46659759
Optimal Broadcast in All-Port Wormhole-Routed Hypercubes
We give an optimal algorithm that broadcasts on an n-dimensional hypercube in O(n/ log/sub 2/ (n+1)) routing steps with wormhole, e-cube routing and all-port communication. Previously, the best algorithm of P.K. McKinley and C. Trefftz (1993) requires [n/2] routing steps. We also give routing algorithms that achieve tight time bounds for n /spl les/7. >
10311678
Optimal Broadcast in All-Port Wormhole-Routed Hypercubes
We give an optimal algorithm that broadcasts on an n-dimensional hypercube in Theta(n/ log_{2}(n+ 1)) routing steps with wormhole, e-cube routing and all-port communication. Previously, the best algorithm of McKinley and Trefftz requires [n/2] routing steps. We also give routing algorithms that achieve tight time bounds for n leqslant 7.
38767612
Efficient Submesh Permutations in Wormhole-routed meshes
Abstract This paper studies how to concurrently permute related logical or physical submeshes in a d -dimensional n × … × n physical mesh via wormhole and dimension-ordered routing. Our objective is to minimize the congestion for realizing the permutations, while maximizing the number and dimensionality of permuted submeshes. We show that for d ⩽ 2 α + β , concurrent independent permutations of n β related physical submeshes, each of α dimensions, can be performed in two routing steps without congestion. If the permuted submeshes are logical ones, they can be permuted in one, instead of two, routing step. In addition, any shift operation along any axis of the logical mesh can be performed in the physical mesh without congestion. We also show that if all nodes know the permutation function, any permutation within a submesh of dimensions ⌊2( d − 1)/3⌋ can be realized in three routing steps without congestion.
28384403
Simple and Efficient Graph Compression Schemes for Dense and Complement Graphs
We present two graph compression schemes for solving problems on dense graphs and complement graphs. They compress a graph or its complement graph into two kinds of succinct representations based on adjacency intervals and adjacency integers, respectively. These two schemes complement each other for different ranges of density. Using these schemes, we develop optimal or near optimal algorithms for fundamental graph problems. In contrast to previous graph compression schemes, ours are simple and efficient for practical applications.
1437223
A Fast General Methodology for Information - Theoretically Optimal Encodings of Graphs
We propose a fast methodology for encoding graphs with informationtheoretically minimum numbers of bits. The methodology is applicable to general classes of graphs; this paper focuses on simple planar graphs. Specifically, a graph with property ? is called a ?-graph. If ? satisfies certain properties, then an n-node ?-graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most s(n)+o(s(n)) bits for any function s(n) = ?(n) so that there are at most 2s(n)+o(s(n)) distinct n-node ?-graphs. Examples of such ? include all conjunctions of the following sets of properties: (1) G is a planar graph or a plane graph; (2) G is directed or undirected; (3) G is triangulated, triconnected, biconnected, merely connected, or not required to be connected; and (4) G has at most l1 (respectively, l2) distinct node (respectively, edge) labels. These examples are novel applications of small cycle separators of planar graphs and settle several problems that have been open since Tutte's census series were published in 1960's.
30190247
A Fast General Methodology for Information-Theoretically Optimal Encodings of Graphs
We propose a fast methodology for encoding graphs with information-theoretically minimum numbers of bits. Specifically, a graph with property $\pi$ is called a {\em $\pi$-graph}. If $\pi$ satisfies certain properties, then an n-node m-edge $\pi$-graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most $\beta(n)+o(\beta(n))$ bits for any continuous superadditive function $\beta(n)$ so that there are at most $2^{\beta(n)+o(\beta(n))}$ distinct $n$-node $\pi$-graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such $\pi$ include all conjunctions over the following groups of properties: (1) G is a planar graph or a plane graph; (2) $G$ is directed or undirected; (3) $G$ is triangulated, triconnected, biconnected, merely connected, or not required to be connected; (4) the nodes of G are labeled with labels from $\{1,\ldots, \ell_1\}$ for $\ell_1\leq n$; (5) the edges of G are labeled with labels from $\{1,\ldots, \ell_2\}$ for $\ell_2\leq m$; and (6) each node (respectively, edge) of G has at most $\ell_3=O(1)$ self-loops (respectively, $\ell_4=O(1)$ multiple edges). Moreover, $\ell_3$ and $\ell_4$ are not required to be O(1) for the cases of $\pi$ being a plane triangulation. These examples are novel applications of small cycle separators of planar graphs and are the only nontrivial classes of graphs, other than rooted trees, with known polynomial-time information-theoretically optimal coding schemes.