DFS

Notes

• Breadth first search uses a queue to explore the graph in rings going away from the start.
• Depth first uses a stack, (Recursive call) to get as far away from the start as possible.
• Input as before
• We will use π and color as before
• They introduce d and f as the discovery and finishing time.
  • We will use these in topological sort later.
• They don't give a start node, but we can discuss how to accomplish this as well.

• DFS(G)
  
   for each vertex u ∈ G.V
       u.color = WHITE
       u.π = NIL
   time = 0
   for each vertex u ∈ G.V
       if u.color = WHITE
           DFS-VISIT(G, u)
  
  
  DFS-VISIT(G, u)
  
   ++time
   u.d = time
   u.color = GRAY
   for each v ∈ G.Adj(u)
       if v.color = WHITE
           v.π = u
           DFS-VISIT(G,v)
   u.color = BLACK
   ++time
   u.f = time
  
      

• Trace with
• This is Θ(|V| + |E|), often written Θ(V + E)
• Some observations
  • R_i = [u.d, u.f]
  • If u.d < v.d < v.f < u.f,
    • then u is the parent of v in the DFS tree
    • and v is a descend of u in the DFS tree.
  • If u.d > v.f or v.d > u.f then they do not have a parent/child relationship in the DFS tree.
• CLRS classifies edges as
  • Tree edges are part of the DFS tree
  • Back edges connect back to a previous part of the tree
  • Forward : connect a parent in the DFS tree to a child, but are not part of the DFS tree
  • Cross: all other edges: everything else
    • Between nodes in the DFS tree that have no relation.
    • Between different DFS trees
• A Directed Acyclic Graph (DAG) is a very used structure in CS.
• DFS can be used to determine if a graph is a dag or not, no back or cross edges.