What algorithm can I use to find the shortest path between specified node types in a graph?

Question

This is the problem:

I have n points (p1, p2, p3, .. pn), each of them can connect to any other with a determined cost x.

Each point belongs to one of a set of p

Answer 1

There are many algorithms that will do better than calculating all the possible paths. Breadth-first search is the basic starting point for the family of algorithms I have in mind, Best-first search is appropriate because you have vertex costs defined, and if you can get more information about your problem space, you may be able to use A* or Dijkstra's algorithm. (In each case, finding the paths from the set of allowed starting nodes.)

Re your edit: Your path constraint (the array of node types you need to satisfy) doesn't prevent you from working with these algorithms; quite the opposite, it helps them all work better. You just need to implement them in a way that allows the path constraint to be incorporated, limiting the vertices available at each step in the search to those that are valid given the constraint.

Answer 2

Here is pseudocode with dynamic programming solution:

n - length of desired path
m - number of vertices
types[n] // desired type of ith node
vertice_types[m]
d[n][m] // our DP tab initially filled with infinities

d[0][0..m] = 0
for length from 1 to n 
  for b from 0 to m
    if types[length] == vertice_types[b]
      for a from 0 to m
        if types[length-1] == vertice_types[a]
          d[length][b] = min(d[length][b], d[length-1][a] + cost(a,b))

your minimum cost path is min(d[n][0..m])

you can reduce size of d table to 2 rows, but it would obfuscate the solution