# 巨连通分支

Graph with strongly connected components marked

Graph with strongly connected components marked

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 are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(V+E)).

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 are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(V+E)).

## Definitions

A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. That is, a path exists from the first vertex in the pair to the second, and another path exists from the second vertex to the first.

A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. That is, a path exists from the first vertex in the pair to the second, and another path exists from the second vertex to the first.

In a directed graph G that may not itself be strongly connected, a pair of vertices u and v are said to be strongly connected to each other if there is a path in each direction between them.

In a directed graph G that may not itself be strongly connected, a pair of vertices u and v are said to be strongly connected to each other if there is a path in each direction between them.

The binary relation of being strongly connected is an equivalence relation, and the induced subgraphs of its equivalence classes are called strongly connected components.

The binary relation of being strongly connected is an equivalence relation, and the induced subgraphs of its equivalence classes are called strongly connected components.

Equivalently, a strongly connected component of a directed graph G is a subgraph that is strongly connected, and is maximal with this property: no additional edges or vertices from G can be included in the subgraph without breaking its property of being strongly connected. The collection of strongly connected components forms a partition of the set of vertices of G.

Equivalently, a strongly connected component of a directed graph G is a subgraph that is strongly connected, and is maximal with this property: no additional edges or vertices from G can be included in the subgraph without breaking its property of being strongly connected. The collection of strongly connected components forms a partition of the set of vertices of G.

The yellow directed acyclic graph is the condensation of the blue directed graph. It is formed by contracting each strongly connected component of the blue graph into a single yellow vertex.

If each strongly connected component is contracted to a single vertex, the resulting graph is a directed acyclic graph, the condensation of G. A directed graph is acyclic if and only if it has no strongly connected subgraphs with more than one vertex, because a directed cycle is strongly connected and every nontrivial strongly connected component contains at least one directed cycle.

If each strongly connected component is contracted to a single vertex, the resulting graph is a directed acyclic graph, the condensation of G. A directed graph is acyclic if and only if it has no strongly connected subgraphs with more than one vertex, because a directed cycle is strongly connected and every nontrivial strongly connected component contains at least one directed cycle.

## DFS-based linear-time algorithms

Several algorithms based on depth first search compute strongly connected components in linear time.

Several algorithms based on depth first search compute strongly connected components in linear time.

• Kosaraju's algorithm uses two passes of depth first search. The first, in the original graph, is used to choose the order in which the outer loop of the second depth first search tests vertices for having been visited already and recursively explores them if not. The second depth first search is on the transpose graph of the original graph, and each recursive exploration finds a single new strongly connected component.[1][2] It is named after S. Rao Kosaraju, who described it (but did not publish his results) in 1978; Micha Sharir later published it in 1981.[3]

Kosaraju算法 Kosaraju's algorithm 使用了两次深度优先搜索。第一次，在原始图中，用于选择第二次深度优先搜索的外循环测试顶点是否已经被访问过的顺序，如果没有，则递归探索它们。第二次深度优先搜索是在原图的转置图上进行的，每次递归探索都会发现一个新的巨连接分支.[4][2] It is named after S. Rao Kosaraju, who described it (but did not publish his results) in 1978; Micha Sharir later published it in 1981.[5]

• Tarjan's strongly connected components algorithm, published by Robert Tarjan in 1972,[6] performs a single pass of depth first search. It maintains a stack of vertices that have been explored by the search but not yet assigned to a component, and calculates "low numbers" of each vertex (an index number of the highest ancestor reachable in one step from a descendant of the vertex) which it uses to determine when a set of vertices should be popped off the stack into a new component.

Tarjan巨连通算法 Tarjan's strongly connected components algorithm ，由Robert Tarjan于1972年提出，[7]执行单次深度优先搜索。它维护一个堆栈（抽象数据类型）Stack 的顶点，这些顶点已经被搜索探索过，但还没有被分配到一个组件中，并计算每个顶点的 "低数"（从顶点的后裔一步步到达的最高祖先的索引号），它用来决定何时应该将一组顶点从堆栈中弹出到一个新的组件中。

• The path-based strong component algorithm uses a depth first search, like Tarjan's algorithm, but with two stacks. One of the stacks is used to keep track of the vertices not yet assigned to components, while the other keeps track of the current path in the depth first search tree. The first linear time version of this algorithm was published by Edsger W. Dijkstra in 1976.[8]

Although Kosaraju's algorithm is conceptually simple, Tarjan's and the path-based algorithm require only one depth-first search rather than two.

The expected sequential running time of this algorithm is shown to be O(n log n), a factor of O(log n) more than the classic algorithms. The parallelism comes from: (1) the reachability queries can be parallelized more easily (e.g. by a BFS, and it can be fast if the diameter of the graph is small); and (2) the independence between the subtasks in the divide-and-conquer process.

This algorithm performs well on real-world graphs, but does not have theoretical guarantee on the parallelism (consider if a graph has no edges, the algorithm requires O(n) levels of recursions).

## Reachability-based Algorithms

Blelloch et al. in 2016 shows that if the reachability queries are applied in a random order, the cost bound of O(n log n) still holds. Furthermore, the queries then can be batched in a prefix-doubling manner (i.e. 1, 2, 4, 8 queries) and run simultaneously in one round. The overall span of this algorithm is log2 n reachability queries, which is probably the optimal parallelism that can be achieved using the reachability-based approach.

Blelloch等人在2016年的研究表明，如果按随机顺序应用可达性查询，O(n log n)的成本约束仍然成立。 此外，查询则可以以前缀加倍的方式进行分批（即1，2，4，8个查询），并在一轮中同时运行。 该算法的总体跨度为log2 n个可达性查询，这可能是使用基于可达性的方法所能达到的最佳并行性。

Previous linear-time algorithms are based on depth-first search which is generally considered hard to parallelize. Fleischer et al.[10] in 2000 proposed a divide-and-conquer approach based on reachability queries, and such algorithms are usually called reachability-based SCC algorithms. The idea of this approach is to pick a random pivot vertex and apply forward and backward reachability queries from this vertex. The two queries partition the vertex set into 4 subsets: vertices reached by both, either one, or none of the searches. One can show that a strongly connected component has to be contained in one of the subsets. The vertex subset reached by both searches forms a strongly connected components, and the algorithm then recurses on the other 3 subsets.

The expected sequential running time of this algorithm is shown to be O(n log n), a factor of O(log n) more than the classic algorithms. The parallelism comes from: (1) the reachability queries can be parallelized more easily (e.g. by a BFS, and it can be fast if the diameter of the graph is small); and (2) the independence between the subtasks in the divide-and-conquer process.

This algorithm performs well on real-world graphs,[2] but does not have theoretical guarantee on the parallelism (consider if a graph has no edges, the algorithm requires O(n) levels of recursions).

Peter M. Maurer describes an algorithm for generating random strongly connected graphs, based on a modification of Tarjan's algorithm to create a spanning tree and adding a minimum of edges such that the result becomes strongly connected. When used in conjunction with the Gilbert or Erdős-Rényi models with node relabelling, the algorithm is capable of generating any strongly connected graph on n nodes, without restriction on the kinds of structures that can be generated.

Peter M. Maurer描述了一种生成随机强连接图的算法，该算法基于对Tarjan算法的修改，以创建一棵生成树，并增加最小的边，从而使结果成为强连接图。 当与Gilbert或Erdős-Rényi模型结合使用时，该算法能够在节点上生成任何强连接图，而不限制可以生成的结构种类。

Blelloch et al.[11] in 2016 shows that if the reachability queries are applied in a random order, the cost bound of O(n log n) still holds. Furthermore, the queries then can be batched in a prefix-doubling manner (i.e. 1, 2, 4, 8 queries) and run simultaneously in one round. The overall span of this algorithm is log2 n reachability queries, which is probably the optimal parallelism that can be achieved using the reachability-based approach.

Blelloch等人[11]在2016年的研究表明，如果以随机顺序应用可达性查询，O(n log n)的成本约束仍然成立。 此外，查询就可以以前缀加倍的方式进行分批（即1，2，4，8次查询），并在一轮中同时运行。 该算法的总体并行算法分析是log2n可达性查询，这可能是使用基于可达性的方法可以达到的最佳并行性。

Algorithms for finding strongly connected components may be used to solve 2-satisfiability problems (systems of Boolean variables with constraints on the values of pairs of variables): as showed, a 2-satisfiability instance is unsatisfiable if and only if there is a variable v such that v and its complement are both contained in the same strongly connected component of the implication graph of the instance.

## Generating random strongly connected graphs

Strongly connected components are also used to compute the Dulmage–Mendelsohn decomposition, a classification of the edges of a bipartite graph, according to whether or not they can be part of a perfect matching in the graph.

Peter M. Maurer describes an algorithm for generating random strongly connected graphs,[12] based on a modification of Tarjan's algorithm to create a spanning tree and adding a minimum of edges such that the result becomes strongly connected. When used in conjunction with the Gilbert or Erdős-Rényi models with node relabelling, the algorithm is capable of generating any strongly connected graph on n nodes, without restriction on the kinds of structures that can be generated.

Peter M. Maurer描述了一种生成随机强连接图的算法,[13]基于对Tarjan算法的修改，创建一棵生成树，并添加最少的边，使结果成为强连接图。 当与Gilbert或Erdős-Rényi模型结合使用时，该算法能够在 n 节点上生成任何强连接图，而不限制可以生成的结构种类。

## Applications

A directed graph is strongly connected if and only if it has an ear decomposition, a partition of the edges into a sequence of directed paths and cycles such that the first subgraph in the sequence is a cycle, and each subsequent subgraph is either a cycle sharing one vertex with previous subgraphs, or a path sharing its two endpoints with previous subgraphs.

Algorithms for finding strongly connected components may be used to solve 2-satisfiability problems (systems of Boolean variables with constraints on the values of pairs of variables): as 脚本错误：没有“Footnotes”这个模块。 showed, a 2-satisfiability instance is unsatisfiable if and only if there is a variable v such that v and its complement are both contained in the same strongly connected component of the implication graph of the instance.[14]

Strongly connected components are also used to compute the Dulmage–Mendelsohn decomposition, a classification of the edges of a bipartite graph, according to whether or not they can be part of a perfect matching in the graph.[16]

## Related results

A directed graph is strongly connected if and only if it has an ear decomposition, a partition of the edges into a sequence of directed paths and cycles such that the first subgraph in the sequence is a cycle, and each subsequent subgraph is either a cycle sharing one vertex with previous subgraphs, or a path sharing its two endpoints with previous subgraphs.

According to Robbins' theorem, an undirected graph may be oriented in such a way that it becomes strongly connected, if and only if it is 2-edge-connected. One way to prove this result is to find an ear decomposition of the underlying undirected graph and then orient each ear consistently.<ref>{{citation

```| last = Robbins | first = H. E. | author-link = Herbert Robbins
```

Category:Graph connectivity

```| journal = American Mathematical Monthly
```

Category:Directed graphs

This page was moved from wikipedia:en:Strongly connected component. Its edit history can be viewed at 巨连通分支/edithistory

1. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. . Section 22.5, pp. 552–557.
2. Hong, Sungpack; Rodia, Nicole C.; Olukotun, Kunle (2013), "On fast parallel detection of strongly connected components (SCC) in small-world graphs" (PDF), Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis - SC '13, pp. 1–11, doi:10.1145/2503210.2503246, ISBN 9781450323789
3. Sharir, Micha (1981), "A strong-connectivity algorithm and its applications in data flow analysis", Computers & Mathematics with Applications, 7: 67–72, doi:10.1016/0898-1221(81)90008-0
4. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. . Section 22.5, pp. 552–557.
5. {Sharir, Micha (1981), "A strong-connectivity algorithm and its applications in data flow analysis", Computers & Mathematics with Applications, 7: 67–72, doi:10.1016/0898-1221(81)90008-0
6. Tarjan, R. E. (1972), "Depth-first search and linear graph algorithms", SIAM Journal on Computing, 1 (2): 146–160, doi:10.1137/0201010
7. Tarjan, R. E. (1972), "Depth-first search and linear graph algorithms", SIAM Journal on Computing, 1 (2): 146–160, doi:10.1137/0201010
8. [[Edsger Dijkstra Although Kosaraju's algorithm is conceptually simple, Tarjan's and the path-based algorithm require only one depth-first search rather than two.|Dijkstra, Edsger]] (1976), A Discipline of Programming Previous linear-time algorithms are based on depth-first search which is generally considered hard to parallelize. Fleischer et al. in 2000 proposed a divide-and-conquer approach based on reachability queries, and such algorithms are usually called reachability-based SCC algorithms. The idea of this approach is to pick a random pivot vertex and apply forward and backward reachability queries from this vertex. The two queries partition the vertex set into 4 subsets: vertices reached by both, either one, or none of the searches. One can show that a strongly connected component has to be contained in one of the subsets. The vertex subset reached by both searches forms a strongly connected components, and the algorithm then recurses on the other 3 subsets., Prentice Hall, Ch. 25 `{{citation}}`: line feed character in `|author-link=` at position 16 (help); line feed character in `|title=` at position 28 (help).
9. [[Edsger Dijkstra Although Kosaraju's algorithm is conceptually simple, Tarjan's and the path-based algorithm require only one depth-first search rather than two.|Dijkstra, Edsger]] (1976), A Discipline of Programming Previous linear-time algorithms are based on depth-first search which is generally considered hard to parallelize. Fleischer et al. in 2000 proposed a divide-and-conquer approach based on reachability queries, and such algorithms are usually called reachability-based SCC algorithms. The idea of this approach is to pick a random pivot vertex and apply forward and backward reachability queries from this vertex. The two queries partition the vertex set into 4 subsets: vertices reached by both, either one, or none of the searches. One can show that a strongly connected component has to be contained in one of the subsets. The vertex subset reached by both searches forms a strongly connected components, and the algorithm then recurses on the other 3 subsets., Prentice Hall, Ch. 25 `{{citation}}`: line feed character in `|author-link=` at position 16 (help); line feed character in `|title=` at position 28 (help).
10. Fleischer, Lisa K.; Hendrickson, Bruce; Pınar, Ali (2000), "On Identifying Strongly Connected Components in Parallel" (PDF), Parallel and Distributed Processing, Lecture Notes in Computer Science, vol. 1800, pp. 505–511, doi:10.1007/3-540-45591-4_68, ISBN 978-3-540-67442-9
11. Blelloch, Guy E.; Gu, Yan; Shun, Julian; Sun, Yihan (2016), "Parallelism in Randomized Incremental Algorithms" (PDF), Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures - SPAA '16, pp. 467–478, doi:10.1145/2935764.2935766, ISBN 9781450342100.
12. Maurer, P. M., Generating strongly connected random graphs (PDF), Int'l Conf. Modeling, Sim. and Vis. Methods MSV'17, CSREA Press, ISBN 1-60132-465-0, retrieved December 27, 2019
13. Maurer, P. M., Generating strongly connected random graphs (PDF), Int'l Conf. Modeling, Sim. and Vis. Methods MSV'17, CSREA Press, ISBN 1-60132-465-0, retrieved December 27, 2019
14. Aspvall, Bengt According to Robbins' theorem, an undirected graph may be oriented in such a way that it becomes strongly connected, if and only if it is 2-edge-connected. One way to prove this result is to find an ear decomposition of the underlying undirected graph and then orient each ear consistently.; Plass, Michael F.; Tarjan, Robert E. (1979), "A linear-time algorithm for testing the truth of certain quantified boolean formulas", Information Processing Letters, 8 (3): 121–123, doi:10.1016/0020-0190(79)90002-4 `{{citation}}`: line feed character in `|first1=` at position 6 (help).
15. Aspvall, Bengt According to Robbins' theorem, an undirected graph may be oriented in such a way that it becomes strongly connected, if and only if it is 2-edge-connected. One way to prove this result is to find an ear decomposition of the underlying undirected graph and then orient each ear consistently.; Plass, Michael F.; Tarjan, Robert E. (1979), "A linear-time algorithm for testing the truth of certain quantified boolean formulas", Information Processing Letters, 8 (3): 121–123, doi:10.1016/0020-0190(79)90002-4 `{{citation}}`: line feed character in `|first1=` at position 6 (help).
16. Dulmage, A. L.; Mendelsohn, N. S. (1958), "Coverings of bipartite graphs", Can. J. Math., 10: 517–534, doi:10.4153/cjm-1958-052-0 `{{citation}}`: Unknown parameter `|lastauthoramp=` ignored (help).