Optimum Path Algorithms for Labelled.AdjacencyMap.Algorithm #225
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| -- | A generic Dijkstra algorithm that relaxes the list of edges | ||
| -- based on the 'Dioid'. | ||
| -- | ||
| -- If the 'Dioid' is 'Distance' (negative 'Dioid') the relaxation |
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Are "negative" and "positive" dioids common terms? What is their precise definition?
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Couldn't you just use ShortestDistance and WidestPath from Algebra.Graph.Label?
https://hackage.haskell.org/package/algebraic-graphs-0.4/docs/Algebra-Graph-Label.html#t:ShortestPath
https://hackage.haskell.org/package/algebraic-graphs-0.4/docs/Algebra-Graph-Label.html#t:WidestPath
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Are "negative" and "positive" dioids common terms? What is their precise definition?
I apologize, I actually was talking about the order of the underlying semi-ring.
If 1 > 0, it would be a positive semi-ring. These definitions assume that there is a partial order defined on the set of elements which I believe I did not mention.
Couldn't you just use ShortestDistance and WidestPath from Algebra.Graph.Label
This is not a generic algorithm but a Dijkstra algorithm. Dijkstra requires a priority heap depending on the order of the semi-ring. Even if one would use ShortestDistance the order of relaxation is decided by the heap.
Using ShortestDistance with a max-heap or WidestPath with a min-heap would result in improper results. The type of heap is decided using the order of the semi-ring.
A generic single source shortest path algorithm does not require a priority heap but has a higher runtime complexity.
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@snowleopard I think one cannot define the order of the relaxation (min-heap vs max-heap) based on the type of Semiring. I believe we should take into account how + and * function as well.
I'm trying to find some good papers to read which describe a generic single source shortest path with the time complexity of Dijkstra. Please let me know if you know any good books/papers.
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These definitions assume that there is a partial order defined on the set of elements which I believe I did not mention.
Right. And I guess you are using Ord as this partial order?
Perhaps, you could use the following partial order defined for any idempotent semiring?
po x y = (x + y == y)
This is the order I use for algebraic graphs: isSubgraphOf x y = (x + y == y).
Please let me know if you know any good books/papers.
I think you already identified some very good papers on this topic. Not sure there is anything I can add to your list.
adithyaov
changed the title
adithyaov
changed the title
…rtest-path-labelled
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@adithyaov I guess this PR is blocked by #232, right? |
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@snowleopard Yes. I'll address #232 and this as soon as possible. I need to write the tests and improve the docs in this PR I believe. |
Added shortest path algorithm for a general labeled graph.