José Luis López-Presa

Fast Algorithm for Graph Isomorphism Testing

In this paper we present a novel approach to the graph isomorphism problem. We combine a direct approach, that tries to find a mapping between the two input graphs using backtracking, with a (possibly partial) automorphism precomputing that allows to prune the search tree. We propose an algorithm, conauto , that has a space complexity of O (n 2 logn ) bits. It runs in time O (n 5) with high probability if either one of the input graphs is a G (n ,p ) random graph, for p *** [*** (ln 4 n / n ln ln n ), 1 *** *** (ln 4 n / n ln ln n )]. We compare the practical performance of conauto with other popular algorithms, with an extensive collection of problem instances. Our algorithm behaves consistently for directed, undirected, positive, and negative cases. Additionally, when it is slower than any of the other algorithms, it is only by a small factor.

Containment Properties of Product and Power Graphs

Abstract Abstract In this paper we study containment properties of graphs in relation with the Cartesian product operation. We show that the isomorphism of two Cartesian powers Gr and Hr implies the isomorphism of G and H, while Gr ⊆ Hr does not imply G ⊆ H, even for the special cases when G and H are connected or have the same number of nodes. Then, we find a simple sufficient condition under which the containment of products implies the containment of the factors: if Πin=1 Gii ⊆ Πin=1 Hj, where all graphs Gi are connected and no graph Hj has 4-cycles, then each Gi is a subgraph of a different graph Hj. Hence, if G is connected and H has no 4-cycles, then Gr ⊆ Hr implies G ⊆ H. These results can be used to derive embedding results for interconnection networks for parallel architectures.

Biased selection for building small-world networks

Small-world networks are currently present in many distributed applications and can be built augmenting a base network with long-range links using a probability distribution. Currently available distributed algorithms to select these long-range neighbors are designed ad hoc for specific probability distributions. In this paper we propose a new algorithm called Biased Selection (BS) that, using a uniform sampling service (that could be implemented with, for instance, a gossip-based protocol), allows to select long-range neighbors with any arbitrary distribution in a distributed way. This algorithm is of iterative nature and has a parameter r that gives its number of iterations. We prove that the obtained sampling distribution converges to the desired distribution as r grows. Additionally, we obtain analytical bounds on the maximum relative error for a given value of this parameter r. Although the BS algorithm is proposed in this paper as a tool to sample nodes in a network, it can be used in any context in which sampling with an arbitrary distribution is required, and only uniform sampling is available. The BS algorithm has been used to choose long-range neighbors in complete and incomplete tori, in order to build Kleinbergs small-world networks. We observe that using a very small number of iterations (1) BS has similar error as a simulation of the Kleinbergs harmonic distribution and (2) the average number of hops with greedy routing is no larger with BS than in a Kleinberg network. Furthermore, we have observed that before converging to the performance of a Kleinberg network, the average number of hops with BS is significantly smaller (up to 14% smaller in a 1000 × 1000 network).

Performance of scheduling policies in adversarial networks with non synchronized clocks

In this paper we generalize the Continuous Adversarial Queuing Theory (CAQT) model [5] by considering the possibility that the router clocks in the network are not synchronized. Clearly, this new extension to the model only affects those scheduling policies that use some form of timing. First, if all clocks run at the same speed, maintaining constant differences, we show that all universally stable policies in CAQT that use the injection time and the remaining path to schedule packets remain universally stable. These policies include, for instance, Shortest in System (SIS) and Longest in System (LIS). Then, if clock differences can vary over time, but difference is bounded, we show the universal stability of SIS and a family of policies related to LIS. The bounds we obtain in this case depend on the maximum difference between clocks. We then present a new policy that we call Longest in Queues (LIQ), which gives priority to the packet that has been waiting the longest in edge queues. This policy is universally stable and, if clocks maintain constant differences, the bounds do not depend on them. To finish, we provide with simulation results that compare the behavior of some of these protocols in a network with stochastic injection of packets.

Novel Techniques to Speed Up the Computation of the Automorphism Group of a Graph

Graph automorphism (GA) is a classical problem, in which the objective is to compute the automorphism group of an input graph. Most GA algorithms explore a search tree using the individualization-refinement procedure. Four novel techniques are proposed which increase the performance of any algorithm of this type by reducing the depth of the search tree and by effectively pruning it. We formally prove that a GA algorithm that uses these techniques correctly computes the automorphism group of an input graph. Then, we describe how these techniques have been incorporated into the GA algorithm conauto, as conauto-2.03, with at most an additive polynomial increase in its asymptotic time complexity. Using a benchmark of different graph families, we have evaluated the impact of these techniques on the size of the search tree, observing a significant reduction both when they are applied individually and when all of them are applied together. This is also reflected in a reduction of the running time, which is substantial for some graph families. Finally, we have compared the search tree size of conauto-2.03 against those of other popular GA algorithms, observing that, in most cases, conauto explores less nodes than these algorithms.