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 Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
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
In computer science, a tree is a widely used abstract data type that represents a hierarchical tree structure with a set of connected nodes. Each node in the tree can be connected to many children (depending on the type of tree), but must be con ...
(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
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was " ...
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, to ...
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 consider ...
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
is the number of
vertices and
the number of
edges. This is linear in the size of the graph. In these applications it also uses space
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
Breadth-first search (BFS) is an algorithm for searching a tree data structure for a node that satisfies a given property. It starts at the tree root and explores all nodes at the present depth prior to moving on to the nodes at the next de ...
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 machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech re ...
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 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 immediate, ...
methods for choosing a likely-looking branch. When an appropriate depth limit is not known a priori,
iterative deepening depth-first search
In computer science, iterative deepening search or more specifically iterative deepening depth-first search (IDS or IDDFS) is a state space/graph search strategy in which a depth-limited version of depth-first search is run repeatedly with inc ...
applies DFS repeatedly with a sequence of increasing limits. In the artificial intelligence mode of analysis, with a
branching factor
In computing, tree data structures, and game theory, the branching factor is the number of children at each node, the outdegree. If this value is not uniform, an ''average branching factor'' can be calculated.
For example, in chess, if a "node" i ...
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
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of s ...
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 mathematics
...
.
Example
![Depth-First-Search](https://upload.wikimedia.org/wikipedia/commons/7/7f/Depth-First-Search.gif)
For the following graph:
![graph](https://upload.wikimedia.org/wikipedia/commons/6/61/Graph.traversal.example.svg)
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
In graph theory, a Trémaux tree of an undirected graph G is a type of spanning tree, generalizing depth-first search trees.
They are defined by the property that every edge of G connects an ancestor–descendant pair in the tree. Trémaux trees ar ...
, 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 conne ...
.
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
In computer science, iterative deepening search or more specifically iterative deepening depth-first search (IDS or IDDFS) is a state space/graph search strategy in which a depth-limited version of depth-first search is run repeatedly with inc ...
is one technique to avoid this infinite loop and would reach all nodes.
Output of a depth-first search
![Tree edges](https://upload.wikimedia.org/wikipedia/commons/5/57/Tree_edges.svg)
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 not ...
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
A binary expression tree is a specific kind of a binary tree used to represent Expression (mathematics), expressions. Two common types of expressions that a binary expression tree can represent are algebraic and boolean algebra, boolean. These tre ...
is the expression in
Polish notation
Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators ''precede'' their operands, in contrast t ...
.
* 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
Reverse Polish notation (RPN), also known as reverse Łukasiewicz notation, Polish postfix notation or simply postfix notation, is a mathematical notation in which operators ''follow'' their operands, in contrast to Polish notation (PN), in whi ...
.
* 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
In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary tr ...
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.
:
![If-then-else-control-flow-graph](https://upload.wikimedia.org/wikipedia/commons/6/68/If-then-else-control-flow-graph.svg)
Reverse postordering produces a
topological sorting
In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge ''uv'' from vertex ''u'' to vertex ''v'', ''u'' comes before ''v'' in the ordering. For ins ...
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 ve ...
. This ordering is also useful in
control-flow analysis
In computer science, control-flow analysis (CFA) is a static-code-analysis technique for determining the control flow of a program. The control flow is expressed as a control-flow graph (CFG). For both functional programming languages and object- ...
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
, 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'')
![Graph](https://upload.wikimedia.org/wikipedia/commons/6/61/Graph.traversal.example.svg)
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
Breadth-first search (BFS) is an algorithm for searching a tree data structure for a node that satisfies a given property. It starts at the tree root and explores all nodes at the present depth prior to moving on to the nodes at the next de ...
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
In computer programming, an iterator is an object that enables a programmer to traverse a container, particularly lists. Various types of iterators are often provided via a container's interface. Though the interface and semantics of a given iterat ...
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
In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge ''uv'' from vertex ''u'' to vertex ''v'', ''u'' comes before ''v'' in the ordering. For ins ...
.
* Finding 2-(edge or vertex)-connected components.
* Finding 3-(edge or vertex)-connected components.
* Finding the
bridges
A bridge is a structure built to Span (engineering), span a physical obstacle (such as a body of water, valley, road, or rail) without blocking the way underneath. It is constructed for the purpose of providing passage over the obstacle, whic ...
of a graph.
* Generating words in order to plot the
limit set
In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they ca ...
of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
.
* Finding
strongly connected components
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that a ...
.
* Determining whether a species is closer to one species or another in a phylogenetic tree.
*
Planarity testing
In graph theory, the planarity testing problem is the algorithmic problem of testing whether a given graph is a planar graph (that is, whether it can be drawn in the plane without edge intersections). This is a well-studied problem in computer sci ...
.
* Solving puzzles with only one solution, such as
maze
A maze is a path or collection of paths, typically from an entrance to a goal. The word is used to refer both to branching tour puzzles through which the solver must find a route, and to simpler non-branching ("unicursal") patterns that lea ...
s. (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
Maze generation algorithms are automated methods for the creation of mazes.
Graph theory based methods
A maze can be generated by starting with a predetermined arrangement of cells (most commonly a rectangular grid but other arrangements ...
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 Commonwealt ...
.
Complexity
The
computational complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) ...
of DFS was investigated by
John Reif
John H. Reif (born 1951) is an American academic, and Professor of Computer Science at Duke University, who has made contributions to large number of fields in computer science: ranging from algorithms and computational complexity theory to robotic ...
. More precisely, given a graph
, let
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 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. 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
In computer science, tree traversal (also known as tree search and walking the tree) is a form of graph traversal and refers to the process of visiting (e.g. retrieving, updating, or deleting) each node in a tree data structure, exactly once. S ...
(for details about pre-order, in-order and post-order depth-first traversal)
*
Breadth-first search
Breadth-first search (BFS) is an algorithm for searching a tree data structure for a node that satisfies a given property. It starts at the tree root and explores all nodes at the present depth prior to moving on to the nodes at the next de ...
*
Iterative deepening depth-first search
In computer science, iterative deepening search or more specifically iterative deepening depth-first search (IDS or IDDFS) is a state space/graph search strategy in which a depth-limited version of depth-first search is run repeatedly with inc ...
*
Search game
A search game is a two-person zero-sum game which takes place in a set called the search space. The searcher can choose any continuous trajectory subject to a maximal velocity constraint. It is always assumed that neither the searcher nor the hider ...
Notes
References
*
Thomas H. Cormen
Thomas H. Cormen is the co-author of ''Introduction to Algorithms'', along with Charles Leiserson, Ron Rivest, and Cliff Stein. In 2013, he published a new book titled '' Algorithms Unlocked''. He is a professor of computer science at Dartmout ...
,
Charles E. Leiserson
Charles Eric Leiserson is a computer scientist, specializing in the theory of parallel computing and distributed computing, and particularly practical applications thereof. As part of this effort, he developed the Cilk multithreaded language. ...
,
Ronald L. Rivest
Ronald Linn Rivest (; born May 6, 1947) is a cryptographer and an Institute Professor at MIT. He is a member of MIT's Department of Electrical Engineering and Computer Science (EECS) and a member of MIT's Computer Science and Artificial Inte ...
, and
Clifford Stein
Clifford Seth Stein (born December 14, 1965), a computer scientist, is a professor of industrial engineering and operations research at Columbia University in New York, NY, where he also holds an appointment in the Department of Computer Scie ...
. ''
Introduction to Algorithms
''Introduction to Algorithms'' is a book on computer programming by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The book has been widely used as the textbook for algorithms courses at many universities and is co ...
'', 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
Patrick Ryan Morin is a Canadian computer scientist specializing in computational geometry and data structures. He is a professor in the School of Computer Science at Carleton University.
Education and career
Morin was educated at Carleton Univers ...
C++ Boost Graph Library: Depth-First SearchQuickGraph depth first search example for .Net
Depth-first search algorithm illustrated explanation (Java and C++ implementations)YAGSBPL – A template-based C++ library for graph search and planning
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