# A transitive closure algorithm

@article{Purdom1970ATC, title={A transitive closure algorithm}, author={Paul Walton Purdom}, journal={BIT Numerical Mathematics}, year={1970}, volume={10}, pages={76-94} }

An algorithm is given for computing the transitive closure of a directed graph in a time no greater thana1N1n+a2n2 for largen wherea1 anda2 are constants depending on the computer used to execute the algorithm,n is the number of nodes in the graph andN1 is the number of arcs (not counting those arcs which are part of a cycle and not counting those arcs which can be removed without changing the transitive closure). For graphs where each arc is selected at random with probabilityp, the average…

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## 102 Citations

An Efficient Algorithm for the Transitive Closure and a Linear Worst-Case Complexity Result for a Class of Sparse Graphs

- Mathematics, Computer ScienceInf. Process. Lett.
- 1986

A new algorithm is described to compute the transitive closure of a circuit-free graph, the complexity of which (in the worst-case sense) is better than O(MN) (M being the number of arcs and N the number-degrees bounded by some fixed constant K).

On computing the transitive closure of a relation

- Mathematics, Computer ScienceActa Informatica
- 2004

An algorithm is presented for computing the transitive closure of an arbitrary relation which is based upon Tarjan's algorithm for finding the strongly connected components of a directed graph, and combines the advantages of Warshall's algorithm, namely, succinctness, a single traversal of the directed graph and ability to exploit the availability of Boolean vector operations.

Transitive closure algorithms based on graph traversal

- Computer ScienceTODS
- 1993

The results indicate that all graph-based algorithms significantly outperform other types of algorithms such as Seminaive and Warren and to the extent possible, adapt these algorithms to perform path computations.

Computing transitive closure on systolic arrays of fixed size

- Computer ScienceDistributed Computing
- 2005

This work proposes two novel ways of computing the transitive closure of an arbitrarily big graph on a systolic array of fixed size using a block-structured algorithm and a simple partitioning of a well-known syStolic algorithm.

A Fully Polynomial Parameterized Algorithm for Counting the Number of Reachable Vertices in a Digraph

- Computer ScienceInf. Process. Lett.
- 2021

This paper presents an O( f 3n)-time exact algorithm, where n is the number of vertices in G and f is the feedback edge number of G, which runs in truly subquadratic time for digraphs of f = O(n 3−e) for any e > 0.

The Transitive Reduction of a Directed Graph

- Mathematics, Computer ScienceSIAM J. Comput.
- 1972

It is shown that the time complexity of the best algorithm for finding the transitive reduction of a graph is the same as the time to compute the transitives closure of agraph or to perform Boolean matrix multiplication.

A new algorithm for transitive closures and computation of recursion in relational databases

- Mathematics, Computer ScienceProceedings on Seventh International Conference on Information Visualization, 2003. IV 2003.
- 2003

The main idea behind this algorithm is tree labeling and graph decomposition, based on which the transitive closure of a directed graph can be computed in O time and in O space, can be used to efficiently compute recursive relationships of adirected graph in a relational environment.

On the calculation of transitive reduction - closure of orders

- Computer Science, MathematicsDiscret. Math.
- 1993

Classes of directed acyclic graphs for which problems can be solved in linear time complexity are proposed, namely: generalized N-free graphs, graphs such that the external or internal degree of any vertex is bounded in the transitive reduction, and bounded decomposition width orders, strongly W-free orders.

On the transitive closure representation and adjustable compression

- Computer ScienceSAC
- 2006

A new encoding method to label a digraph is presented, which reduces the footprints of all previous strategies for recognizing recursive relationships and leads to a new algorithm for computing transitive closures for DAGs (directed acyclic graph) in O(e·b) time and O(n· b) space.

Efficient Parallel Implementation of Transitive Closure of Digraphs

- Computer SciencePVM/MPI
- 2003

Based on a CGM/BSP parallel algorithm for computing the transitive closure of an acyclic directed graph (digraph), a modified version that works for any digraph and does not use the linear extension labeling is presented.

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