A Functional Approach to External Graph Algorithms

We present a new approach for designing external graph algorithms and use it to design simple, deterministic and randomized external algorithms for computing connected components, minimum spanning forests, bottleneck minimum spanning forests, maximal independent sets (randomized only), and maximal matchings in undirected graphs. Our I/ O bounds compete with those of previous approaches. We also introduce a semi-external model, in which the vertex set but not the edge set of a graph fits in main memory. In this model we give an improved connected components algorithm, using new results for external grouping and sorting with duplicates. Unlike previous approaches, ours is purely functional—without side effects—and is thus amenable to standard checkpointing and programming language optimization techniques. This is an important practical consideration for applications that may take hours to run.     

A Randomized Linear-Work EREW PRAM Algorithm to Find a Minimum Spanning Forest

We present a randomized EREW PRAM algorithm to find a minimum spanning forest in a weighted undirected graph. On an n -vertex graph the algorithm runs in o(( log n) ^{1+ ɛ} ) expected time for any ɛ >0 and performs linear expected work. This is the first linear-work, polylog-time algorithm on the EREW PRAM for this problem. This also gives parallel algorithms that perform expected linear work on two general-purpose models of parallel computation—the QSM and the BSP.     

Property Testing in Bounded Degree Graphs

Abstract. We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when it is ``far' from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure the distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k -connected (for k>1 ), cycle-free and Eulerian. Our algorithms work in time polynomial in 1/ɛ , always accept the graph when it has the tested property, and reject with high probability if the graph is ɛ -far from having the property. For example, the 2-connectivity algorithm rejects (with high probability) any N -vertex d -degree graph for which more than ɛ dN edges need to be added in order to make the graph 2-edge-connected. In addition we prove lower bounds of Ω(\sqrt N ) on the query complexity of testing algorithms for the bipartite and expander properties.     

Approximating Biconnectivity in Parallel

Consider the following NP-hard problems: Given a graph G , find minimum 2-edge connected and 2-vertex connected subgraphs spanning all vertices of G . Over the past few years, exciting sequential algorithms for approximating such minimum subgraphs have been produced [6],[10]. The approximation factors are improved from 2 to 3/2 for both of the problems. Yet the techniques involved are all based on augmenting depth-first-search trees and no similar progress has been carried to the parallel context. This paper presents NC algorithms to achieve approximation factors of 3/2 + ε for these two problems without computing depth-first-search trees. Received June 21, 1995; revised December 24, 1996, and June 2, 1997.     

Efficient Graph-Theoretic Algorithms on a Linear Array with a Reconfigurable Pipelined Bus System

We present efficient algorithms for solving several fundamental graph-theoretic problems on a Linear Array with a Reconfigurable Pipelined Bus System (LARPBS), one of the recently proposed models of computation based on optical buses. Our algorithms include finding connected components, minimum spanning forest, biconnected components, bridges and articulation points for an undirected graph. We compute the connected components and minimum spanning forest of a graph in O(log n) time using O(m+n) processors where m and n are the number of edges and vertices in the graph and m=O(n ²) for a dense graph. Both the processor and time complexities of these two algorithms match the complexities of algorithms on the Arbitrary and Priority CRCW PRAM models which are two of the strongest PRAM models. The algorithms for these two problems published by Li et al. [7] have been considered to be the most efficient on the LARPBS model till now. Their algorithm [7] for these two problems require O(log n) time and O(n ³/log n) processors. Hence, our algorithms have the same time complexity but require less processors. Our algorithms for computing biconnected components, bridges and articulation points of a graph run in O(log n) time on an LARPBS with O(n ²) processors. No previous algorithm was known for these latter problems on the LARPBS.     

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.     

Erratum: An Approximation Algorithm for Minimum-Cost Vertex-Connectivity Problems

Abstract. There is an error in our paper ``An Approximation Algorithm for Minimum-Cost Vertex- Connectivity Problems' (Algorithmica (1997), 18:21—43). In that paper we considered the following problem: given an undirected graph and values r _ij for each pair of vertices i and j , find a minimum-cost set of edges such that there are r _ij vertex-disjoint paths between vertices i and j . We gave approximation algorithms for two special cases of this problem. Our algorithms rely on a primal—dual approach which has led to approximation algorithms for many edge-connectivity problems. The algorithms work in a series of stages; in each stage an augmentation subroutine augments the connectivity of the current solution. The error is in a lemma for the proof of the performance guarantee of the augmentation subroutine. In the case r _ij = k for all i,j , we described a polynomial-time algorithm that claimed to output a solution of cost no more than 2 H (k) times optimal, where H = 1 + 1/2 + · · · + 1/n . This result is erroneous. We describe an example where our primal—dual augmentation subroutine, when augmenting a k -vertex connected graph to a (k+1) -vertex connected graph, gives solutions that are a factor Ω(k) away from the minimum. In the case r _ij ∈ {0,1,2} for all i,j , we gave a polynomial-time algorithm which outputs a solution of cost no more than three times the optimal. In this case we prove that the statement in the lemma that was erroneous for the k -vertex connected case does hold, and that the algorithm performs as claimed.     

Algorithmic Graph Minor Theory: Improved Grid Minor Bounds and Wagner’s Contraction

We explore three important avenues of research in algorithmic graph-minor theory, which all stem from a key min-max relation between the treewidth of a graph and its largest grid minor. This min-max relation is a keystone of the Graph Minor Theory of Robertson and Seymour, which ultimately proves Wagner’s Conjecture about the structure of minor-closed graph properties. First, we obtain the only known polynomial min-max relation for graphs that do not exclude any fixed minor, namely, map graphs and power graphs. Second, we obtain explicit (and improved) bounds on the min-max relation for an important class of graphs excluding a minor, namely, K _3,k -minor-free graphs, using new techniques that do not rely on Graph Minor Theory. These two avenues lead to faster fixed-parameter algorithms for two families of graph problems, called minor-bidimensional and contraction-bidimensional parameters, which include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal parameters, dominating set, edge dominating set, R-dominating set, connected dominating set, connected edge dominating set, connected R-dominating set, and unweighted TSP tour. Third, we disprove a variation of Wagner’s Conjecture for the case of graph contractions in general graphs, and in a sense characterize which graphs satisfy the variation. This result demonstrates the limitations of a general theory of algorithms for the family of contraction-closed problems (which includes, for example, the celebrated dominating-set problem). If this conjecture had been true, we would have had an extremely powerful tool for proving the existence of efficient algorithms for any contraction-closed problem, like we do for minor-closed problems via Graph Minor Theory.     

Expected parallel time and sequential space complexity of graph and digraph problems

This paper determines upper bounds on the expected time complexity for a variety of parallel algorithms for undirected and directed random graph problems. For connectivity, biconnectivity, transitive closure, minimum spanning trees, and all pairs minimum cost paths, we prove the expected time to beO(log logn) for the CRCW PRAM (this parallel RAM machine allows resolution of write conflicts) andO(logn · log logn) for the CREW PRAM (which allows simultaneous reads but not simultaneous writes). We also show that the problem of graph isomorphism has expected parallel timeO(log logn) for the CRCW PRAM andO(logn) for the CREW PRAM. Most of these results follow because of upper bounds on the mean depth of a graph, derived in this paper, for more general graphs than was known before.For undirected connectivity especially, we present a new probabilistic algorithm which runs on a randomized input and has an expected running time ofO(log logn) on the CRCW PRAM, withO(n) expected number of processors only.Our results also improve known upper bounds on the expected space required for sequential graph algorithms. For example, we show that the problems of finding connected components, transitive closure, minimum spanning trees, and minimum cost paths have expected sequential spaceO(logn · log logn) on a deterministic Turing Machine. We use a simulation of the CRCW PRAM to get these expected sequential space bounds.This research was supported by National Science Foundation Grant DCR-85-03251 and Office of Naval Research Contract N00014-80-C-0647.This research was partially supported by the National Science Foundation Grants MCS-83-00630, DCR-8503497, by the Greek Ministry of Research and Technology, and by the ESPRIT Basic Research Actions Project ALCOM.     

Approximation Algorithms for Connected Dominating Sets

The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(Δ)+2 and H(Δ)+2 are presented, where Δ is the maximum degree and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited or has at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (c _n +1) \ln n where c _n ln k is the approximation factor for the node weighted Steiner tree problem (currently c _n = 1.6103 ). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c+1) H(Δ) +c-1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644 ). Received June 22, 1996; revised February 28, 1997.     

Efficient Parallel Graph Algorithms for Coarse-Grained Multicomputers and BSP

In this paper we present deterministic parallel algorithms for the coarse-grained multicomputer (CGM) and bulk synchronous parallel (BSP) models for solving the following well-known graph problems: (1) list ranking, (2) Euler tour construction in a tree, (3) computing the connected components and spanning forest, (4) lowest common ancestor preprocessing, (5) tree contraction and expression tree evaluation, (6) computing an ear decomposition or open ear decomposition, and (7) 2-edge connectivity and biconnectivity (testing and component computation). The algorithms require O(log p) communication rounds with linear sequential work per round (p = no. processors, N = total input size). Each processor creates, during the entire algorithm, messages of total size O(log (p) (N/p)) . The algorithms assume that the local memory per processor (i.e., N/p ) is larger than p ^ε , for some fixed ε > 0 . Our results imply BSP algorithms with O(log p) supersteps, O(g log (p) (N/p)) communication time, and O(log (p) (N/p)) local computation time. It is important to observe that the number of communication rounds/ supersteps obtained in this paper is independent of the problem size, and grows only logarithmically with respect to p . With growing problem size, only the sizes of the messages grow but the total number of messages remains unchanged. Due to the considerable protocol overhead associated with each message transmission, this is an important property. The result for Problem (1) is a considerable improvement over those previously reported. The algorithms for Problems (2)—(7) are the first practically relevant parallel algorithms for these standard graph problems. Received July 5, 2000; revised April 16, 2001.     

An artificial bee colony algorithm for the minimum routing cost spanning tree problem

Given a connected, weighted, and undirected graph, the minimum routing cost spanning tree problem seeks a spanning tree of minimum routing cost on this graph, where routing cost of a spanning tree is defined as the sum of the costs of the paths connecting all possible pairs of distinct vertices in that spanning tree. This problem has several important applications in networks design and computational biology. In this paper, we have proposed an artificial bee colony (ABC) algorithm-based approach for this problem. We have compared our approach against four best methods reported in the literature—two genetic algorithms, a stochastic hill climber and a perturbation-based local search. Computational results show the superiority of our ABC approach over other approaches.     

An Online Algorithm for the Dynamic Maximal Dense Tree Problem

Abstract. The Online Maximal Dense Tree problem is as follows: given a weighted directed graph and a source node, users issue online requests for connection to the source node. A request can either be accepted or rejected (the admission control decision). If the connection request is accepted, it must be connected to the source or to a node previously connected to the source (the routing decision). The objective is to maximize the total number of connections while keeping the connection density , i.e. the ratio of accepted requests to the weight of the spanning tree, sufficiently high. The primary motivation for the Maximal Dense Tree problem is the Online Capacitated Multicast admission control and routing problem. In the Online Capacitated Multicast problem, we are given a communication network with limited link capacities and a set of signal source nodes. Users generate online requests for connection to the signal sources, and the network administrator has to make the admission control and routing decisions. The goal of the network administrator is to maximize the total number of users connected subject to the network capacity constraints. The Online Maximal Dense Tree problem is also faced by a cable TV operator who wishes to connect as many customers as possible while keeping down the amount of wiring per customer. Informally, the Online Maximal Dense Tree algorithm must ``gamble' on certain geographic areas, connecting nodes which are unprofitable to start with, in the hope that eventually enough requests will arrive in its vicinity to make the investment profitable. In this paper we present a randomized online algorithm for the Maximal Dense Tree problem that guarantees acceptance of a (1- ɛ) factor of the requests accepted by the optimum offline algorithm with the expectation of density being at most polylogarithmically lower than that of the offline algorithm. This yields an online capacitated multicast algorithm whose throughput is only poly-logarithmically lower than that of the optimum offline algorithm. Previous work on multicast routing and maximal dense tree problems either made probabilistic assumptions or resulted in linear performance gaps with the offline algorithm. Attempts to solve the Online Maximal Dense Tree problem have also lead to the development of the first polylogarithmic approximation algorithms for the k -MST and the Prize Collecting Salesman problems [AABV].     

Splitting a Delaunay Triangulation in Linear Time

Abstract. Computing the Delaunay triangulation of n points requires usually a minimum of Ω(n log n) operations, but in some special cases where some additional knowledge is provided, faster algorithms can be designed. Given two sets of points, we prove that, if the Delaunay triangulation of all the points is known, the Delaunay triangulation of each set can be computed in randomized expected linear time.     

Polynomial Time Algorithms for 2-Edge-Connectivity Augmentation Problems

Abstract. Given a 2-edge-connected, real weighted graph G with n vertices and m edges, the 2-edge-connectivity augmentation problem is that of finding a minimum weight set of edges of G to be added to a spanning subgraph H of G to make it 2-edge-connected. While the general problem is NP-hard and 2 -approximable, in this paper we prove that it becomes polynomial time solvable if H is a depth-first search tree of G . More precisely, we provide an efficient algorithm for solving this special case which runs in O (M · α(M,n)) time, where α is the classic inverse of Ackermann's function and M=m · α(m,n) . This algorithm has two main consequences: first, it provides a faster 2 -approximation algorithm for the general 2 -edge-connectivity augmentation problem; second, it solves in O (m · α(m,n)) time the problem of restoring, by means of a minimum weight set of replacement edges, the 2 -edge-connectivity of a 2-edge-connected communication network undergoing a link failure.     

Improved heuristics for the bounded-diameter minimum spanning tree problem

Given an undirected, connected, weighted graph and a positive integer k, the bounded-diameter minimum spanning tree (BDMST) problem seeks a spanning tree of the graph with smallest weight, among all spanning trees of the graph, which contain no path with more than k edges. In general, this problem is NP-Hard for 4 ≤ k < n − 1, where n is the number of vertices in the graph. This work is an improvement over two existing greedy heuristics, called randomized greedy heuristic (RGH) and centre-based tree construction heuristic (CBTC), and a permutation-coded evolutionary algorithm for the BDMST problem. We have proposed two improvements in RGH/CBTC. The first improvement iteratively tries to modify the bounded-diameter spanning tree obtained by RGH/CBTC so as to reduce its cost, whereas the second improves the speed. We have modified the crossover and mutation operators and the decoder used in permutation-coded evolutionary algorithm so as to improve its performance. Computational results show the effectiveness of our approaches. Our approaches obtained better quality solutions in a much shorter time on all test problem instances considered.     

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.     

并行图论算法研究进展

在这篇综述文章中，我们将重点介绍并行图论处近年来的发展概况及主要成果，并给出一些可能的发展方向。具体内容包括：基于共享存储模型上的图搜索技术、连发支及最小生成树算法、增值并行图论算法、最短路径算法、极大独立集算法、极大匹配与最大匹配算法，图着色算法、求欧拉回路及哈密尔顿回路算法，图同构算法、图Ｋ连通算法以及最大流最小割算法等。     

Maintenance of a Piercing Set for Intervals with Applications

   Abstract. We show how to maintain efficiently a minimum piercing set for a set S of intervals on the line, under insertions and deletions to/from S. A linear-size dynamic data structure is presented, which enables us to compute a new minimum piercing set following an insertion or deletion in time O(c( S) log |S|), where c (S) is the size of the new minimum piercing set. We also show how to maintain a piercing set for S of size at most (1+ɛ)c (S), for 0 < ɛ ≤ 1 , in

Quasi-Fully Dynamic Algorithms for Two-Connectivity and Cycle Equivalence

We introduce a new class of dynamic graph algorithms called quasi-fully dynamic algorithms , which are much more general than backtracking algorithms and are much simpler than fully dynamic algorithms. These algorithms are especially suitable for applications in which a certain core connected portion of the graph remains fixed, and fully dynamic updates occur on the remaining edges in the graph. We present very simple quasi-fully dynamic algorithms with O(log n) worst-case time per operation for 2-edge connectivity and O(log n) amortized time per operation for cycle equivalence. The former is deterministic while the latter is Monte-Carlo-type randomized. For 2-vertex connectivity, we give a deterministic quasi-fully dynamic algorithm with O(log ³ n) amortized time per operation. The quasi-fully dynamic algorithm we present for cycle equivalence (which has several applications in optimizing compilers) is of special interest since the algorithm is quite simple, and no special-purpose incremental or backtracking algorithm is known for this problem. Received October 26, 1998; revised October 1, 1999, and April 15, 2001.