Unidirectional TSP
Problem Description
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.
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.
This problem deals with finding a minimal path through a grid of points while traveling only from left to right.
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.
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 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
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
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
题目大意：给定一个n*m的矩阵，从第一列的任何一个位置出发，到最后一列的任何一个位置终止，每一次只能往下一列走，走直线或者对角线，求出最小权值，以及打印字典序最小路径。
分析：状态转移很简单，就是取右上，右，右下的最小值。但因为要打印路径，从前往后打印，所以最左边应该是最终结果，就要从右往左循环，这样状态转移也就变成了取左上，左，左下的最小值。具体见代码。
代码如下：
1 # include<stdio.h> 2 # include<string.h> 3 # include<iostream> 4 using namespace std; 5 # define inf 0xffffff 6 int map[12][102]; 7 int p[12][102]; 8 int s[102]; 9 int main() 10 { 11 int m,n,i,j; 12 while(scanf("%d%d",&m,&n)!=EOF) 13 { 14 memset(p,-1,sizeof(p)); 15 for(i=1; i<=m; i++) 16 for(j=1; j<=n; j++) 17 scanf("%d",&map[i][j]); 18 for(j=n-1; j>=1; j--) //从后往前计算，因为要打印路径 19 for(i=1; i<=m; i++) 20 { 21 int b=i,a,c,d,e=-1; //a，b，c分别为向左上，向左，向左下的坐标 22 if(i>=2) a=i-1; 23 else a=m; 24 if(i<m) c=i+1; 25 else c=1; 26 d = min(min(map[a][j+1],map[b][j+1]),map[c][j+1]); //取最小值 27 map[i][j] += d; 28 if(d==map[a][j+1]) e=a; 29 if(d==map[b][j+1]&&(e==-1||(e!=-1&&b<e))) e=b; //e的作用当两行的数字相同时去小的一个 30 if(d==map[c][j+1]&&(e==-1||(e!=-1&&c<e))) e=c; 31 p[i][j] = e; //记录路径 32 } 33 int flag = 0; 34 int sum =inf; 35 for(i=1; i<=m; i++) 36 if(map[i][1] < sum) 37 { 38 sum = map[i][1]; 39 flag=i; 40 } 41 printf("%d",flag); 42 flag = p[flag][1]; 43 for(i=2; i<=n && flag != -1; i++) 44 { 45 printf(" %d",flag); 46 flag = p[flag][i]; 47 } 48 printf(" %d ",sum); 49 } 50 return 0; 51 }