Finding all the shortest paths between two nodes in unweighted undirected graph
I need help finding all the shortest paths between two nodes in an unweighted undirected graph.
I am able to find one of the shortest paths using BFS, but so far I am lost as to how I could find and print out all of them.
Any idea of the algorithm / pseudocode I could use?
Answer
As a caveat, remember that there can be exponentially many shortest paths between two nodes in a graph. Any algorithm for this will potentially take exponential time.
That said, there is a relatively straightforward modification to BFS that you can use as a preprocessing step to speed up generation of all possible paths. Remember that as BFS runs, it proceeds outwards in "layers," getting a single shortest path to all nodes at distance 0, then distance 1, then distance 2, etc. The motivating idea behind BFS is that any node at distance k + 1 from the start node must be connected by an edge to some node at distance k from the start node. BFS discovers this node at distance k + 1 by finding some path of length k to a node at distance k, then extending it by some edge.
If your goal is to find all shortest paths, then you can modify BFS by extending every path to a node at distance k to all the nodes at distance k + 1 that they connect to, rather than picking a single edge. To do this, modify BFS in the following way: whenever you process an edge by adding its endpoint in the processing queue, don't immediately mark that node as being done. Instead, insert that node into the queue annotated with which edge you followed to get to it. This will potentially let you insert the same node into the queue multiple times if there are multiple nodes that link to it. When you remove a node from the queue, then you mark it as being done and never insert it into the queue again. Similarly, rather than storing a single parent pointer, you'll store multiple parent pointers, one for each node that linked into that node.
If you do this modified BFS, you will end up with a DAG where every node will either be the start node and have no outgoing edges, or will be at distance k + 1 from the start node and will have a pointer to each node of distance k that it is connected to. From there, you can reconstruct all shortest paths from some node to the start node by listing of all possible paths from your node of choice back to the start node within the DAG. This can be done recursively:
- There is only one path from the start node to itself, namely the empty path.
- For any other node, the paths can be found by following each outgoing edge, then recursively extending those paths to yield a path back to the start node.
Hope this helps!