1);    for(int i=path[s][0],j=1;j<n;i=path[i][j],++j)        printf(" %d",i+1);    printf("\n");    printf("%d\n",ans);}int main(){    while(~scanf("%d%d",&m,&n))    {        for(int i=0;i<m;++i)            for(int j=0;j<n;++j)            scanf("%d",&matrix[i][j]);        solve();    }    return 0;}