Question

I am working on solving this problem:

Professor Gaedel has written a program that he claims implements Dijkstra’s algorithm.
The program produces v.d and v.π for each vertex v in V. Give an
O.(V + E)-time algorithm to check the output of the professor’s program. It should
determine whether the d and π attributes match those of some shortest-paths tree.
You may assume that all edge weights are nonnegative.

v.d is the shortest distance from starting node to v.
v.π is v’s predecessor in the shortest path from starting node to v

My idea is:
For every vertex (i), compare i.d with (i.π).d. If i’s predecessor has a larger d value then we cannot have a shortest-path tree.

I believe this can check if the professor’s output is not a shortest-path tree, but I don’t think it can confirm that the output is a shortest-path tree. I cannot think of a way to do this without more information.

Am I on the right track?

Answer 1

I think this would work

Do a DFS, but instead of following the regular graph edges, follow only the π value for each vertex. You’re doing this to produce a topological ordering, so that the first vertex to finish will be the first vertex in the topological ordering. Note that the first vertex in the topo sort you produce will be the “source” vertex that was given to Gaedel’s algorithm.

Now that you have a topo ordering, you can relax edges in the most efficient order, just like how you would do it on a DAG.

for each v in topoSortedVerts
    if v.d_verify != v.d_Gaedel
        //fail

    for each u in v.adjacencies
        relax(v, u)

if v.d_verify != v.d_Gaedel
    //fail

I think you may also need to make sure all V verts are considered, and that the source vertex matches. Maybe. Also, I guess Gaedel’s predecessor subgraph induced by the π values could be real jacked up and have all kinds of crazy things wrong with it, but I assume it doesn’t.

It is O(V + E) because the outer loop runs V times, and the inner loop runs E times using aggregate analysis.