Topological Sort

• Topological sort allows us to sort a directed graph.
• This is useful for anything where one item depends on another.
  • Scheduling
  • Building (scheduling)
  • Running inter-related tasks (scheduling)
  • Deciding what order to take classes in (scheduling)
• Some more graph terminology first.
  • As we said, a directed graph or digraph has directed edges.
  • Stored the same way as an undirected graph.
  • If we run DFS on a directed graph we can classify the edges
    • A Tree Edge: are the reverse of π edges, ie part of the dfs tree.
    • A Back Edge: an edge from a descendant to an ancestor in the DFS tree.
    • A Forward Edge: an edge which goes from an ancestor to a descendant, but is not a Tree Edge.
    • A Cross Edge: an edge which goes between nodes in the dfs tree, but not an ancestor descendant relationship.
  • During the DFS, we can classify edges:
    • Going to a white node indicates a tree edge.
    • Going to a grey node indicates a back edge.
    • Going to a black node indicates a cross or forward edge.
      • Going from u to v,
      • if d[u] < d[v], it is forward edge
      • if d[u] > d[v] it is a cross edge.
  • Try this
  • Back edges indicate a cycle.
  • Frequently we need to deal with a special form of a directed graph called a directed acyclic graph.
    • DAG
    • DAGs have no cycles
    • Or no back edges.
    • The existence of a cycle in a scheduling problem
      • At best means co-requisites
      • Or there is a real problem! (IE you need to do two things at once)
  • Applications of DAGs
    • Workflow (as we have seen)
    • Compilers, dag is needed for evaluation of expressions.
• A topological sort will produce an ordering of a DAG
• Two Topological sort algorithms:
  • From CLRS

    • TOPOLOGICAL_SORT(G)
      
       Call DFS(g) to compute f[v] for each vertex.  
       As each vertex is finished, insert it onto the front of a linked list.
       Return the list.
      
      	

    • Finish?
    • Work?
    • Time?
  • From Levitin
    • A decrease and conquer algorithm
    • A source vertex is a vertex with no incoming edges.

    • TOPOLOGICAL_SORTII(G)
      
        L ← [] 
       while there are verticies in G
           Locate a source vertex v.
           L.push_back(v)
           Remove v from G.
      
      	    

    • Finish?
    • Work?
    • Time?