Edge weights #4: shortest path(s) #656
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mtorpey
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Another quick partial review: it looks like there's another implementation of Floyd-Warshall in this PR, is there a reason not to use the existing implementation? Iirc the existing implementation is pretty flexible, it should be possible to use it for this too, but I might be mistaken
I've taken a look at FLOYD_WARSHALL in digraphs.c, and it would need quite a lot of reworking to use here, for 2 reasons:
Obviously we could refactor the C code, but perhaps that's a future issue? |
| # The "source and destination" problem calls the "single source" problem and | ||
| # extracts information for the given destination. | ||
| # | ||
| # Justification and benchmarks are in Raiyan's MSci thesis, Chapter 6. |
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Any chance of adding a link to this thesis?
| local all_paths; | ||
| if not source in DigraphVertices(digraph) then | ||
| ErrorNoReturn("the 2nd argument <source> must be a vertex of the ", | ||
| "digraph <digraph> that is the 1st argument,"); |
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| "digraph <digraph> that is the 1st argument,"); | |
| "of the 1st argument <digraph> (a digraph),"); |
or
| "digraph <digraph> that is the 1st argument,"); | |
| "of the 1st argument <digraph>,"); |
be a bit shorter?
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Assuming that the algorithms work as intended (I haven't really checked this, but I'm hoping @mtorpey and @RaiyanC did), this looks good, there are a number of minor things that should be changed, and one other thing too:
- can you please add to the doc something about the complexity of each of the functions?
We have tried to do this elsewhere in the documentation, and it's often really helpful.
| function(digraph, source) | ||
| if not source in DigraphVertices(digraph) then | ||
| ErrorNoReturn("the 2nd argument <source> must be a vertex of the ", | ||
| "digraph <digraph> that is the 1st argument,"); |
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Same comment as above?
| local paths, v, a, current, edge_index; | ||
| if not source in DigraphVertices(digraph) then | ||
| ErrorNoReturn("the 2nd argument <source> must be a vertex of the ", | ||
| "digraph <digraph> that is the 1st argument,"); |
| "digraph <digraph> that is the 1st argument,"); | ||
| elif not dest in DigraphVertices(digraph) then | ||
| ErrorNoReturn("the 3rd argument <dest> must be a vertex of the ", | ||
| "digraph <digraph> that is the 1st argument,"); |
| od; | ||
| od; | ||
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| # try every triple: distance from u to v via k |
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Can you please add a comment about why we aren't using the C Floyd-Warshall here? Just to understand again in the future, doesn't have to be very elaborate comment.
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| maxNodes := nrVertices * (nrVertices - 1); | ||
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| # the boundary for performance is edge weight 0.125 |
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Can you possibly reflow this comment, so that it uses more of the width, and possibly add some punctuation if applicable? Thanks!
| # detect negative cycles | ||
| for i in vertices do | ||
| if distances[i][i] < 0 then | ||
| ErrorNoReturn("1st arg <digraph> contains a negative-weighted cycle,"); |
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This error message could be more helpful, possibly explain why a negative-weighted cycle is not ok, and how to compute the correct answer in this case?
| if nrEdges <= threshold then | ||
| return DIGRAPHS_Edge_Weighted_Johnson(digraph); | ||
| else | ||
| return DIGRAPHS_Edge_Weighted_FloydWarshall(digraph); |
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Isn't there an issue here if the graph contains a negative cycle? In particular, if it does contain a negative cycle, then shouldn't we use the other algorithm? So, wouldn't it be better to return fail from DIGRAPHS_Edge_Weighted_FloydWarshall in the case digraph contains a negative cycle, and then just run DIGRAPHS_Edge_Weighted_Johnson anyway?
| to itself is considered to be 0, and so both <C>parents[<A>source</A>]</C> and | ||
| <C>edges[<A>source</A>]</C> are <K>fail</K>. | ||
| Edge weights can have negative values, but this operation will fail with an | ||
| error if a negative-weighted cycle exists. <P/> |
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Not sure that this is accurate, isn't it only the case that it'll fail when the input digraph is "dense", and Floyd-Warshall is applied?
james-d-mitchell
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This adds algorithms for finding shortest paths in edge-weighted digraphs, where "shortest" means "lowest total weight".
Three different "shortest path" problems are solved:
DigraphShortestPaths(digraph)DigraphShortestPaths(digraph, source)DigraphShortestPath (digraph, source, dest)The "all pairs" problem has two algorithms:
The "single source" problem has three algorithms:
The "source and destination" problem calls the "single source" problem and
extracts information for the given destination.
Justification and benchmarks are in @RaiyanC's MSci thesis, Chapter 6.