path ← wgraph ##.wpath (from to) ⍝ Quickest path from/to ⍵ in weighted graph ⍺.

[wgraph] is a 2-row matrix representing a weighted graph. The result [path] is a
vector  that  represents  a  lowest-cost path between the given vertices. Notice
that the return path (to from) may not be the reverse of the forward path.

NB:  All  of the weights in the 2nd row of [wgraph] must be non-negative, other-
wise the result is unpredictable.

Technical notes:

Shortest  paths  through  _unweighted_  graphs, result immediately from a simple
breadth-first  search:  the  target  vertex is discovered in the fewest possible
steps, as the search radiates outwards from the starting vertex.

However,  with  a _weighted_ graph, the search can't stop on finding the target,
as  there may yet be lower-cost paths. The technique used here, is to maintain a
cost-to-reach  value  (which  is initialised to: ⌊/⍬), for each vertex.  At each
point  in  the  search,  vertices  are pursued only if the cost of reaching them
would be less than their current cost-to-reach value. In this case, the cost-to-
reach value for the vertices is adjusted accordingly and the search continued.

As  the  "wave" front of visited vertices radiates from the starting vertex, the
code uses primitive operator @ to generate the new tree and cost vectors:

    new←back cvec decr⍨@wave¨⍵  ⍝ successor tree & cost

A  problem arises if the wave converges on the same vertex from two or more dir-
ections.  The  following  diagram  shows the wave propogating left-to-right from
vertices A, B and C with cost-to-reach values of 8 9 and 7, respectively.


The  edge  costs  A→D, B→D and C→D are 4 2 and 6, so the cost-to-reach D must be
set to:
            ⌊8 9 7+4 2 6

The @ can be made to work if the links and costs are presented in _decreasing_
cost order:

        @(··D··D··D··)⊢  ··  C  A  B ·· ⍝ parent with link
        @(··D··D··D··)←  ·· 13 12 11 ·· ⍝ updated cost vector.

@ is defined to process its subscripts left-to-right, so that the right-most
assignment to D is the only effective one. Here is the code:

        cvec←⊃,/mask/¨accm          ⍝ cost vector
        decr←(⍒cvec)I⊢              ⍝ in decreasing order of cost
        wave←decr↑,/next            ⍝ vertex wave front
        new←back cvec decr⍨@wave¨⍵  ⍝ successor tree & cost
        (∪wave)∇ new                ⍝ wave spreads to adjacent vertices

Thanks to Mike Day for suggesting a fix to the original coding.

A small but important refinement, is to keep track of the cost-to-reach value of
the _target_ node. Any path that would be more expensive than this is ignored:

        best←adjv I¨⊂cost           ⍝ costs to beat
        mask←accm<best⌊to⊃cost      ⍝ mask of better routes
See also →bfs← for notes on parallel breadth-first search.


    aa                      ⍝ weighted graph "aa"
│2 3│3│2 4│1 5│3│
│1 3│1│4 1│1 1│1│

    aa wpath 3 2            ⍝ best path 3→2.
3 4 1 2

    aa[1;] path 3 2         ⍝ compare: best path 3→2 _ignoring_ weights.
3 2

See also: wGraphs Graphs bfs wspan wcost path

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