Depth-first search

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 expansion.

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.

When searching large graphs that can not 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.

Pseudocode (recursive)

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

Python implementation (iterative)

def depthFirstSearch( start, goal ): stack = Stack() start.setVisited() stack.push( start ) while not stack.empty(): node = stack.top() if node == goal: return stack # stack contains path to solution else: child = node.findUnvisitedChild() if child == none: stack.pop() else: child.setVisited() stack.push( child )

Another version

dfs(graph G) { list L = empty tree T = empty choose a starting vertex x search(x) while(L is not full) 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 }

External links

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 0262032937. Section 22.3: Depth-first search, pp.540–549.

Search algorithms | Trees (structure)

Gourt Home::Gourt :: Depth-first search

Pseudocode (recursive)

Python implementation (iterative)

Another version

External links

References