Tarjan's strongly connected components algorithm: Difference between revisions

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'''Tarjan's strongly connected components algorithm''' is an [[algorithm]] in [[graph theory]] for finding the [[strongly connected component]]s (SCCs) of a [[directed graph]]. It runs in [[linear time]], matching the time bound for alternative methods including [[Kosaraju's algorithm]] and the [[path-based strong component algorithm]]. Tarjan'sThe algorithm is named for its inventor, [[Robert Tarjan]].<ref name=Tarjan>{{citation|first=R. E.|last=Tarjan|authorlinkauthor-link=Robert Tarjan|title=Depth-first search and linear graph algorithms|journal=[[SIAM Journal on Computing]]|volume=1|year=1972|issue=2|pages=146–160|doi=10.1137/0201010|citeseerx=10.1.1.327.8418|url=http://www.cs.ucsb.edu/~gilbert/cs240a/old/cs240aSpr2011/slides/TarjanDFS.pdf|access-date=2024-04-07|archive-date=2017-08-29|archive-url=https://web.archive.org/web/20170829214726/http://www.cs.ucsb.edu/~gilbert/cs240a/old/cs240aSpr2011/slides/TarjanDFS.pdf|url-status=bot: unknown}}</ref>

== Overview ==

The algorithm takes a [[directed graph]] as input, and produces a [[Partition of a set|partition]] of the graph's [[Vertex (graph theory)|vertices]] into the graph's strongly connected components. Each vertex of the graph appears in exactly one of the strongly connected components. Any vertex that is not on a directed cycle forms a strongly connected component all by itself: for example, aany vertex whose in-degree or out-degree is 0, or anyevery vertex of ana [[directed acyclic graph]].

The basic idea of the algorithm is this: a depth-first search (DFS) begins from an arbitrary start node (and subsequent depth-first searches are conducted on any nodes that have not yet been found). As usual with depth-first search, the search visits every node of the graph exactly once, decliningrefusing to revisit any node that has already been visited. Thus, the collection of search trees is a [[Spanning forest#Spanning forests|spanning forest]] of the graph. The strongly connected components will be recovered as certain subtrees of this forest. The roots of these subtrees are called the "roots" of the strongly connected components. Any node of a strongly connected component might serve as thea root, if it happens to be the first node of thea component that is discovered by the search.

=== Stack invariant ===

Nodes are placed on a [[Stack (data structure)|stack]] in the order in which they are visited. When the depth-first search recursively visits a node <ttcode>v</ttcode> and its descendants, those nodes are not all necessarily popped from the stack when this recursive call returns. The crucial [[Invariant (computer science)|invariant property]] is that a node remains on the stack after it has been visited if and only if there exists a path in the input graph from it to some node earlier on the stack. In other words, a node is only removed from the DFS stack when all of its connected paths have been traversed.

At the end of the call that visits <ttcode>v</ttcode> and its descendants, we know whether <ttcode>v</ttcode> itself has a path to any node earlier on the stack. If so, the call returns, leaving <ttcode>v</ttcode> on the stack to preserve the invariant. If not, then <ttcode>v</ttcode> must be the root of its strongly connected component, which consists of <ttcode>v</ttcode> together with any nodes later on the stack than <ttcode>v</ttcode> (such nodes all have paths back to <ttcode>v</ttcode> but not to any earlier node, because if they had paths to earlier nodes then <ttcode>v</ttcode> would also have paths to earlier nodes which is false). The connected component rooted at <ttcode>v</ttcode> is then popped from the stack and returned, again preserving the invariant.

=== Bookkeeping ===

Each node <code>v</code> is assigned a unique integer <ttcode>v.index</ttcode>, which numbers the nodes consecutively in the order in which they are discovered. It also maintains a value <ttcode>v.lowlink</ttcode> that represents the smallest index of any node on the stack known to be reachable from <ttcode>v</ttcode> through <ttcode>v</ttcode>'s DFS subtree, including <ttcode>v</ttcode> itself. Therefore <ttcode>v</ttcode> must be left on the stack if <ttcode>v.lowlink < v.index</ttcode>, whereas v must be removed as the root of a strongly connected component if <ttcode>v.lowlink == v.index</ttcode>. The value <ttcode>v.lowlink</ttcode> is computed during the depth-first search from <ttcode>v</ttcode>, as this finds the nodes that are reachable from <ttcode>v</ttcode>.

The lowlink is different from the lowpoint, which is the smallest index reachable from <code>v</code> through any part of the graph.<ref name=Tarjan/>{{rp|156}}<ref name="CMU2018"/>

== The algorithm in pseudocode ==

'''algorithm''' tarjan '''is'''

'''input:''' graph ''G'' = (''V'', ''E'')

'''output:''' set of strongly connected components (sets of vertices)

''index'' := 0

''S'' := empty arraystack

'''for each''' ''v'' '''in''' ''V'' '''do'''

'''if''' (''v''.index is undefined) '''then'''

strongconnect(''v'')

'''end if'''

'''endfunction''' forstrongconnect(''v'')

''// Set the depth index for v to the smallest unused index''

'''function'v''.index := strongconnect(''vindex'')

''// Set the depth index for ''v''.lowlink to the smallest unused:= ''index''

''vindex''.index := ''index'' + 1

''vS''.lowlink := push(''indexv'')

''indexv''.onStack := ''index'' + 1true

''S''.push(''v'')

''// Consider successors of v''

''v''.onStack := true

'''for each''' (''v'', ''w'') '''in''' ''E'' '''do'''

'''if''' (''w''.index is undefined) '''then'''

''// Consider successors of v''

''// Successor w has not yet been visited; recurse on it''

'''for each''' (''v'', ''w'') '''in''' ''E'' '''do'''

strongconnect(''w'')

'''if''' (''w''.index is undefined) '''then'''

''// Successor w has not yet been visited; recurse''v''.lowlink on:= itmin(''v''.lowlink, ''w''.lowlink)

strongconnect( '''else if''' ''w'').onStack '''then'''

''// Successor w is in stack S and hence in the current SCC''

''v''.lowlink := min(''v''.lowlink, ''w''.lowlink)

''// If ''w'' is not on stack, then (''v'', ''w'') is an edge pointing to an SCC already found and must be ignored

'''else if''' (''w''.onStack) '''then'''

''// Successor w is in stack S and hence''// See below inregarding the currentnext SCCline''

''// If ''wv''.lowlink is not on stack, then:= min(''v''.lowlink, ''w''.index) is a cross-edge in the DFS tree and must be ignored

''// Note: The next line may look odd - but is correct.''

''// ItIf saysv w.indexis nota w.lowlink;root thatnode, ispop deliberatethe stack and from thegenerate originalan paperSCC''

'''if''' ''v''.lowlink := min(''v''.lowlink,index ''w'then'''.index)

start a new strongly connected component

'''end if'''

'''end forrepeat'''

''w'' := ''S''.pop()

''// If v is a root node, pop the stack and generate an SCC''w''.onStack := false

add ''w'' to current strongly connected component

'''if''' (''v''.lowlink = ''v''.index) '''then'''

'''else ifwhile''' (''w''.onStack) ≠ '''then'v''

start a new strongly connected component

output the current strongly connected component

'''repeat'''

''w'' := ''S''.pop()

''w''.onStack := false

add ''w'' to current strongly connected component

'''while''' (''w'' != ''v'')

output the current strongly connected component

'''end if'''

'''end function'''

The <ttcode>index</ttcode> variable is the depth-first search node number counter. <ttcode>S</ttcode> is the node stack, which starts out empty and stores the history of nodes explored but not yet committed to a strongly connected component. Note that thisThis is not the normal depth-first search stack, as nodes are not popped as the search returns up the tree; they are only popped when an entire strongly connected component has been found.

The outermost loop searches each node that has not yet been visited, ensuring that nodes which are not reachable from the first node are still eventually traversed. The function <ttcode>strongconnect</ttcode> performs a single depth-first search of the graph, finding all successors from the node <ttcode>v</ttcode>, and reporting all strongly connected components of that subgraph.

When each node finishes recursing, if its lowlink is still set to its index, then it is the root node of a strongly connected component, formed by all of the nodes above it on the stack. The algorithm pops the stack up to and including the current node, and presents all of these nodes as a strongly connected component.

In Tarjan's paper, when <code>''w''</code> is on the stack, <code>''v''.lowlink</code> is updated with the assignment <code>''v''.lowlink := min(''v''.lowlink, ''w''.index)</code>.<ref name=Tarjan/>{{rp|157}} A common variation is to instead use <code>''v''.lowlink := min(''v''.lowlink, ''w''.lowlink)</code>.<ref>{{cite conference

Note that <tt>''v''.lowlink := min(''v''.lowlink, ''w''.index) </tt> is the correct way to update ''v.lowlink'' if ''w'' is on stack. Because ''w'' is on the stack already, ''(v, w)'' is a back-edge in the DFS tree and therefore ''w'' is not in the subtree of ''v''. Because ''v.lowlink'' takes into account nodes reachable only through the nodes in the subtree of ''v'' we must stop at ''w'' and use ''w.index'' instead of ''w.lowlink''.

| last1 = Kordy | first1 = Piotr

| last2 = Langerak | first2 = Rom

| last3 = Mauw | first3 = Sjouke

| last4 = Polderman | first4 = Jan Willem

| editor1-last = Jones | editor1-first = Cliff B.

| editor2-last = Pihlajasaari | editor2-first = Pekka

| editor3-last = Sun | editor3-first = Jun

| contribution = A symbolic algorithm for the analysis of robust timed automata

| contribution-url = https://satoss.uni.lu/members/sjouke/papers/KLMP14.pdf

| doi = 10.1007/978-3-319-06410-9_25

| isbn = 978-3-319-06409-3

| pages = 351–366

| publisher = Springer

| series = Lecture Notes in Computer Science

| title = FM 2014: Formal Methods – 19th International Symposium, Singapore, May 12–16, 2014. Proceedings

| volume = 8442

| year = 2014}}</ref><ref>{{cite web |title=Lecture 19: Tarjan's Algorithm for Identifying Strongly Connected Components in the Dependency Graph |url=http://courses.cms.caltech.edu/cs130/lectures-2024wi/CS130-Wi2024-Lec19.pdf |website=CS130 Software Engineering |publisher=Caltech |date=Winter 2024}}</ref> This modified algorithm does not compute the lowlink numbers as Tarjan defined them, but the test <code>''v''.lowlink = ''v''.index</code> still identifies root nodes of strongly connected components, and therefore the overall algorithm remains valid.<ref name="CMU2018">{{cite web |title=Lecture #19: Depth First Search and Strong Components |url=https://www.cs.cmu.edu/~15451-f18/lectures/lec19-DFS-strong-components.pdf |website=15-451/651: Design & Analysis of Algorithms |publisher=Carnegie Mellon |date=1 November 2018}}</ref>

== Complexity ==

''Time Complexity'': The Tarjan procedure is called once for each node; the forall statement considers each edge at most once. The algorithm's running time is therefore linear in the number of edges and nodes in G, i.e. <math>O(|V|+|E|)</math>.

In order to achieve this complexity, the test for whether <ttcode>w</ttcode> is on the stack should be done in constant time.

This maycan be done, foras example,in bythe storingpseudocode above: store a flag on each node that indicates whether it is on the stack, and performing this test by examining the flag.

''Space Complexity'': The Tarjan procedure requires two words of supplementary data per vertex for the <ttcode>index</ttcode> and <ttcode>lowlink</ttcode> fields, along with one bit for <ttcode>onStack</ttcode> and another for determining when <ttcode>index</ttcode> is undefined. In addition, one word is required on each stack frame to hold <ttcode>v</ttcode> and another for the current position in the edge list. Finally, the worst-case size of the stack <ttcode>S</ttcode> must be <math>|V|</math> (i.e. when the graph is one giant component). This gives a final analysis of <math>O(|V|\cdot(2+5w))</math> where <math>w</math> is the machine word size. The variation of Nuutila and Soisalon-Soininen reduced this to <math>O(|V|\cdot(1+4w))</math> and, subsequently, that of Pearce requires only <math>O(|V|\cdot(1+3w))</math>.<ref>{{cite webjournal|last=Nuutila|first=Esko|title=On Finding the Strongly Connected Components in a Directed Graph|url=https://doi.org/10.1016/0020-0190(94)90047-7|journal=Information Processing Letters|pages=9-149–14|volume=49|number=1|accessdatedoi=13 December 201710.1016/0020-0190(94)90047-7|year=1994}}</ref><ref>{{cite webjournal|last=Pearce|first=David|title=A Space Efficient Algorithm for Detecting Strongly Connected Components|url=https://doi.org/10.1016/j.ipl.2015.08.010|journal=Information Processing Letters|pages=47-5247–52|number=1|volume=116|accessdatedoi=13 December 201710.1016/j.ipl.2015.08.010}}</ref>

==Additional remarks==

While there is nothing special about the order of the nodes within each strongly connected component, one useful property of the algorithm is that no strongly connected component will be identified before any of its successors. Therefore, the order in which the strongly connected components are identified constitutes a reverse [[Topological sorting|topological sort]] of the [[Directed acyclic graph|DAG]] formed by the strongly connected components.<ref>{{cite web|last=Harrison|first=Paul|title=Robust topological sorting and Tarjan's algorithm in Python|url=http://www.logarithmic.net/pfh/blog/01208083168|accessdateaccess-date=9 February 2011}}</ref>

[[Donald Knuth]] described Tarjan's SCC algorithm as one of Knuth'shis favorite implementations in histhe book ''The Stanford GraphBase''.<ref>Knuth, ''The Stanford GraphBase'', pages 512–519.</ref>

He also wrote:<ref>{{cite webbook|last=Knuth|first=Donald|title=Twenty Questions for Donald Knuth|url=http://www.informit.com/articles/article.aspx?p=2213858&WT.mc_id=Author_Knuth_20Questions|date=2014-05-20}}</ref> {{quote|The data structures that he devised for this problem fit together in an amazingly beautiful way, so that the quantities you need to look at while exploring a directed graph are always magically at your fingertips. And his algorithm also does topological sorting as a byproduct.}}

== References ==

<references />

== External links ==

*[https://stackoverflow.com/questions/6643076/tarjan-cycle-detection-help-c#sca Implementation of Tarjan's Algorithm in .NET]

*[https://github.com/danielrbradley/CycleDetection Implementation of Tarjan's Algorithm in .NET (GitHub)]

*[http://www.vacilando.org/article/php-implementation-tarjans-cycle-detection-algorithm Implementation of Tarjan's Algorithm in PHP]

*[https://github.com/bwesterb/py-tarjan/ Another implementation of Tarjan's Algorithm in Python]

*[https://gist.github.com/1440602 Implementation of Tarjan's Algorithm in Javascript]

*[http://clj-me.cgrand.net/2013/03/18/tarjans-strongly-connected-components-algorithm/ Implementation of Tarjan's Algorithm in Clojure]

*[http://www.geeksforgeeks.org/tarjan-algorithm-find-strongly-connected-components/ Implementation of Tarjan's Algorithm in C++]

*[https://github.com/pallas/libite/blob/master/tarjan.cc Iterative version of Tarjan's Algorithm in C++]

*[https://github.com/t1/graph/blob/master/src/main/java/com/github/t1/graph/StronglyConnectedComponentsFinder.java Implementation of Tarjan's Algorithm in Java]

*[https://github.com/looplab/tarjan Implementation of Tarjan's Algorithm in Go]

[[Category:Graph algorithms]]

[[Category:Graph connectivity]]

[[Category:Articles with example pseudocode]]

== External links ==

* [https://rosettacode.org/wiki/Tarjan Rosetta Code], showing implementations in different languages

* [https://github.com/bwesterbVacilando/pyphp-tarjan/ AnotherPHP implementation of Tarjan's Algorithmstrongly connected incomponents Pythonalgorithm]

* [https://github.com/Vacilando/js-tarjan JavaScript implementation of Tarjan's strongly connected components algorithm]