Tarjan's Algorithm Research Articles

On finding the strongly connected components in a directed graph

We present two improved versions of Tarjan's algorithm for the detection of strongly connected components in a directed graph. Our new algorithms handle sparse graphs and graphs with many trivial components (containing only one node) more economically than Tarjan's original algorithm. As an application we present an efficient transitive closure algorithm.

Solving the Euclidean bottleneck matching problem byk-relative neighborhood graphs

Given a set of pointsV in the plane, the Euclidean bottleneck matching problem is to match each point with some other point such that the longest Euclidean distance between matched points, resulting from this matching, is minimized. To solve this problem, we definek-relative neighborhood graphs, (kRNG) which are derived from Toussaint's relative neighborhood graphs (RNG). Two points are calledk-relative neighbors if and only if there are less thank points ofV which are closer to both of the two points than the two points are to each other. AkRNG is an undirected graph (V,E r k) whereE r k is the set of pairs of points ofV which arek-relative neighbors. We prove that there exists an optimal solution of the Euclidean bottleneck matching problem which is a subset ofE r 17. We also prove that ¦E r k¦ < 18kn wheren is the number of points in setV. Our algorithm would construct a 17RNG first. This takesO(n2) time. We then use Gabow and Tarjan's bottleneck maximum cardinality matching algorithm for general graphs whose time-complexity isO((n logn)0.5m), wherem is the number of edges in the graph, to solve the bottleneck maximum cardinality matching problem in the 17RNG. This takesO(n1.5 log0.5n) time. The total time-complexity of our algorithm for the Euclidean bottleneck matching problem isO(n2 +n1.5 log0.5n).

New scaling algorithms for the assignment and minimum mean cycle problems

In this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems. Our assignment algorithm is based on applying scaling to a hybrid version of the recentauction algorithm of Bertsekas and the successive shortest path algorithm. The algorithm proceeds by relaxing the optimality conditions, and the amount of relaxation is successively reduced to zero. On a network with 2n nodes,m arcs, and integer arc costs bounded byC, the algorithm runs in O(\(\sqrt n \) m log(nC)) time and uses very simple data structures. This time bound is comparable to the time taken by Gabow and Tarjan's scaling algorithm, and is better than all other time bounds under thesimilarity assumption, i.e.,C = O(n k) for somek. We next consider the minimum mean cycle problem. Themean cost of a cycle is defined as the cost of the cycle divided by the number of arcs it contains. Theminimum mean cycle problem is to identify a cycle whose mean cost is minimum. We show that by using ideas of the assignment algorithm in an approximate binary search procedure, the minimum mean cycle problem can also be solved in O(\(\sqrt n \) m lognC) time. Under the similarity assumption, this is the best available time bound to solve the minimum mean cycle problem.

Scaling algorithms for network problems

This paper gives algorithms for network problems that work by scaling the numeric parameters. Assume all parameters are integers. Let n, m, and N denote the number of vertices, number of edges, and largest parameter of the network, respectively. A scaling algorithm for maximum weight matching on a bipartite graph runs in O(n 3 4 m log N) time. For appropriate N this improves the traditional Hungarian method, whose most efficient implementation is O( n( m + n log n)). The speedup results from finding augmenting paths in batches. The matching algorithm gives similar improvements for the following problems: single-source shortest paths for arbitrary edge lengths (Bellman's algorithm); maximum weight degree-constrained subgraph; minimum cost flow on a 0–1 network. Scaling gives a simple maximum value flow algorithm that matches the best known bound (Sleator and Tarjan's algorithm) when log N= O(log n). Scaling also gives a good algorithm for shortest paths on a directed graph with nonnegative edge lengths (Dijkstra's algorithm).

An improved transitive closure algorithm

Several efficient transitive closure algorithms operate on the strongly connected components of a digraph, some of them using Tarjan's algorithm [17]. Exploiting facts from graph theory and the special properties of Tarjan's algorithm we develop a new, improved algorithm. The transitive reduction of a digraph defined in [1] may be obtained as a byproduct.

A note on finding optimum branchings

AbstractThe subject of this note is Tarjan's algorithm for finding an optimum branching in a directed graph. Two errors are pointed out, namely (i) an incorrect claim involving branching uniqueness, and (ii) an imprecise way of updating edge values in each iteration. These two inaccuracies do not affect the basic validity of the algorithm. It is shown here that they may be fixed via a simple modification, which leaves unchanged the overall time and space performances.

An Implementation of Tarjan's Algorithm for the Block Triangularization of a Matrix

article Free Access Share on An Implementation of Tarjan's Algorithm for the Block Triangularization of a Matrix Authors: I. S. Duff Computer Science and Systems Division, Building 8.9, AERE Harwell, Didcot, Oxfordshire, OX11 ORA, England Computer Science and Systems Division, Building 8.9, AERE Harwell, Didcot, Oxfordshire, OX11 ORA, EnglandView Profile , J. K. Reid Computer Science and Systems Division, Building 8.9, AERE Harwell, Didcot, Oxfordshire, OX11 ORA, England Computer Science and Systems Division, Building 8.9, AERE Harwell, Didcot, Oxfordshire, OX11 ORA, EnglandView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 4Issue 2June 1978 pp 137–147https://doi.org/10.1145/355780.355785Online:01 June 1978Publication History 101citation2,004DownloadsMetricsTotal Citations101Total Downloads2,004Last 12 Months70Last 6 weeks10 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

On edge‐disjoint branchings

Abstract : Edmonds has given a complicated algorithmic proof of a theorem characterizing directed graphs that contain k edge-disjoint branchings having specified root sets. Tarjan has described a conceptually simple and good algorithm for finding such branchings when they exist. Tarjan's algorithm is based on a lemma implicit in Edmonds' results. A simple direct proof of this lemma is given, thereby providing a simpler proof of Edmonds' theorem and a simpler proof that Tarjan's algorithm works.

Note on Hopcroft and Tarjan's Planarity Algorithm

Errata are given for “Efficient Planarity Testing” by John Hopcroft and Robert Tarjan [ J. ACM 21 , 4 (Oct. 1974), 549-568].