Let G = (V; E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of Aingworth, Chekuri and Motwani, we describe g) time algorithm APASP 2 for computing all distances in G with an additive one-sided error of at most 2. The algorithm APASP 2 is simple, easy to implement, and faster than the fastest known matrix multiplication algorithm. Furthermore, for every even k ? 2, we describe an g) time algorithm APASP k for computing all distances in G with an additive one-sided error of at most k.

Recent approaches to large vocabulary decoding with finite state graphs have focused on the use of state minimization algorithms to produce relatively compact graphs. This paper extends the finite state approach by developing complementary arc-minimization techniques. The use of these techniques in concert with state minimization allows us to statically compile decoding graphs in which the acoustic models utilize a full word of cross-word context. This is in significant contrast to typical systems which use only a single phone. We show that the particular arc-minimization problem that arises is in fact an NP-complete combinatorial optimization problem, and describe the reduction from 3-SAT. We present experimental results that illustrate the moderate sizes and runtimes of graphs for the Switchboard task. 1.

We demonstrate how to find a perfect matching in a bipartite graph containing {\sqrt n \sigma^3} disjoint perfect matchings in time O(\sqrt n m/\sigma ).

Complex graphs, ones containing thousands of nodes of high degree, are difficult to visualize. Displaying all of the nodes and edges of these graphs can create an incomprehensible cluttered output. This paper presents a simplification algorithm that may be applied to a complex graph in order to produce a controlled thinning of the graph. Using importance metrics, the simplification process removes nodes from the graph, leaving the central structure for visualization and evaluation. The simplification algorithm consists of two steps, calculation of the importance metrics and pruning. Several metrics based on various topological graph properties are described. The metrics are then used in a pruning process to simplify the graph. Nodes, along with their corresponding edges, are removed from the graph, while maintaining the graph’s overall connectivity. This simplified graph provides a cleaner, more meaningful visual representation of the graph’s structure; thus aiding the analysis of the graph’s underlying data. 1

Research article On the origin of distribution patterns of motifs in biological networks