Minimal Weight Paths

This problem was of the 2nd programming competition of Turabo's University.
Here is the description of the problem. Thanks to Luis Ojeda. He helped me in how to save the optimal values and the sum.

Given an m×n matrix of integers, you are to write a program that computes a path of
minimal weight from left to right across the matrix. A path starts anywhere in column
1 and consists of a sequence of steps terminating in column n. Each 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.



The weight of a path is the sum of the integers in each of the n cells of the matrix that
are visited.
The minimum paths through two slightly different 5 × 6 matrices are shown below.
The matrix values differ only in the bottom row. The path for the matrix on the right
takes advantage of the adjacency between the first and last rows.

Input
The input consists of a sequence of matrix specifications. Each matrix consists of the
row and column dimensions on a line, denoted m and n, respectively. This is followed
by m · n integers, appearing 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.
11.6. Problems 261
Output
Two lines should be output for each matrix specification. The first line represents a
minimal-weight path, and the second line is the cost of this 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 lexicographically smallest path 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 2
19

Here is my solution in C#.