A simple and fast parallel coloring for distance-hereditary graphs

In the literature, there are quite a few sequential and parallel algorithms to solve problems on distance-hereditary graphs. Two well-known classes of graphs, which contain trees and cographs, belong to distance-hereditary graphs. We consider the vertex-coloring problem on distance-hereditary graphs. Let T/sub d/(|V|, |E|) and P/sub d/d(|V|, |E|) denote the time and processor complexities, respectively, required to construct a decomposition tree representation of a distance-hereditary graph G=(V,E) on a PRAM model M/sub d/. Our algorithm runs in O(T/sub d/(|V|, |E|)+log|V|) time using O(P/sub d/(|V|, |E|)+|V|/log|V|) processors on M/sub d/. The best known result for constructing a decomposition tree needs O(log/sup 2/ |V|) time using O(|V|+|E|) processors on a CREW PRAM. If a decomposition tree is provided as input, we solve the problem in O(log |V|) time using O(|V|/log |V|) processors on an EREW PRAM. To the best of our knowledge, there is no parallel algorithm for this problem on distance-hereditary graphs.     

A Linear Time Algorithm for the Arc Disjoint Menger Problem in Planar Directed Graphs

Given a graph G=(V,E) and two vertices s,t ∈ V , s\neq t , the Menger problem is to find a maximum number of disjoint paths connecting s and t . Depending on whether the input graph is directed or not, and what kind of disjointness criterion is demanded, this general formulation is specialized to the directed or undirected vertex, and the edge or arc disjoint Menger problem, respectively. For planar graphs the edge disjoint Menger problem has been solved to optimality [W2], while the fastest algorithm for the arc disjoint version is Weihe's general maximum flow algorithm for planar networks [W1], which has running time \bf O (|V| log |V|) . Here we present a linear time, i.e., asymptotically optimal, algorithm for the arc disjoint version in planar directed graphs. Received August 1997; revised January 1999.     

Parallel Algorithms for Maximum Matching in Complements of Interval Graphs and Related Problems

Given a set of n intervals representing an interval graph, the problem of finding a maximum matching between pairs of disjoint (nonintersecting) intervals has been considered in the sequential model. In this paper we present parallel algorithms for computing maximum cardinality matchings among pairs of disjoint intervals in interval graphs in the EREW PRAM and hypercube models. For the general case of the problem, our algorithms compute a maximum matching in O( log ³ n) time using O(n/ log ² n) processors on the EREW PRAM and using n processors on the hypercubes. For the case of proper interval graphs, our algorithm runs in O( log n ) time using O(n) processors if the input intervals are not given already sorted and using O(n/ log n ) processors otherwise, on the EREW PRAM. On n -processor hypercubes, our algorithm for the proper interval case takes O( log n log log n ) time for unsorted input and O( log n ) time for sorted input. Our parallel results also lead to optimal sequential algorithms for computing maximum matchings among disjoint intervals. In addition, we present an improved parallel algorithm for maximum matching between overlapping intervals in proper interval graphs. Received November 20, 1995; revised September 3, 1998.     

A Generalization of AT-Free Graphs and a Generic Algorithm for Solving Triangulation Problems

Abstract. A subset A of the vertices of a graph G is an asteroidal set if for each vertex a ∈ A a connected component of G-N[a] exists containing A\backslash{a} . An asteroidal set of cardinality three is called asteriodal triple and graphs without an asteriodal triple are called AT-free . The maximum cardinality of an asteroidal set of G , denoted by \an(G) , is said to be the asteriodal number of G . We present a scheme for designing algorithms for triangulation problems on graphs. As a consequence, we obtain algorithms to compute graph parameters such as treewidth, minimum fill-in and vertex ranking number. The running time of these algorithms is a polynomial (of degree asteriodal number plus a small constant) in the number of vertices and the number of minimal separators of the input graph.     

Fast Parallel Reordering and Isomorphism Testing of k -Trees

Abstract. In this paper two problems on the class of k -trees, a subclass of the class of chordal graphs, are considered: the fast reordering problem and the isomorphism problem. An O(log ² n) time parallel algorithm for the fast reordering problem is described that uses O(nk(n-k)/\kern -1ptlog n) processors on a CRCW PRAM proving membership in the class NC for fixed k . An O(nk(k+1)!) time sequential algorithm for the isomorphism problem is obtained representing an improvement over the O(n ² k(k+1)!) algorithm of Sekharan (the second author) [10]. A parallel version of this sequential algorithm is presented that runs in O(log ² n) time using O((nk((k+1)!+n-k))/log n) processors improving on a parallel algorithm of Sekharan for the isomorphism problem [10]. Both the sequential and parallel algorithms use a concept introduced in this paper called the kernel of a k -tree.     

Minimum Cycle Bases for Network Graphs

The minimum cycle basis problem in a graph G = (V,E) is the task to construct a minimum length basis of its cycle vector space. A well-known algorithm by Horton of 1987 needs running time O(|V||E|^2.376). We present a new combinatorial approach which generates minimum cycle bases in time O(\max{|E|³,|E||V|²log |V|}) with a space requirement of (|E|²). This method is especially suitable for large sparse graphs of electric engineering applications since there, typically, |E| is close to linear in |V|.     

Efficient Parallel Algorithms for Permutation Graphs

In this paper, we present optimal O(log n) time, O(n/log n) processor EREW PRAM parallel algorithms for finding the connected components, cut vertices, and bridges of a permutation graph. We also present an O(log n) time, O(n) processor, CREW PRAM model parallel algorithm for finding a Breadth First Search (BFS) spanning tree of a permutation graph rooted at vertex 1 and use the same to derive an efficient parallel algorithm for the All Pairs Shortest Path problem on permutation graphs.     

Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

Abstract. We present an algorithm that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) , where c=4^ 6\sqrt 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O(\sqrt \rule 0pt 4pt \smash γ (G) ) , and that such a tree decomposition can be found in O(\sqrt \rule 0pt 4pt \smash γ (G) n) time. The same technique can be used to show that the k -FACE COVER problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c ₁ ^ \sqrt k n) time, where c ₁ =3^ 36\sqrt 34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k -DOMINATING SET, e.g., k -INDEPENDENT DOMINATING SET and k -WEIGHTED DOMINATING SET.     

无向图的边极大匹配并行算法及其应用*

在EREW PRAM(exclusive-read and exclusive-write parallel random access machine)并行计算模型上,对范围很广的一类无向图的边极大匹配问题,给出时间复杂性为O(logn),使用O((n+m)/logn)处理器的最佳、高速并行算法.     

Reduction Algorithms for Graphs of Small Treewidth

This paper presents a number of new ideas and results on graph reduction applied to graphs of bounded treewidth. S. Arnborg, B. Courcelle, A. Proskurowski, and D. Seese (J. Assoc. Comput. Mach.40, 1134–1164 (1993)) have shown that many decision problems on graphs can be solved in linear time on graphs of bounded treewidth, using a finite set of reduction rules. These algorithms can be used to solve problems on graphs of bounded treewidth without the need to obtain a tree decomposition of the input graph first. We show that the reduction method can be extended to solve the construction variants of many decision problems on graphs of bounded treewidth, including all problems definable in monadic second order logic. We also show that a variant of these reduction algorithms can be used to solve (constructive) optimization problems in O(n) time. For example, optimization and construction variants of I S and H C N can be solved in this way on graphs of small treewidth. Additionally, we show that the results of H. L. Bodlaender and T. Hagerup (SIAM J. Comput.27, 1725–1746 (1998)) can be applied to our reduction algorithms, which results in parallel reduction algorithms that use O(n) operations and O(log n log* n) time on an EREW PRAM, or O(log n) time on a CRCW PRAM.     

An Optimal Parallel Co-Connectivity Algorithm

In this paper we consider the problem of computing the connected components of the complement of a given graph. We describe a simple sequential algorithm for this problem, which works on the input graph and not on its complement, and which for a graph on n vertices and m edges runs in optimal O(n+m) time. Moreover, unlike previous linear co-connectivity algorithms, this algorithm admits efficient parallelization, leading to an optimal O(log n)-time and O((n+m)log n)-processor algorithm on the EREW PRAM model of computation. It is worth noting that, for the related problem of computing the connected components of a graph, no optimal deterministic parallel algorithm is currently available. The co-connectivity algorithms find applications in a number of problems. In fact, we also include a parallel recognition algorithm for weakly triangulated graphs, which takes advantage of the parallel co-connectivity algorithm and achieves an O(log² n) time complexity using O((n+m²) log n) processors on the EREW PRAM model of computation.     

Optimal Sublogarithmic Time Parallel Algorithms on Rooted Forests

The problem of finding a sublogarithmic time optimal parallel algorithm for 3 -colouring rooted forests has been open for long. We settle this problem by obtaining an O(( log log n) log ^* ( log ^* n)) time optimal parallel algorithm on a TOLERANT Concurrent Read Concurrent Write (CRCW) Parallel Random Access Machine (PRAM). Furthermore, we show that if f(n) is the running time of the best known algorithm for 3 -colouring a rooted forest on a COMMON or TOLERANT CRCW PRAM, a fractional independent set of the rooted forest can be found in O(f(n)) time with the same number of processors, on the same model. Using these results, it is shown that decomposable top-down algebraic computation and, hence, depth computation (ranking), 2 -colouring and prefix summation on rooted forests can be done in O( log n) optimal time on a TOLERANT CRCW PRAM. These algorithms have been obtained by proving a result of independent interest, one concerning the self-simulation property of TOLERANT: an N -processor TOLERANT CRCW PRAM that uses an address space of size O(N) only, can be simulated on an n -processor TOLERANT PRAM in O(N/n) time, with no asymptotic increase in space or cost, when n=O(N/ log log N) . Received May 20, 1997; revised June 15, 1998.     

Parallel Algorithms for Maximal Acyclic Sets

Given a graph G=(V,E), the well-known spanning forest problem of G can be viewed as the problem of finding a maximal subset F of edges in G such that the subgraph induced by F is acyclic. Although this problem has well-known efficient NC algorithms, its vertex counterpart, the problem of finding a maximal subset U of vertices in G such that the subgraph induced by U is acyclic, has not been shown to be in NC (or even in RNC) and is not believed to be parallelizable in general. In this paper we present NC algorithms for solving the latter problem for two special cases. First, we show that, for a planar graph with n vertices, the problem can be solved in time with O(n) processors on an EREW PRAM. Second, we show that the problem is solvable in NC if the input graph G has only vertex-induced paths of length polylogarithmic in the number of vertices of G. As a consequence of this result, we show that certain natural extensions of the well-studied maximal independent set problem remain solvable in NC. Moreover, we show that, for a constant-degree graph with n vertices, the problem can be solved in time with O(n ² ) processors on an EREW PRAM. Received July 3, 1995; revised April 1, 1996.     

Uniquely Restricted Matchings

A matching in a graph is a set of edges no two of which share a common vertex. In this paper we introduce a new, specialized type of matching which we call uniquely restricted matchings, originally motivated by the problem of determining a lower bound on the rank of a matrix having a specified zero/ non-zero pattern. A uniquely restricted matching is defined to be a matching M whose saturated vertices induce a subgraph which has only one perfect matching, namely M itself. We introduce the two problems of recognizing a uniquely restricted matching and of finding a maximum uniquely restricted matching in a given graph, and present algorithms and complexity results for certain special classes of graphs. We demonstrate that testing whether a given matching M is uniquely restricted can be done in O(|M||E|) time for an arbitrary graph G=(V,E) and in linear time for cacti, interval graphs, bipartite graphs, split graphs and threshold graphs. The maximum uniquely restricted matching problem is shown to be NP-complete for bipartite graphs, split graphs, and hence for chordal graphs and comparability graphs, but can be solved in linear time for threshold graphs, proper interval graphs, cacti and block graphs. Received April 12, 1998; revised June 21, 1999.     

I/O-Efficient Algorithms for Graphs of Bounded Treewidth

We present an algorithm that takes I/Os (sort(N)=Θ((N/(DB))log  _M/B (N/B)) is the number of I/Os it takes to sort N data items) to compute a tree decomposition of width at most k, for any graph G of treewidth at most k and size N, where k is a constant. Given such a tree decomposition, we use a dynamic programming framework to solve a wide variety of problems on G in I/Os, including the single-source shortest path problem and a number of problems that are NP-hard on general graphs. The tree decomposition can also be used to obtain an optimal separator decomposition of G. We use such a decomposition to perform depth-first search in G in  I/Os. As important tools that are used in the tree decomposition algorithm, we introduce flippable DAGs and present an algorithm that computes a perfect elimination ordering of a k-tree in I/Os. The second contribution of our paper, which is of independent interest, is a general and simple framework for obtaining I/O-efficient algorithms for a number of graph problems that can be solved using greedy algorithms in internal memory. We apply this framework in order to obtain an improved algorithm for finding a maximal matching and the first deterministic I/O-efficient algorithm for finding a maximal independent set of an arbitrary graph. Both algorithms take I/Os. The maximal matching algorithm is used in the tree decomposition algorithm. An abstract of this paper was presented at the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, Proceedings, pp. 89–90, 2001. Research of A. Maheshwari supported by NSERC. Part of this work was done while the second author was a Ph.D. student at the School of Computer Science of Carleton University.     

An Efficient Output-Size Sensitive Parallel Algorithm for Hidden-Surface Removal for Terrains

We describe an efficient parallel algorithm for hidden-surface removal for terrain maps. The algorithm runs in O(log ⁴ n) steps on the CREW PRAM model with a work bound of O((n+k) \polylog ( n)) where n and k are the input and output sizes, respectively. In order to achieve the work bound we use a number of techniques, among which our use of persistent data structures is somewhat novel in the context of parallel algorithms. Received July 29, 1998; revised October 5, 1999.     

Isomorphism for Graphs of Bounded Distance Width

In this paper we study the GRAPH ISOMORPHISM problem on graphs of bounded treewidth, bounded degree, or bounded bandwidth. GRAPH ISOMORPHISM can be solved in polynomial time for graphs of bounded treewidth, pathwidth, or bandwidth, but the exponent depends on the treewidth, pathwidth, or bandwidth. Thus, we look for special cases where ``fixed parameter tractable' polynomial time algorithms can be established. We introduce some new and natural graph parameters: the (rooted) path distance width, which is a restriction of bandwidth, and the (rooted) tree distance width, which is a restriction of treewidth. We give algorithms that solve GRAPH ISOMORPHISM in O(n ² ) time for graphs with bounded rooted path distance width, and in O(n ³ ) time for graphs with bounded rooted tree distance width. Additionally, we show that computing the path distance width of a graph is NP-hard, but both path and tree distance width can be computed in O(n ^k+1 ) time, when they are bounded by a constant k; the rooted path or tree distance width can be computed in O(ne) time. Finally, we study the relationships between the newly introduced parameters and other existing graph parameters. Received February 18, 1997; revised February 23, 1998.     

A Generalization of AT-Free Graphs and a Generic Algorithm for Solving Triangulation Problems

A subset A of the vertices of a graph G is an asteroidal set if for each vertex a ∈ A a connected component of G-N[a] exists containing A\backslash{a} . An asteroidal set of cardinality three is called asteriodal triple and graphs without an asteriodal triple are called AT-free . The maximum cardinality of an asteroidal set of G , denoted by \an(G) , is said to be the asteriodal number of G . We present a scheme for designing algorithms for triangulation problems on graphs. As a consequence, we obtain algorithms to compute graph parameters such as treewidth, minimum fill-in and vertex ranking number. The running time of these algorithms is a polynomial (of degree asteriodal number plus a small constant) in the number of vertices and the number of minimal separators of the input graph.     

Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

An axis-parallel k-dimensional box is a Cartesian product R ₁×R ₂×???×R _k where R _i (for 1≤i≤k) is a closed interval of the form [a _i ,b _i ] on the real line. For a graph G, its boxicity box?(G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a $\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1}An axis-parallel k-dimensional box is a Cartesian product R ₁×R ₂×⋅⋅⋅×R _k where R _i (for 1≤i≤k) is a closed interval of the form [a _i ,b _i ] on the real line. For a graph G, its boxicity box (G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a ?1+\frac1clogn?^d-1\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1} approximation ratio for any constant c≥1 when d≥2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard.     

A Neural Algorithm for the Maximum Clique Problem: Analysis,Experiments, and Circuit Implementation

{In this paper we design and analyze a neural approximation algorithm for the Maximum Clique problem. This algorithm, having as input an arbitrary undirected graph G = \langle V, E\rangle , constructs a finite sequence of Hopfield networks such that the attractor of the last network in the sequence represents a maximal clique of G . We prove that D(G) ⋅ |E ^{\rm c} | , where D(G) = max _{{i,j}\notin E} \min{d _i , d _j } , d _i is the degree of the vertex i of G , and |E ^{\rm c} | denotes the cardinality of the set of edges in the complement graph, is an upper bound to the number of the networks in the sequence. Some experiments made on the second DIMACS benchmark and on random graphs show that: 1. The quality of the solutions found by the algorithm is satisfactory. 2. The theoretical upper bound D(G) ⋅ |E ^{\rm c} | is quite pessimistic. For random graphs we propose an empirical formula that gives a better estimate of the number of networks in the sequence. Moreover, thanks to the simplicity of the algorithm, we are able to design a uniform family of circuits of small size (\approx 10n ² log ₂ n ) that implements it. The circuit, which solves the problems for graphs of at most 32 vertices, has then been programmed on FPGAs (Field Programmable Gate Arrays). An analysis in terms of size and time complexity is given. Received November 10, 1998; revised December 2000.