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I have created an undirected graph with 6 vertices, visually represented like this: https://i.stack.imgur.com/EtIbO.png

I would like to write a script that can find all paths from a starting point without revisiting the same node, with no given end point.

Every example of the BFS, DFS, A* algorithm I have looked at require an end destination node. However, on a larger graph it may be NP-hard to find all possible pathways from point A to point Z. For this reason, I want to find all paths to all destinations that are achievable within a set number of moves (on this graph for example -- 3 moves == 4 max vertices in path)

I coded the graph using PHP arrays with each key being a vertice and its array containing the adjacent points:

<?php

$graph[1] = array(2,6);
$graph[2] = array(1,4);
$graph[3] = array(4,5);
$graph[4] = array(2,3,6);
$graph[5] = array(6,3);
$graph[6] = array(1,5,4);

I don't know of an algorithm though, that performs a path search in this manner. My desired output would be something like this:

Path 1: 1,2
Path 2: 1,2,4
Path 3: 1,2,4,3
Path 4: 1,6
Path 5: 1,6,4,3
Path 6: 1,6,5
Path 7: 1,6,5,3

I have no problem writing the required code, but the necessary steps for the algorithm/function (assuming tree traversal recursion?) are difficult to understand.

Question: What approach/algorithm should be used to do this, and do you have an example (or at least pseudocode) that shows how it works given the graph input array?

server_kitten
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2 Answers2

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Single starting point and branching tells you to build a tree. This is just tree search with limited depth. At each level you need to prune nodes you've already visited. Pseudo code is in the wiki article

stark
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Well, in a dense graph (|E| ~ O(|V|^2)) the size of the output is exponential in the amount of maximum moves, and any program will take time at least linear in the size of the data it has to output.

The easiest way to do what you ask is to take a vanilla recursive DFS algorithm and modify it so that

1) the stack of vertices is maintained in an actual stack (I mean array/vector/Python list/etc whatever equivalent you have in PHP). insert target vertex to it before calling, pop it after the call returns.

2) when reaching depth k, the function prints the stack to the output and returns

3) recursive function "unmarks" current vertex just before returning

I can't help you with PHP specifically though.

yonil
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