Spanners of bounded degree graphs

A k-spanner of a graph G is a spanning subgraph of G in which the distance between any pair of vertices is at most k times the distance in G. We prove that for fixed k,w, the problem of deciding if a given graph has a k-spanner of treewidth w is fixed-parameter tractable on graphs of bounded degree. In particular, this implies that finding a k-spanner that is a tree (a tree k-spanner) is fixed-parameter tractable on graphs of bounded degree. In contrast, we observe that if the graph has only one vertex of unbounded degree, then Treek-Spanner is NP-complete for k?4.     

Packing triangles in bounded degree graphs

We consider the two problems of finding the maximum number of node disjoint triangles and edge disjoint triangles in an undirected graph. We show that the first (respectively second) problem is polynomially solvable if the maximum degree of the input graph is at most 3 (respectively 4), whereas it is APX-hard for general graphs and NP-hard for planar graphs if the maximum degree is 4 (respectively 5) or more.     

Dynamic maintenance of the transitive closure in disjunctive graphs

We propose an approach with feasible space requirement to maintain the transitive closure of a class of hypergraphs called OR-graphs. OR-graphs are equivalent to disjunctive deductive databases where disjunctions are limited to one attribute in each OR-table. It has been shown that query processing in disjunctive deductive databases grows into CoNP with very simple examples, but few attempts have been made, as is done in this paper, to obtain classes of disjunctive databases and queries for which efficient algorithms exist. Polynomial time algorithms are presented to compute the transitive closure of OR-graphs and to handle dynamic insertions and deletions. With algorithms for insertions and deletions, we provide a simple but efficient technique to solve the failure set problem in reliability models, which is equivalent to finding the closure of an arbitrary non-empty set of simple nodes. We also show that a minimal extension to OR-graphs makes the computational complexity of the transitive closure CoNP-complete.Research supported in part by NSF under IRI-9210220 and IRI-9111988, Omron Corporation and Omron Management Center of America.     

Algorithms for transitive closure

Let σ′(n) denote the number of all strongly connected graphs on the n-element set. We prove that σ′(n)?2ⁿ²·(1−n(n−1)/2ⁿ⁻¹). Hence the algorithm computing a transitive closure by a reduction to acyclic graphs has the expected time O(n²), under the assumption of uniform distribution of input graphs. Furthermore, we present a new algorithm constructing the transitive closure of an acyclic graph.     

Algebraic structures for transitive closure

Closed semi-rings and the closure of matrices over closed semi-rings are defined and studied. Closed semi-rings are structures weaker than the structures studied by Conway [3] and Aho, Hopcroft and Ullman [1]. Examples of closed semi-rings and closure operations are given, including the case of semi-rings on which the closure of an element is not always defined. Two algorithms are proved to compute the closure of a matrix over any closed semi-ring; the first one based on Gauss–Jordan elimination is a generalization of algorithms by Warshall, Floyd and Kleene; the second one based on Gauss elimination has been studied by Tarjan [11, 12], from the complexity point of view in a slightly different framework. Simple semi-rings, where the closure operation for elements is trivial, are defined and it is shown that the closure of an n × n-matrix over a simple semi-ring is the sum of its powers of degree less than n. Dijkstra semi-rings are defined and it is shown that the rows of the closure of a matrix over a Dijkstra semi-ring, can be computed by a generalized version of Dijkstra's algorithm.     

Speeding up dynamic time warping distance for sparse time series data

Dynamic time warping (DTW) distance has been effectively used in mining time series data in a multitude of domains. However, in its original formulation DTW is extremely inefficient in comparing long sparse time series, containing mostly zeros and some unevenly spaced nonzero observations. Original DTW distance does not take advantage of this sparsity, leading to redundant calculations and a prohibitively large computational cost for long time series. We derive a new time warping similarity measure (AWarp) for sparse time series that works on the run-length encoded representation of sparse time series. The complexity of AWarp is quadratic on the number of observations as opposed to the range of time of the time series. Therefore, AWarp can be several orders of magnitude faster than DTW on sparse time series. AWarp is exact for binary-valued time series and a close approximation of the original DTW distance for any-valued series. We discuss useful variants of AWarp: bounded (both upper and lower), constrained, and multidimensional. We show applications of AWarp to three data mining tasks including clustering, classification, and outlier detection, which are otherwise not feasible using classic DTW, while producing equivalent results. Potential areas of application include bot detection, human activity classification, search trend analysis, seismic analysis, and unusual review pattern mining.     

A processor-time-minimal systolic array for transitive closure

Using a directed acyclic graph (DAG) model of algorithms, the authors focus on processor-time-minimal multiprocessor schedules: time-minimal multiprocessor schedules that use as few processors as possible. The Kung, Lo, and Lewis (KLL) algorithm for computing the transitive closure of a relation over a set of n elements requires at least 5n-4 parallel steps. As originally reported. their systolic array comprises n² processing elements. It is shown that any time-minimal multiprocessor schedule of the KLL algorithm's dag needs at least n²/3 processing elements. Then a processor-time-minimal systolic array realizing the KLL dag is constructed. Its processing elements are organized as a cylindrically connected 2-D mesh, when n=0 mod 3. When n≠0 mod 3, the 2-D mesh is connected as a torus     

Flooding in dynamic graphs with arbitrary degree sequence

This paper addresses the flooding problem in dynamic graphs, where flooding is the basic mechanism in which every node becoming aware of a piece of information at step t$t$ forwards this information to all its neighbors at all forthcoming steps t^′>t$t^{\prime }>t$. We show that a technique developed in a previous paper, for analyzing flooding in a Markovian sequence of Erdös–Rényi graphs, is robust enough to be used also in different contexts. We establish this fact by analyzing flooding in a sequence of graphs drawn independently at random according to a model of random graphs with given expected degree sequence. In the prominent case of power-law degree distributions, we prove that flooding takes almost surely O(logn)$O(logn)$ steps even if, almost surely, none of the graphs in the sequence is connected. In the general case of graphs with an arbitrary degree sequence, we prove several upper bounds on the flooding time, which depend on specific properties of the degree sequence.     

Spanners for bounded tree-length graphs

This paper concerns construction of additive stretched spanners with few edges for n$n$-vertex graphs having a tree-decomposition into bags of diameter at most δ$\delta$, i.e., the tree-length δ$\delta$ graphs. For such graphs we construct additive 2δ$2\delta$-spanners with O(δn+nlogn)$O(\delta n+nlogn)$ edges, and additive 4δ$4\delta$-spanners with O(δn)$O\left(\delta n\right)$ edges. This provides new upper bounds for chordal graphs for which δ=1$\delta =1$. We also show a lower bound, and prove that there are graphs of tree-length δ$\delta$ for which every multiplicative δ$\delta$-spanner (and thus every additive (δ−1)$(\delta -1)$-spanner) requires Ω(n^1+1/Θ(δ))$\Omega \left(n^{1+1/\Theta \left(\delta \right)}\right)$ edges.     

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.     

#BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region

Counting independent sets on bipartite graphs (#BIS) is considered a canonical counting problem of intermediate approximation complexity. It is conjectured that #BIS neither has an FPRAS nor is as hard as #Sat to approximate. We study #BIS in the general framework of two-state spin systems on bipartite graphs. We define two notions, nearly-independent phase-correlated spins and unary symmetry breaking. We prove that it is #BIS-hard to approximate the partition function of any 2-spin system on bipartite graphs supporting these two notions. Consequently, we classify the complexity of approximating the partition function of antiferromagnetic 2-spin systems on bounded-degree bipartite graphs.     

Decomposition of sparse graphs into two forests, one having bounded maximum degree

Let G be a graph. The maximum average degree of G, written Mad(G), is the largest average degree among the subgraphs of G. It was proved in Montassier et al. (2010) [11] that, for any integer k?0, every simple graph with maximum average degree less than admits an edge-partition into a forest and a subgraph with maximum degree at most k; furthermore, when k?3 both subgraphs can be required to be forests. In this note, we extend this result proving that, for k=4,5, every simple graph with maximum average degree less than mk admits an edge-partition into two forests, one having maximum degree at most k (i.e. every graph with maximum average degree less than (resp. ) admits an edge-partition into two forests, one having maximum degree at most 4 (resp. 5)).     

Extending SQL with generalized transitive closure

We present SQL/TC, an extension of SQL, to allow the expression of generalized transitive closure queries. The extension permits the user to pose queries that compute paths between two points and information associated with these paths. Such queries may specify selections on arcs, paths, or sets of paths, The output of a query may include the aggregation of information for different paths between the same endpoints. Our notation is declarative, preserves the spirit of SQL, and allows a declarative and concise formulation of transitive closure queries     

An efficient database transitive closure algorithm

The integration of logic rules and relational databases has recently emerged as an important technique for developing knowledge management systems. An important class of logic rules utilized by these systems is the so-called transitive closure rules, the processing of which requires the computation of the transitive closure of database relations referenced by these rules. This article presents a new algorithm suitable for computing the transitive closure of very large database relations. This algorithm proceeds in two phases. In the first phase, a general graph is condensed into an acyclic one, and at the same time a special sparse matrix is formed from the acyclic graph. The second phase is the main one, in which all the page I/O operations are minimized by removing most of the redundant operations that appear in previous algorithms. Using simulation, this article also studies and examines the performance of this algorithm and compares it with the previous algorithms.     

Practical parallel Union-Find algorithms for transitive closure and clustering

Practical parallel algorithms, based on classical sequential Union-Find algorithms for computing transitive closures of binary relations, are described and implemented for both shared memory and distributed memory parallel computers. By practical algorithms, we mean algorithms that are efficient for parallel systems with bounded numbers of processors as opposed to algorithms where the number of processors grows with the problem size. Transitive closures are useful for decomposing many applications problems into independent subproblems. The implementations were on an ENCORE Multimax shared memory machine and an NCUBE hypercube. Our implementations indicate that transitive closure computations are intrinsically difficult for distributed memory parallel machines because of the need for global information. By contrast, our results for shared memory machines exhibited excellent speedups.Supported in part by NSF Grant DCR-8619103, ONR contract N000-86-G-0202 and DOE Grant DE-FG02-85ER25001.Supported in part by RADC contract F30602-85-C-0303.Supported in part by RADC contract F30602-85-C-0303.