Modular decomposition

Abstract: The problem of computing k-edge connected components (k-ECCs) of a graph G for a specific k is a fundamental graph problem and has been investigated recently. In this paper, we study the problemof ECC decomposition, which computes the k-ECCs of a graph G for all k values. ECC decomposition can be widely applied in a variety of applications such as graph-topology analysis, community detection, Steiner component search, and graph visualization. A straightforward solution for ECC decomposition is to apply the existing k-ECC computation algorithm to compute the k-ECCs for all k values. However, this solution is not applicable to large graphs for two challenging reasons. First, all existing k-ECC computation algorithms are highly memory intensive due to the complex data structures used in the algorithms. Second, the number of possible k values can be very large, resulting in a high computational cost when each k value is independently considered. In this paper, we address the above challenges, and study I/O efficient ECC decomposition via graph reduction. We introduce two elegant graph reduction operators which aim to reduce the size of the graph loaded in memory while preserving the connectivity information of a certain set of edges to be computed for a specific k. We also propose three novel I/O efficient algorithms, Bottom-Up, Top-Down, and Hybrid, that explore the k values in different orders to reduce the redundant computations between different k values. We analyze the I/O and memory costs for all proposed algorithms. In our experiments, we evaluate our algorithms using seven real large datasets with various graph properties, one of which contains 1.95 billion edges. The experimental results show that our proposed algorithms are scalable and efficient.

Abstract: Parallel cellular algorithms for recognizing adjacency and incidence matrices of several classes of graphs are given. These classes include cubic graphs, complete graphs, connected graphs, and trees.

Abstract: This paper introduces QFrag, a distributed system for graph search on top of bulk synchronous processing (BSP) systems such as MapReduce and Spark. Searching for patterns in graphs is an important and computationally complex problem. Most current distributed search systems scale to graphs that do not fit in main memory by partitioning the input graph. For analytical queries, however, this approach entails running expensive distributed joins on large intermediate data. In this paper we explore an alternative approach: replicating the input graph and running independent parallel instances of a sequential graph search algorithm. In principle, this approach leads us to an embarrassingly parallel problem, since workers can complete their tasks in parallel without coordination. However, the skew present in natural graphs makes this problem a deceitfully parallel one, i.e., an embarrassingly parallel problem with poor load balancing. We therefore introduce a task fragmentation technique that avoids stragglers but at the same time minimizes coordination. Our evaluation shows that QFrag outperforms BSP-based systems by orders of magnitude, and performs similar to asynchronous MPI-based systems on simple queries. Furthermore, it is able to run computationally complex analytical queries that other systems are unable to handle.