Algorithm or approach to path problem, shortest path with n points for n <= 12

Question

i have n points on a 2d plane, with n <= 12, and i need the distance of the shortest path available including all points, starting on any of them, but not making a closed circuit

i've been trying floyd-marshal, travel salesman problem and other algorithms without success.

the problem is considered EASY for my teacher so i don't think it would require arora approximations or so, but i dont know whats the best approach to solve this, but maybe some dynamic algorithm and something like

for i = 0 to n
    for j = 0 to n
    if path_distance(i,j) < mininum
        set minimum

any help?

Are you allowed to revisit points more than once? Or do you have to visit every point once and exactly once? — templatetypedef, Mar 03 '11 at 01:18
This is an O(n!) problem, not an O(n^2) problem like you tried. Half a billion possible paths. Do it on a piece of paper first. — Hans Passant, Mar 03 '11 at 02:00
i know that, path_distance(i,j) would compute the shortest path between i and j using all points — Daniel, Mar 03 '11 at 02:15
It's not the Travelling Salesman Problem - because that has the additional restriction that you have to finish back where you started. — GrahamS, Mar 09 '11 at 09:55

score 1 · Answer 1 · answered Mar 03 '11 at 01:34

If you know that you only have twelve points and want to find the shortest path that visits every node exactly once, then you can always just brute-force the solution by trying all possible permutations of the nodes and computing the length of the path along that permutation. This doesn't scale well at all, but if you have a fixed upper bound on the number of nodes then it should be reasonable.

score 1 · Answer 2 · answered Mar 09 '11 at 08:28

1

If and only if n <= 12, I 'd recommend the Branch and Bound algorithm, which is an improved version of the brute force algorithm.

answered Mar 09 '11 at 08:28

Geoffrey De Smet

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score 1 · Answer 3 · answered Mar 09 '11 at 10:05

I'll only provide a hint since this is homework:

Recursion is very useful when breaking down these kinds of problems.

You could write a function that takes a list of the route so far, the best/minimum distance so far, and a list of unvisited nodes. For-each unvisited node it would try adding it to the route so far and if that route was still shorter than the best/minimum distance so far then it would call itself with the new route and list of unvisited nodes.

Algorithm or approach to path problem, shortest path with n points for n <= 12

3 Answers3