Question

Consider the following pseudocode for the A* search algorithm:

A*(G, s, h)
  for each vertex u in V
    d[u] := f[u] := infinity
    color[u] := WHITE
    p[u] := u
  end for
  color[s] := GRAY
  d[s] := 0
  f[s] := h(s)
  INSERT(Q, s)
  while (Q != Ø)
    u := EXTRACT-MIN(Q)
    for each vertex v in Adj[u]
      if (w(u,v) + d[u] < d[v])
        d[v] := w(u,v) + d[u]
    f[v] := d[v] + h(v)
    p[v] := u
    if (color[v] = WHITE)
      color[v] := GRAY
      INSERT(Q, v)
    else if (color[v] = BLACK)
      color[v] := GRAY
      INSERT(Q, v)
    end if
      else
        ...
    end for
    color[u] := BLACK
  end while

Now – do I understand correctly that if we want to find a path from the source vertex (s) to some destination vertex (let’s name it d) then we can simply add a check right after the u := EXTRACT-MIN(Q) statement like this:

    u := EXTRACT-MIN(Q)
    if (u = d) RETURN PATH

This is obviously correct in case we don’t need to reopen vertices (else if (color[v] = BLACK), but is it correct in case we have to reopen them (for example, if the heuristic function is not monotonic)?

Answer 1

This is correct. If you find the destination node, then you’ll never have to reopen anything; you can just return the path. By the properties of the A* algorithm (including an admissible heuristic), the first time you pop the destination node off the priority queue, you’ll have a shortest path to it.