Q矩阵单元之间的最短路径

Question

I saw this problem in a local competition and I am trying to solve it,

I am given a matrix with ' r ' rows and ' c ' columns. Then I am given ' Q ' cells,

My task is to find the shortest path: Starting at the top left corner ' (0,0) ' , ending at the bottom right corner ' (row-1,column-1) ' and travessing all ' Q ' cells . I can move either right, left, up and down.

Brute Force is out of option (has I only have 1 second of CPU time and I can be given at most 50 columns and rows).

Here is a Sample Input:

6 3   
3        
0 1    
4 2  
5 2 

Here is the Sample Output:

7

I am thinking about some sort of BFS, but I have no idea how to apply it to this specific problem.

Any help or advise would be much appreciated !

Answer 1

Calculate the shortest path between every Q and starting and finishing point. Create a graph with these distances.

Since you can only move orthogonally, the shortest distance between two points is the sum of the absolute differences between their co-ordinates.

Select an algorithm to visit all nodes in your distance graph. Possible starting point Bottom line is that this has become the "travelling salesman problem" and you will need to decide which of the many available algorithms suits you best.

If I am understanding your input, the distance graph looks like this

Locations

B  0 0
Q1 0 1    
Q2 4 2  
Q3 5 2 
E  3 3

Distances

B  0  1  6  7  6
Q1 0  0  5  6  5
Q2 0  0  0  1  2
Q3 0  0  0  0  3
E  0  0  0  0  0