Topological Sort

Notes

• Sometimes graphs represent a list of tasks, with the edges representing a dependancey
• Look the first graph in the suppliment
  • This represents the way someone gets dressed (not me, I don't use ties or jackets, asd soon perhaps not socks!)
  • You need to put your pants on before your shoes
  • And it doesn't matter when you put your watch on.
• We can use Topological Sort to produce such an ordering.
• It is important that there are no directed cycles (a->b->c->a)
  • The topological sort can not be done in this case.
  • The DFS will produce the DFS tree that has no cycles.
• The first idea is to use Depth-First-Search to form the DFS graph.
  • We can then use the finsh time to determine the tasks that must be done first.

• TOPLOGICAL-SORT(G)
  
   call DFS(G)
   As each vertex finishes, insert it into a list L
   Return L
   

• Wikipedia has a second one:
  Kahn's algorithm, 1962

  KHAN-TOPO(G)
  
   L ← {}  // an  empty list
   // make S the set of all vertexes with no incoming edges
   S ← {v | v ∈ G.V  and there is no e = (x,v) ∈ G.E}
   While S is not empty do
      remove a node n from S
      add n to L
      for v ∈ Adj(n)
        remove (v,n) from G
        if v has no other incoming edges
           insert m into S
   If G.V ≠ {}
       return error (the graph has a cycle)
   else
       return l