Implicit Decomposition for Write-Efficient Connectivity Algorithms

The future of main memory appears to lie in the direction of new technologies that provide strong capacity-to-performance ratios, but have write operations that are much more expensive than reads in terms of latency, bandwidth, and energy. Motivated by this trend, we propose sequential and parallel algorithms to solve graph connectivity problems using significantly fewer writes than conventional algorithms. Our primary algorithmic tool is the construction of an $o(n)$-sized "implicit decomposition" of a bounded-degree graph $G$ on $n$ nodes, which combined with read-only access to $G$ enables fast answers to connectivity and biconnectivity queries on $G$. The construction breaks the linear-write "barrier", resulting in costs that are asymptotically lower than conventional algorithms while adding only a modest cost to querying time. For general non-sparse graphs on $m$ edges, we also provide the first $o(m)$ writes and $O(m)$ operations parallel algorithms for connectivity and biconnectivity. These algorithms provide insight into how applications can efficiently process computations on large graphs in systems with read-write asymmetry

Approximating the Minimum Equivalent Digraph

The MEG (minimum equivalent graph) problem is, given a directed graph, to find a small subset of the edges that maintains all reachability relations between nodes. The problem is NP-hard. This paper gives an approximation algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its analysis are based on the simple idea of contracting long cycles. (This result is strengthened slightly in ``On strongly connected digraphs with bounded cycle length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local improvement'' algorithm, showing that its performance guarantee is 1.75.Comment: conference version in ACM-SIAM Symposium on Discrete Algorithms (1994

An Experimental Study of Parallel Biconnected Components Algorithms on Symmetric Multiprocessors (SMPs)

We present an experimental study of parallel biconnected components algorithms employing several fundamental parallel primitives, e.g., prefix sum, list ranking, sorting, connectivity, spanning tree, and tree computations. Previous experimental studies of these primitives demonstrate reasonable parallel speedups. However, when these algorithms are used as subroutines to solve higher-level problems, there are two factors that hinder fast parallel implementations. One is parallel overhead, i.e., the large constant factors hidden in the asymptotic bounds; the other is the discrepancy among the data structures used in the primitives that brings non-negligible conversion cost. We present various optimization techniques and a new parallel algorithm that significantly improve the performance of finding biconnected components of a graph on symmetric multiprocessors (SMPs). Finding biconnected components has application in fault-tolerant network design, and is also used in graph planarity testing. Our parallel implementation achieves speedups up to 4 using 12 processors on a Sun E4500 for large, sparse graphs, and the source code is freely-available at our web site http://www.ece.unm.edu/~dbader.This work was supported in part by NSF Grants CAREER ACI-00-93039, ITR ACI-00-81404, DEB-99- 10123, ITR EIA-01-21377, Biocomplexity DEB-01-20709, DBI-0420513, ITR EF/BIO 03-31654 and DBI-04- 20513; and DARPA Contract NBCH30390004

Solving the Maximally Balanced Connected Partition Problem in Graphs by Using Genetic Algorithm

This paper exposes a research of the NP-hard Maximally Balanced Connected Partition problem (MBCP). The proposed solution comprises of a genetic algorithm (GA) that uses: binary representation, fine-grained tournament selection, one-point crossover, simple mutation with frozen genes and caching technique. In cases of unconnected partitions, penalty functions are successfully applied in order to obtain the feasible individuals. The effectiveness of presented approach is demonstrated on the grid graph instances and on random instances with up to 300 vertices and 2 000 edges

An implementation of a generic memetic algorithm for the edge biconnectivity augmentation problem

In this paper we present an implementation of a generic memetic algorithm for the edge bi-connectivity augmentation problem --the problem of augmenting a given graph by a cheapest possible set of additional edges in order to make the graph edge bi-connected. This problem is known for its applications to communication network design --the extension of an existing communication network to become robust against single link failures-- as well as in VLSI floor planning. We provide a C++ implementation of a generic memetic algorithm for the problem, as a good alternative for approximately solving it. We use known benchmarks in the literature for the problem as to experimentally evaluate how good the generic memetic algorithm works for the problem.Postprint (published version

Approximating minimum cost connectivity problems

We survey approximation algorithms of connectivity problems. The survey presented describing various techniques. In the talk the following techniques and results are presented. 1)Outconnectivity: Its well known that there exists a polynomial time algorithm to solve the problems of finding an edge k-outconnected from r subgraph [EDMONDS] and a vertex k-outconnectivity subgraph from r [Frank-Tardos] . We show how to use this to obtain a ratio 2 approximation for the min cost edge k-connectivity problem. 2)The critical cycle theorem of Mader: We state a fundamental theorem of Mader and use it to provide a 1+(k-1)/n ratio approximation for the min cost vertex k-connected subgraph, in the metric case. We also show results for the min power vertex k-connected problem using this lemma. We show that the min power is equivalent to the min-cost case with respect to approximation. 3)Laminarity and uncrossing: We use the well known laminarity of a BFS solution and show a simple new proof due to Ravi et al for Jain\u27s 2 approximation for Steiner network