Find ALL paths in a grid between 2 nodes

Question

I'm trying to find all the paths between 2 nodes in a grid, and the path has to go through all the node from start to end. Example (Start = S, End = E)

0 0 0
0 S 0
0 0 E

The answer for the above is 2 paths: (ignore the '.''s)

0-0-0
|.......|
0 S-0
|
0-0-E

0-0-0
|......|
0 S 0
|...|...|
0-0 E

I thought of using recursing, but gave that up due to the high overhead of each call...and decided to go with an iterative approach using a stack. (kind of like recursing but not...fixed memory overhead). The solution works great for grids of size (4x7), but tried it on an 8x8 grid...and it didn't finish in 4 hours...which sort of makes sense since the total number of possibilities is around 3**62 (approx), as each node that is not on the edge has 3 possible ways of traversal...and hence that solution fails.

I thought of splitting the 8x8 grid into 2, using a vertical and horizontal split...but this would lead to finding less than ideal paths.

Is there something that I'm missing???? Can something be done to make it go faster? I will post the solution that I have tomorrow (done in C).

score 0 · Answer 1 · answered Nov 12 '14 at 14:36

0

Well, there is a linear-time algorithm for finding a longest path between any two given vertices in a rectangular grid graph, which seems to be exactly what you're looking for.

answered Nov 12 '14 at 14:36

vmg

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score 0 · Answer 2 · answered Oct 25 '10 at 08:36

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Have a look at the shortest paths problem it should get you started on this. The Bellmann-Ford or Dijkstra's algorithm are worth trying. You might be able to improve on the efficiency of those if your grid has some properties you can exploit.

answered Oct 25 '10 at 08:36

Matti Lyra

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Matti...I've already found a solution...it works on small grids...based off of recursion... – GK12 Oct 25 '10 at 09:34
Recursions is only for shortest path....and neither Dijkstra or Bellman ford will help since they're for shortest path...the problem isn't that i can't find a path...the problem is that i need ALL the paths.... – GK12 Oct 25 '10 at 09:36

score 0 · Answer 3 · answered Oct 25 '10 at 08:37

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Nope, you do not miss anything and nothing can be done faster.

This is the Longest Path Problem which is NP-Hard.

answered Oct 25 '10 at 08:37

tafa

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That's not entirely accurate - having specified start and end nodes reduces the complexity immensely. – Sam Dufel Oct 25 '10 at 08:45
@Sam, fair enough. Haven't thought of that. – tafa Oct 25 '10 at 10:50

Find ALL paths in a grid between 2 nodes

3 Answers3