Depth-first search

• DPV: Ch 3.2–3.3
• Throughout this handout \(G=(V, E)\) can be directed or undirected.
• Depth-first search (DFS) is a method of graph searching or graph exploration that is suited for computing graph connectivity properties, testing whether a graph is planar, solving many other graph problems, and serving as a subroutine to other algorithms.
• A graph search visits all the vertices and scans all the edges.

Idea

• DFS idea: Repeatedly scan an edge that is incident to the most recently discovered vertex with unscanned edges.

• Recursive Implementation of this idea.

  dfs(v) {  
      D:  
      for each edge (v,w) do {  
          S:  
          if vertex w has not been discovered then  
              dfs(w)  
      }  
      F:  
  }

• Label D is where vertex \(v\) is discovered.
• Label S is where each edge incident to \(v\) is being scanned.
• Label F is where the visit of vertex \(v\) is about to finish.
• All depth-first search algorithms have this same structure. They differ in their codes for labels D, S and F.

• The main procedure (function) starts the search by calling dfs(s) from some starting vertex \(s\).

Conceptualization of DFS

• Two ways to conceptualize DFS:
  1. Path view of DFS.
  2. Tree view of DFS
• A DFS path is the path of edges the search traverses to discover a vertex \(v\).
  [EXAMPLE PICTURE OF DFS PATHS HERE]
• At any given point in time, the sequence of vertices in the DFS path corresponds to the vertices on the recursion stack of .
• All the DFS paths together can be represented by a forest (tree if G is connected).
• The DFS forest consists of all the edges that lead to the discovery of some new vertex.
  [EXAMPLE PICTURE OF DFS FOREST HERE]
• A DFS tree is the recursion tree of dfs. Node \(v\) is the parent of node \(w\) if dfs(w) is called from dfs(v).

Notes

• In general, to ensure that the search explores the entire graph, the main procedure is written

  for each vertex v do  
      if v has not been discovered then  
          dfs(v)

• DFS can be implemented in time \(O(m+n)\).
• We can test whether a graph is connected in time \(O(m+n)\) by DFS.
• For an unconnected graph, we can find the connected components in this same time.

• We can test if all vertices of a digraph are reachable from a source vertex \(s\) in time \(O(m+n)\) by DFS.

Exercises

1. Write a DFS algorithm to find the connected components of an undirected graph.
2. Write a DFS algorithm to solve the Reachability Problem.
3. Write a DFS algorithm to test whether a given undirected graph is bipartite.
4. Write a DFS algorithm to test whether a given directed graph contains a cycle.
5. Write a DFS algorithm to test whether a given undirected graph contains a cycle.