Efficient parallel algorithms for path problems in directed graphs

In this paper we describe a technique for finding efficient parallel algorithms for problems on directed graphs that involve checking the existence of certain kinds of paths in the graph. This technique provides efficient algorithms for finding dominators in flow graphs, performing interval and loop analysis on reducible flow graphs, and finding the feedback vertices of a digraph. Each of these algorithms takesO(log² n) time using the same number of processors needed for fast matrix multiplication. All of these bounds are for an EREW PRAM.     

Overlapping community detection in labeled graphs

We present a new approach for the problem of finding overlapping communities in graphs and social networks. Our approach consists of a novel problem definition and three accompanying algorithms. We are particularly interested in graphs that have labels on their vertices, although our methods are also applicable to graphs with no labels. Our goal is to find k communities so that the total edge density over all k communities is maximized. In the case of labeled graphs, we require that each community is succinctly described by a set of labels. This requirement provides a better understanding for the discovered communities. The proposed problem formulation leads to the discovery of vertex-overlapping and dense communities that cover as many graph edges as possible. We capture these properties with a simple objective function, which we solve by adapting efficient approximation algorithms for the generalized maximum-coverage problem and the densest-subgraph problem. Our proposed algorithm is a generic greedy scheme. We experiment with three variants of the scheme, obtained by varying the greedy step of finding a dense subgraph. We validate our algorithms by comparing with other state-of-the-art community-detection methods on a variety of performance measures. Our experiments confirm that our algorithms achieve results of high quality in terms of the reported measures, and are practical in terms of performance.     

Sublogarithmic distributed MIS algorithm for sparse graphs using Nash-Williams decomposition

We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing an MIS on graphs of bounded arboricity. This is a large family of graphs that includes graphs of bounded degree, planar graphs, graphs of bounded genus, graphs of bounded treewidth, graphs that exclude a fixed minor, and many other graphs. We also devise efficient algorithms for coloring graphs from these families. These results are achieved by the following technique that may be of independent interest. Our algorithm starts with computing a certain graph-theoretic structure, called Nash-Williams forests-decomposition. Then this structure is used to compute the MIS or coloring. Our results demonstrate that this methodology is very powerful. Finally, we show nearly-tight lower bounds on the running time of any distributed algorithm for computing a forests-decomposition.     

Forests,frames, and games: Algorithms for matroid sums and applications

This paper presents improved algorithms for matroid-partitioning problems, such as finding a maximum cardinality set of edges of a graph that can be partitioned intok forests, and finding as many disjoint spanning trees as possible. The notion of a clump in a matroid sum is introduced, and efficient algorithms for clumps are presented. Applications of these algorithms are given to problems arising in the study of the structural rigidity of graphs, the Shannon switching game, and others.     

Fast Algorithms for max independent set

We first propose a method, called “bottom-up method” that, informally, “propagates” improvement of the worst-case complexity for “sparse” instances to “denser” ones and we show an easy though non-trivial application of it to the min set cover problem. We then tackle max independent set. Here, we propagate improvements of worst-case complexity from graphs of average degree?d to graphs of average degree greater than?d. Indeed, using algorithms for max independent set in graphs of average degree 3, we successively solve max independent set in graphs of average degree 4, 5 and?6. Then, we combine the bottom-up technique with measure and conquer techniques to get improved running times for graphs of maximum degree?5 and?6 but also for general graphs. The computation bounds obtained for max independent set are?O ^?(1.1571 ⁿ ), O ^?(1.1895 ⁿ ) and?O ^?(1.2050 ⁿ ), for graphs of maximum (or more generally average) degree?4, 5 and?6 respectively, and?O ^?(1.2114 ⁿ ) for general graphs. These results improve upon the best known results for these cases for polynomial space algorithms.     

Communication-efficient parallel algorithms for distributed random-access machines

This paper introduces a model for parallel computation, called thedistributed randomaccess machine (DRAM), in which the communication requirements of parallel algorithms can be evaluated. A DRAM is an abstraction of a parallel computer in which memory accesses are implemented by routing messages through a communication network. A DRAM explicitly models the congestion of messages across cuts of the network. We introduce the notion of aconservative algorithm as one whose communication requirements at each step can be bounded by the congestion of pointers of the input data structure across cuts of a DRAM. We give a simple lemma that shows how to “shortcut” pointers in a data structure so that remote processors can communicate without causing undue congestion. We giveO(lgn)-step, linear-processor, linear-space, conservative algorithms for a variety of problems onn-node trees, such as computing treewalk numberings, finding the separator of a tree, and evaluating all subexpressions in an expression tree. We giveO(lg² n)-step, linear-processor, linear-space, conservative algorithms for problems on graphs of sizen, including finding a minimum-cost spanning forest, computing biconnected components, and constructing an Eulerian cycle. Most of these algorithms use as a subroutine a generalization of the prefix computation to trees. We show that any suchtreefix computation can be performed inO(lgn) steps using a conservative variant of Miller and Reif's tree-contraction technique.     

Trading uninitialized space for time

The design of efficient graph algorithms usually precludes the test of edge existence, because an efficient support of that operation already requires time for the initialization of an adjacency-matrix representation. We describe an alternative representation of static directed graphs taking Θ(n+m) initialization time and using Θ(n²) space, which supports the efficient implementation of all usual operations on static graphs. The sparse graph representation allows the design of efficient graph algorithms using both iteration over all vertices adjacent with a given vertex and edge-existence operations, although at the expense of additional (uninitialized) space which may, nevertheless, be used for other purposes. To the best of our knowledge, the representation leads to the first graph algorithms with the disconcerting property that the time complexity is better than the space complexity.     

A refined search tree technique for Dominating Set on planar graphs

We establish a refined search tree technique for the parameterized DOMINATING SET problem on planar graphs. Here, we are given an undirected graph and we ask for a set of at most k vertices such that every other vertex has at least one neighbor in this set. We describe algorithms with running times O(8^kn) and O(8^kk+n³), where n is the number of vertices in the graph, based on bounded search trees. We describe a set of polynomial time data-reduction rules for a more general “annotated” problem on black/white graphs that asks for a set of k vertices (black or white) that dominate all the black vertices. An intricate argument based on the Euler formula then establishes an efficient branching strategy for reduced inputs to this problem. In addition, we give a family examples showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final search tree algorithm is easy to implement; its analysis, however, is involved.     

Serial and Parallel Algorithms for (k,2)-Partite Graphs

We introduce a class of layered graphs which we call (k,2)-partite and which we argue are an interesting class because of several important applications. We show that testing for (k,2)-partiteness can be done efficiently both on sequential and parallel machines, by showing that membership is in NSPACE(log n) and in NC². We show that (k,2)-partite graphs have bounded path width. We then show that a particular NP-complete problem, namely Maximum Independent Set, is solvable in linear time on bounded pathwidth graphs if the path decomposition is included in the input. Finally, we show that the Maximum Independent Set problem is in NC² for (k,2)-partite graphs. We note that linear time solutions for certain NP-complete problems have been shown for a wider class of graphs, namely partial k-trees. Our linear time algorithm is somewhat simpler in structure. We conjecture that our techniques can be used on many NP-complete problems to yield efficient algorithms for (k,2)-partite graphs.     

Random Walks,Bisections and Gossiping in Circulant Graphs

Circulant graphs are regular graphs based on Cayley graphs defined on the Abelian group \(\mathbb{Z}_{n}\) . They are popular network topologies that arise in distributed computing. Using number theoretical tools, we first prove two main results for random directed k-regular circulant graphs with n vertices, when n is sufficiently large and k is fixed. First, for any fixed ε>0, n=p prime and L≥p ^1/k (logp)^1+1/k+ε , walks of length at most L terminate at every vertex with asymptotically the same probability. Second, for any n, there is a polynomial time algorithm that for almost all undirected 2r-regular circulant graphs finds a vertex bisector and an edge bisector, both of size less than n ^1?1/r+o(1). We then prove that the latter result also holds for all (rather than for almost all) 2r-regular circulant graphs with n=p, prime, vertices, while, in general, it does not hold for composite n. Using the bisection results, we provide lower bounds on the number of rounds required by any gossiping algorithms for any n. We introduce generic distributed algorithms to solve the gossip problem in any circulant graphs. We illustrate the efficiency of these algorithms by giving nearly matching upper bounds of the number of rounds required by these algorithms in the vertex-disjoint and the edge-disjoint paths communication models in particular circulant graphs.     

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.     

Algorithms for maximum weight induced paths

A family of graphs is a k-bounded-hole family if every graph in the family has no holes with more than k vertices. The problem of finding in a graph a maximum weight induced path has applications in large communication and neural networks when worst case communication time needs to be evaluated; unfortunately this problem is NP-hard even when restricted to bipartite graphs. We show that this problem has polynomial time algorithms for k-bounded-hole families of graphs, for interval-filament graphs and for graphs decomposable by clique cut-sets or by splits into prime subgraphs for which such algorithms exist.     

Subexponential algorithms for partial cover problems

Partial Cover problems are optimization versions of fundamental and well-studied problems like Vertex Cover and Dominating Set. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number k of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by k. It was recently shown by Amini et al. (2008) [1] that Partial Vertex Cover and Partial Dominating Set are fixed parameter tractable on large classes of sparse graphs, namely H-minor-free graphs, which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time ²O(k)nO(1).During the last decade there has been an extensive study on parameterized subexponential algorithms. In particular, it was shown that the classical Vertex Cover and Dominating Set problems can be solved in subexponential time on H-minor-free graphs. The techniques developed to obtain subexponential algorithms for classical problems do not apply to partial cover problems. It was left as an open problem by Amini et al. (2008) [1] whether there is a subexponential algorithm for Partial Vertex Cover and Partial Dominating Set. In this paper, we answer the question affirmatively by solving both problems in time not only on planar graphs but also on much larger classes of graphs, namely, apex-minor-free graphs. Compared to previously known algorithms for these problems our algorithms are significantly faster and simpler.     

A Parallel Algorithm for Finding Minimum Cutsets in Reducible Graphs

We present a parallel algorithm for finding minimum cutsets in reducible graphs. For a reducible graph that has N nodes our algorithm runs in O(log³N) time using O(N³/log N) PEs on the EREW P-RAM model of computation. We also present a parallel heuristic for finding minimal cutsets in general graphs.     

A global parallel algorithm for the hypergraph transversal problem

We consider the problem of finding all minimal transversals of a hypergraph H⊆V², given by an explicit list of its hyperedges. We give a new decomposition technique for solving the problem with the following advantages: (i) Global parallelism: for certain classes of hypergraphs, e.g., hypergraphs of bounded edge size, and any given integer k, the algorithm outputs k minimal transversals of H in time bounded by polylog(|V|,|H|,k) assuming poly(|V|,|H|,k) number of processors. Except for the case of graphs, none of the previously known algorithms for solving the same problem exhibit this feature. (ii) Using this technique, we also obtain new results on the complexity of generating minimal transversals for new classes of hypergraphs, namely hypergraphs of bounded dual-conformality, and hypergraphs in which every edge intersects every minimal transversal in a bounded number of vertices.     

Detecting Holes and Antiholes in Graphs

In this paper we study the problems of detecting holes and antiholes in general undirected graphs, and we present algorithms for these problems. For an input graph G on n vertices and m edges, our algorithms run in O(n + m²) time and require O(n m) space; we thus provide a solution to the open problem posed by Hayward et al. asking for an O(n⁴)-time algorithm for finding holes in arbitrary graphs. The key element of the algorithms is the use of the depth-first-search traversal on appropriate auxiliary graphs in which moving between any two adjacent vertices is equivalent to walking along a P₄ (i.e., a chordless path on four vertices) of the input graph or on its complement, respectively. The approach can be generalized so that for a fixed constant k ≥ 5 we obtain an O(n^k-1)-time algorithm for the detection of a hole (antihole resp.) on at least k vertices. Additionally, we describe a different approach which allows us to detect antiholes in graphs that do not contain chordless cycles on five vertices in O(n + m²) time requiring O(n + m) space. Again, for a fixed constant k ≥ 6, the approach can be extended to yield O(n^k-2)-time and O(n²)-space algorithms for detecting holes (antiholes resp.) on at least k vertices in graphs which do not contain holes (antiholes resp.) on k - 1 vertices. Our algorithms are simple and can be easily used in practice. Finally, we also show how our detection algorithms can be augmented so that they return a hole or an antihole whenever such a structure is detected in the input graph; the augmentation takes O(n + m) time and space.     

Efficient Parallel Algorithms for Hierarchical Clustering on Arrays with Reconfigurable Optical Buses

Clustering is a basic operation in image processing and computer vision, and it plays an important role in unsupervised pattern recognition and image segmentation. While there are many methods for clustering, the single-link hierarchical clustering is one of the most popular techniques. In this paper, with the advantages of both optical transmission and electronic computation, we design efficient parallel hierarchical clustering algorithms on the arrays with reconfigurable optical buses (AROB). We first design three efficient basic operations which include the matrix multiplication of two N×N matrices, finding the minimum spanning tree of a graph with N vertices, and identifying the connected component containing a specified vertex. Based on these three data operations, an O(log N) time parallel hierarchical clustering algorithm is proposed using N³ processors. Furthermore, if the connectivity of the AROB with four-port connection is allowed, two constant time clustering algorithms can be also derived using N⁴ and N³ processors, respectively. These results improve on previously known algorithms developed on various parallel computational models.     

On a simple randomized algorithm for finding a 2-factor in sparse graphs

We analyze the performance of a simple randomized algorithm for finding 2-factors in directed Hamiltonian graphs of out-degree at most two and in undirected Hamiltonian graphs of degree at most three. For the directed case, the algorithm finds a 2-factor in O(n²) expected time. The analysis of our algorithm is based on random walks on the line and interestingly resembles the analysis of a randomized algorithm for the 2-SAT problem given by Papadimitriou [On selecting a satisfying truth assignment, in: Proc. 32nd Annual IEEE Symp. on the Foundations of Computer Science (FOCS), 1991, p. 163]. For the undirected case, the algorithm finds a 2-factor in O(n³) expected time. We also analyze random versions of these graphs and show that cycles of length Ω(n/logn) can be found with high probability in polynomial time. This partially answers an open question of Broder et al. [Finding hidden Hamilton cycles, Random Structures Algorithms 5 (1994) 395] on finding hidden Hamiltonian cycles in sparse random graphs and improves on a result of Karger et al. [On approximating the longest path in a graph, Algorithmica 18 (1997) 82].     

Peer-to-peer information retrieval using shared-content clustering

Peer-to-peer (p2p) networks are used by millions for searching and downloading content. Recently, clustering algorithms were shown to be useful for helping users find content in large networks. Yet, many of these algorithms overlook the fact that p2p networks follow graph models with a power-law node degree distribution. This paper studies the obtained clusters when applying clustering algorithms on power-law graphs and their applicability for finding content. Driven by the observed deficiencies, a simple yet efficient clustering algorithm is proposed, which targets a relaxed optimization of a minimal distance distribution of each cluster with a size balancing scheme. A comparative analysis using a song-similarity graph collected from 1.2 million Gnutella users reveals that commonly used efficiency measures often overlook search and recommendation applicability issues and provide the wrong impression that the resulting clusters are well suited for these tasks. We show that the proposed algorithm performs well on various measures that are well suited for the domain.     

Algorithms for multicommodity flows in planar graphs

This paper gives efficient algorithms for the muiticommodity flow problem for two classes C₁₂ and C₀₁ of planar undirected graphs. Every graph in C₁₂ has two face boundaries B₁ and B₂ such that each of the source-sink pairs lies on B₁ or B₂. On the other hand, every graph inC ₀₁ has a face boundaryB ₁ such that some of the source-sink pairs lie onB ₁ and all the other pairs share a common sink lying onB ₁. The algorithms run inO(kn +nT(n)) time if a graph hasn vertices andk source-sink pairs andT(n) is the time required for finding the single-source shortest paths in a planar graph ofn vertices.