Depth-first search

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depth-first search
Order in which the nodes are expanded
General Data
Class: Search Algorithm Data Structure: Graph Time Complexity: O(bm) Space Complexity: O(bm) Optimal: no Complete: yes b - branching factor m - maximum depth of graph

Depth-first search (DFS) is an algorithm for traversing or searching a tree, tree structure, or graph. Intuitively, one starts at the root (selecting some node as the root in the graph case) and explores as far as possible along each branch before backtracking.

Formally, DFS is an uninformed search that progresses by expanding the first child node of the search tree that appears and thus going deeper and deeper until a goal node is found, or until it hits a node that has no children. Then the search backtracks, returning to the most recent node it hadn't finished exploring. In a non-recursive implementation, all freshly expanded nodes are added to a LIFO stack for exploration.

Space complexity of DFS is much lower than BFS (breadth-first search). It also lends itself much better to heuristic methods of choosing a likely-looking branch. Time complexity of both algorithms are proportional to the number of vertices plus the number of edges in the graphs they traverse (O(|V| + |E|)).

When searching large graphs that cannot be fully contained in memory, DFS suffers from non-termination when the length of a path in the search tree is infinite. The simple solution of "remember which nodes I have already seen" doesn't always work because there can be insufficient memory. This can be solved by maintaining an increasing limit on the depth of the tree, which is called iterative deepening depth-first search.

For the following graph:

a depth-first search starting at A, assuming that the left edges in the shown graph are chosen before right edges, and assuming the search remembers previously-visited nodes and will not repeat them (since this is a small graph), will visit the nodes in the following order: A, B, D, F, E, C, G.

Performing the same search without remembering previously visited nodes results in visiting nodes in the order A, B, D, F, E, A, B, D, F, E, etc. forever, caught in the A, B, D, F, E cycle and never reaching C or G.

Iterative deepening prevents this loop and will reach the following nodes on the following depths, assuming it proceeds left-to-right as above:

• 0: A
• 1: A (repeated), B, C, E

(Note that iterative deepening has now seen C, when a conventional depth-first search did not.)

• 2: A, B, D, F, C, G, E, F

(Note that it still sees C, but that it came later. Also note that it sees E via a different path, and loops back to F twice.)

• 3: A, B, D, F, E, C, G, E, F, B

For this graph, as more depth is added, the two cycles "ABFE" and "AEFB" will simply get longer before the algorithm gives up and tries another branch.

Contents 1 Output of a depth-first search 1.1 Vertex orderings 2 Pseudocode (recursive) 2.1 Another version 3 Cycles 4 Applications 5 External links 6 References

[edit] Output of a depth-first search

The most natural result of a depth first search of a graph (if it is considered as a function rather than a procedure) is a spanning tree of the vertices reached during the search. Based on this spanning tree, the edges of the original graph can be divided into three classes: forward edges, which point from a node of the tree to one of its descendants, back edges, which point from a node to one of its ancestors, and cross edges, which do neither. Sometimes tree edges, edges which belong to the spanning tree itself, are classified separately from forward edges. It can be shown that if the graph is undirected then all of its edges are tree edges or back edges.

[edit] Vertex orderings

It is also possible to use the depth-first search to linearly order the vertices (or nodes) of the original graph (or tree). There are three common ways of doing this:

• A preordering is a list of the vertices in the order that they were first visited by the depth-first search algorithm. This is a compact and natural way of describing the progress of the search, as was done earlier in this article. A preordering of an expression tree is the expression in Polish notation.

• A postordering is a list of the vertices in the order that they were last visited by the algorithm. A postordering of an expression tree is the expression in reverse Polish notation.

• A reverse postordering is the reverse of a postordering, i.e. a list of the vertices in the opposite order of their last visit. When searching a tree, reverse postordering is the same as preordering, but in general they are different when searching a graph. For example, when searching the graph

beginning at node A, the possible preorderings are A B D C and A C D B (depending upon whether the algorithm chooses to visit B or C first), while the possible reverse postorderings are A B C D and A C B D. Reverse postordering produces a topological sorting of any directed acyclic graph. This ordering is also useful in control flow analysis as it often represents a natural linearization of the control flow. The graph above might represent the flow of control in a code fragment like

     if (A) then {
       B
     } else {
       C
     }
     D

and it is natural to consider this code in the order A B C D or A C B D, but not natural to use the order A B D C or A C D B.

[edit] Pseudocode (recursive)

dfs(v)
    process(v)
    mark v as visited
    for all vertices i adjacent to v not visited
        dfs(i)

[edit] Another version

dfs(graph G)
{
  list L = empty
  tree T = empty
  choose a starting vertex x
  search(x)
  while(L is not empty)
    remove edge (v, w) from end of L
    if w not yet visited
    {
      add (v, w) to T
      search(w)
    }
}
   
search(vertex v)
{
  visit v
  for each edge (v, w)
    add edge (v, w) to end of L
}

[edit] Cycles

A directed graph G is acyclic if and only if a depth-first search of G yields no back edges. If a back edge (u, v) exists, this means vertex v must be an ancestor of vertex u in the depth-first search tree. Then there is also a path from v to u, and thus the back edge (u, v) must create a cycle.

[edit] Applications

Here are some algorithms where DFS is used:

• Finding connected components.
• Topological sorting.
• Finding 2-(edge or vertex)-connected components.
• Finding strongly connected components.

[edit] External links

• C++ Boost Graph Library: Depth-First Search
• Depth-First Search Animation (for a directed graph)
• Depth First and Breadth First Search: Explanation and Code

[edit] References

• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 22.3: Depth-first search, pp.540–549.