Implementing While loop in module that will iterate correctly

Question

I am attempting to write a module that prints an optimal path from a starting node to a destination node. The first node is 1 and the final node is 100 (though when module prints these they should be 0 and 99 respectively). The module takes as inputs a distance matrix, Q, and a list, J, whose elements are the shortest-path weight from node i to destination node. Below is my best attempt.

findPathandTotalCost3[Q_, J_] := Module[
  {costs, node, v, w, i},
  i = 1;
  costs[i_] := 
   costs[i] = Table[Q[[node[i], w]] + J[[w]], {w, Length[J]}];
  node[1] = 1;
  node[2] = Position[costs[1], Min[costs[1]]];
  node[i] = Position[costs[i - 1], Min[costs[i - 1]];
    While[node[i] < 100,
     (Print[node[i] - 1]; ++i; costs[i]; node[i];)];];]

All my attempts at implementing this module seem to have problems with either correctly iterating or specifying which parts are recursive. Below is a typical error output.

findPathandTotalCost3[Q, J]

Part::pkspec1: The expression node$45807[0] cannot be used as a part specification.

Part::pkspec1: The expression node$45807[0] cannot be used as a part specification.

Part::pkspec1: The expression node$45807[0] cannot be used as a part specification.

General::stop: Further output of Part::pkspec1 will be suppressed during this calculation.

0

As can be seen, the first node prints correctly ("0") but the module subsequently doesn't evaluate properly.

Does anyone have any suggestions as to how to make this work? I can post more of my code attempts if helpful. Thanks

EDIT: Q and J below. Q is too large to paste in full, but is a list of 100 lists, each similar to:

{Infinity, 0.04, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, 11.11, Infinity, Infinity, Infinity, Infinity, Infinity, 72.21, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, 
  Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, 
  Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, 
  Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, 
  Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity}, 
 {Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, 20.59, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, 64.94, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, 
  Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, 
  Infinity, Infinity, Infinity, Infinity, 1247.25, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, 
  Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, 
  Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity, Infinity}

where, e.g. 0.04 is the edgeweight 1->2.

J:

{160.55, 162.26, 88.52, 143.73, 145.12, 147.43, 141.67, 144.1, \
149.44, 140.95, 150.8, 141.99, 148.93, 303.77, 130.85, 107.01, \
128.15, 114.66, 104.44, 124.66, 124.42, 168.62, 200.27, 88.21, \
114.61, 102.74, 112.81, 112.8, 131.97, 70.38, 71.45, 176.51, 66.16, \
65.84, 110.18, 64.7, 156.07, 67.8, 67.44, 63.95, 77.15, 62.61, 58.66, \
149.25, 50.72, 52.26, 67.53, 48.58, 65.21, 46.27, 45.76, 54.36, \
135.03, 44.38, 54.99, 42.16, 40.05, 40.03, 62.47, 30.69, 33.02, 37.5, \
35.56, 38.77, 32.62, 34.98, 34.34, 31.39, 31.68, 30.47, 30.41, 30.02, \
35.96, 22.04, 21.16, 21.45, 20.64, 42.31, 79.71, 8.91, 33.37, 77.12, \
15.27, 10.37, 33.5, 7.46, 85.72, 4.8, 4.59, 37.6, 13.56, 22.8, 11.87, \
3.28, 3.09, 0.27, 1.06, 0.63, 0.33, 0}

Answer 1

For explicitness, I'll generate some fake data and then use the built-in functions.

weightedAdjacencyMatrix =
  With[
    {maxWeightCount = 15, nodeCount = 10, maxWeight = 20},
    With[
      {edgeCoordinates = 
        DeleteCases[
          RandomInteger[{1, nodeCount}, {maxWeightCount, 2}], {a_, a_}]},
      SparseArray[
        MapThread[
          Rule, 
          {edgeCoordinates, 
           RandomReal[{0, maxWeight}, 
             Length@edgeCoordinates]}], 
        {nodeCount, nodeCount}, 
        Infinity]]]

We can create a Graph directly from this.

graph = 
  WeightedAdjacencyGraph[weightedAdjacencyMatrix, 
    VertexLabels -> "Name", EdgeLabels -> "EdgeWeight"]

The options aren't necessary, but can be useful later to understand verify the results.

The following will give you the shortest path (using the edge weights).

FindShortestPath[graph, 1, 10]

And the next will give you the distance.

GraphDistance[graph, 1, 10]

Does that answer your question? Obviously I chose a small graph for clarity/experimentation, but could this be applied to your situation? I've left out how to mulitply the path weightings (contained in J) to the edge weightings (contained in Q).