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Depth-first search (DFS) is an
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
for traversing or searching
tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
or
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible along each branch before backtracking. Extra memory, usually a
stack Stack may refer to: Places * Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group * Blue Stack Mountains, in Co. Donegal, Ireland People * Stack (surname) (including a list of people ...
, is needed to keep track of the nodes discovered so far along a specified branch which helps in backtracking of the graph. A version of depth-first search was investigated in the 19th century by French mathematician Charles Pierre Trémaux as a strategy for solving mazes.


Properties

The
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
and
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...
analysis of DFS differs according to its application area. In theoretical computer science, DFS is typically used to traverse an entire graph, and takes time where , V, is the number of vertices and , E, the number of
edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...
. This is linear in the size of the graph. In these applications it also uses space O(, V, ) in the worst case to store the
stack Stack may refer to: Places * Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group * Blue Stack Mountains, in Co. Donegal, Ireland People * Stack (surname) (including a list of people ...
of vertices on the current search path as well as the set of already-visited vertices. Thus, in this setting, the time and space bounds are the same as for breadth-first search and the choice of which of these two algorithms to use depends less on their complexity and more on the different properties of the vertex orderings the two algorithms produce. For applications of DFS in relation to specific domains, such as searching for solutions in
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machine A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, moveme ...
or web-crawling, the graph to be traversed is often either too large to visit in its entirety or infinite (DFS may suffer from
non-termination In computer science, a computation is said to diverge if it does not terminate or terminates in an exceptional state. Otherwise it is said to converge. In domains where computations are expected to be infinite, such as process calculi, a computatio ...
). In such cases, search is only performed to a limited depth; due to limited resources, such as memory or disk space, one typically does not use data structures to keep track of the set of all previously visited vertices. When search is performed to a limited depth, the time is still linear in terms of the number of expanded vertices and edges (although this number is not the same as the size of the entire graph because some vertices may be searched more than once and others not at all) but the space complexity of this variant of DFS is only proportional to the depth limit, and as a result, is much smaller than the space needed for searching to the same depth using breadth-first search. For such applications, DFS also lends itself much better to
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediat ...
methods for choosing a likely-looking branch. When an appropriate depth limit is not known a priori, iterative deepening depth-first search applies DFS repeatedly with a sequence of increasing limits. In the artificial intelligence mode of analysis, with a branching factor greater than one, iterative deepening increases the running time by only a constant factor over the case in which the correct depth limit is known due to the geometric growth of the number of nodes per level. DFS may also be used to collect a sample of graph nodes. However, incomplete DFS, similarly to incomplete BFS, is
bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group ...
ed towards nodes of high
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...
.


Example

For the following graph: a depth-first search starting at the node 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. The edges traversed in this search form a Trémaux tree, a structure with important applications in
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
. Performing the same search without remembering previously visited nodes results in visiting the 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 is one technique to avoid this infinite loop and would reach all nodes.


Output of a depth-first search

The result of a depth-first search of a graph can be conveniently described in terms of a
spanning tree In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is no ...
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. If the original graph is undirected then all of its edges are tree edges or back edges.


Vertex orderings

It is also possible to use depth-first search to linearly order the vertices of a graph or tree. There are four possible 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 preordering is the reverse of a preordering, i.e. a list of the vertices in the opposite order of their first visit. Reverse preordering is not the same as postordering. * A reverse postordering is the reverse of a postordering, i.e. a list of the vertices in the opposite order of their last visit. Reverse postordering is not the same as preordering. For binary trees there is additionally in-ordering and reverse in-ordering. For example, when searching the directed graph below beginning at node A, the sequence of traversals is either A B D B A C A or A C D C A B A (choosing to first visit B or C from A is up to the algorithm). Note that repeat visits in the form of backtracking to a node, to check if it has still unvisited neighbors, are included here (even if it is found to have none). Thus the possible preorderings are A B D C and A C D B, while the possible postorderings are D B C A and D C B A, and the possible reverse postorderings are A C B D and A B C D. : Reverse postordering produces a topological sorting of any
directed acyclic graph In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one v ...
. This ordering is also useful in control-flow analysis as it often represents a natural linearization of the control flows. The graph above might represent the flow of control in the code fragment below, 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. if (A) then else D


Pseudocode

Input: Output: A recursive implementation of DFS: procedure DFS(''G'', ''v'') is label ''v'' as discovered for all directed edges from ''v'' to ''w that are'' in ''G''.adjacentEdges(''v'') do if vertex ''w'' is not labeled as discovered then recursively call DFS(''G'', ''w'') A non-recursive implementation of DFS with worst-case space complexity O(, E, ), with the possibility of duplicate vertices on the stack: procedure DFS_iterative(''G'', ''v'') is let ''S'' be a stack ''S''.push(''v'') while ''S'' is not empty do ''v'' = ''S''.pop() if ''v'' is not labeled as discovered then label ''v'' as discovered for all edges from ''v'' to ''w'' in ''G''.adjacentEdges(''v'') do ''S''.push(''w'') These two variations of DFS visit the neighbors of each vertex in the opposite order from each other: the first neighbor of ''v'' visited by the recursive variation is the first one in the list of adjacent edges, while in the iterative variation the first visited neighbor is the last one in the list of adjacent edges. The recursive implementation will visit the nodes from the example graph in the following order: A, B, D, F, E, C, G. The non-recursive implementation will visit the nodes as: A, E, F, B, D, C, G. The non-recursive implementation is similar to breadth-first search but differs from it in two ways: # it uses a stack instead of a queue, and # it delays checking whether a vertex has been discovered until the vertex is popped from the stack rather than making this check before adding the vertex. If is a
tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
, replacing the queue of the breadth-first search algorithm with a stack will yield a depth-first search algorithm. For general graphs, replacing the stack of the iterative depth-first search implementation with a queue would also produce a breadth-first search algorithm, although a somewhat nonstandard one. Another possible implementation of iterative depth-first search uses a stack of iterators of the list of neighbors of a node, instead of a stack of nodes. This yields the same traversal as recursive DFS. procedure DFS_iterative(''G'', ''v'') is let ''S'' be a stack label ''v'' as discovered ''S''.push(iterator of ''G''.adjacentEdges(''v'')) while ''S'' is not empty do if ''S''.peek().hasNext() then ''w'' = ''S''.peek().next() if ''w'' is not labeled as discovered then label ''w'' as discovered ''S''.push(iterator of ''G''.adjacentEdges(''w'')) else ''S''.pop()


Applications

Algorithms that use depth-first search as a building block include: * Finding connected components. * Topological sorting. * Finding 2-(edge or vertex)-connected components. * Finding 3-(edge or vertex)-connected components. * Finding the bridges of a graph. * Generating words in order to plot the limit set of a group. * Finding strongly connected components. * Determining whether a species is closer to one species or another in a phylogenetic tree. * Planarity testing. * Solving puzzles with only one solution, such as mazes. (DFS can be adapted to find all solutions to a maze by only including nodes on the current path in the visited set.) * Maze generation may use a randomized DFS. * Finding biconnectivity in graphs. *
Succession Succession is the act or process of following in order or sequence. Governance and politics *Order of succession, in politics, the ascension to power by one ruler, official, or monarch after the death, resignation, or removal from office of ...
to the throne shared by the
Commonwealth realms A Commonwealth realm is a sovereign state in the Commonwealth of Nations whose monarch and head of state is shared among the other realms. Each realm functions as an independent state, equal with the other realms and nations of the Commonw ...
.


Complexity

The computational complexity of DFS was investigated by John Reif. More precisely, given a graph G, let O=(v_1,\dots,v_n) be the ordering computed by the standard recursive DFS algorithm. This ordering is called the lexicographic depth-first search ordering. John Reif considered the complexity of computing the lexicographic depth-first search ordering, given a graph and a source. A
decision version In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whe ...
of the problem (testing whether some vertex occurs before some vertex in this order) is P-complete, meaning that it is "a nightmare for parallel processing". A depth-first search ordering (not necessarily the lexicographic one), can be computed by a randomized parallel algorithm in the complexity class
RNC RNC may refer to: Technology and sciences *Radio Network Controller, a governing element of a mobile phone network *Ribosome-nascent chain complex, in biology *Romanian National R&D Computer Network, registry for the .ro top-level domain * file ex ...
. As of 1997, it remained unknown whether a depth-first traversal could be constructed by a deterministic parallel algorithm, in the complexity class NC..


See also

* Tree traversal (for details about pre-order, in-order and post-order depth-first traversal) * Breadth-first search * Iterative deepening depth-first search * Search game


Notes


References

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


External links


Open Data Structures - Section 12.3.2 - Depth-First-Search
Pat Morin
C++ Boost Graph Library: Depth-First Search





QuickGraph
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