A randomized algorithm that sorts on an N node network with constant valence in O(log N) time is given. More particularly, the algorithm sorts N items on an N-node cube-connected cycles graph, and, for some constant k, for all large enough a, it terminates within ka log N time with probability at least 1 - N-a.

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Computational Aspects of Vlsi

A logarithmic time sort for linear size networks

The authors develop an algebraic framework that exposes the structural kinship among the deBruijn, shujCfle-exchange, butterjy, and cube-connected cycles networks and illustrate algorithmic benefits that ensue from the exposed relationships. The framework builds on two alge- braically specified genres of graphs: A group action graph (GAG, for short) is given by a set V of vertices and a set H of permutations of V: For each v E V and each r EII, there is an arc labeled r from vertex v to vertex v. A Cayley graph is a GAG (V, H), where V is the group Gr(H) generated by H and where each r 6 H acts on each g 6 Gr(H) by right multiplication. The graphs (Gr(H), H) and (V, H) are called associated graphs. It is shown that every GAG is a quotient graph of its associated Cayley graph. By applying such general results, the authors determine the following: The butterfly network (a Cayley graph) and the deBruijn network (a GAG) are associated graphs. The cube-connected cycles network (a Cayley graph) and the shuffle-exchange network (a GAG) are associated graphs. The order-n instance of both the butterfly and the cube-connected cycles share the same underlying group, but have slightly different generator sets II. By analyzing these algebraic results, it is delimited, for any Cayley graph G and associated GAG 7-/, a family of "leveled" algorithms which run as efficiently on T/ as they do on (the much larger) G. Further analysis of the results yields new, surprisingly efficient simulations by the shuffle-oriented networks (the shuffle-exchange and deBruijn networks) of like-sized butterfly-oriented networks (the butterfly and cube-connected cycles networks): An N-vertex butterfly-oriented network can be simulated by the smallest shuffle-oriented network that is big enough to hold it with slowdown O(log log N). This simulation is exponentially faster than the anticipated logarithmic slowdown. The mappings that underlie the simulation can be computed in linear time; and they afford one an algorithmic tech- nique for translating any program developed for a butterfly-oriented architecture into an equivalent program for a shuffle-oriented architecture, the latter program incurring only the indicated slowdown factor.

background or methods

Group action graphs and parallel architectures

In this paper, we prove tight upper and lower bounds on the number or processors, information transfer, wire area and time needed to sort N numbers in a bounded-degree fixed-connection network. Our most important new results are: 1) the construction of an O(N))-node bounded-degree network capable of sorting N numbers in O(log N) word steps, 2) a proof that any network capable of sorting N(7 log N)-bit numbers in T bit-steps requires area A where AT2≥ Ω(N2log2N), and 3) the construction of a “small-constant-factor” bounded-degree network that sorts N θ(log N)-bit numbers in T = θ(log N) bit steps with A = θ(N2) area.

Tight bounds on the complexity of parallel sorting

This book is devoted to the investigation of a special topic in theoretical computer science - communication complexity as an abstract measure of the complexity of computing problems. Its main aim is to show how the theoretical study of communication complexity can be useful in the process of designing effective parallel algorithms. The author shows how to get important information about the parallel complexity (parallel time, the number of processors, the descriptional complexity of the topology of the parallel architecture) of specific computing problems from knowledge of their communication complexity. The book is written as a textbook for undergraduate and graduate students and provides a careful explanation of the subject as well as motivation for further research.

Communication Complexity and Parallel Computing

We introduce a new interconnection network for large-scale distributed memory multiprocessors called the hierarchical cubic network (HCN). We establish that the number of routing steps needed by several data parallel applications running on a HCN-based system and a hypercube-based system are about identical. Further, hypercube connections can be emulated on the HCN in constant time. Simulation of uniform and localized traffic patterns reveal that the normalized average internode distances in a HCN are better than in a comparable hypercube. Additionally, the HCN also has about three-fourths the diameter of a comparable hypercube, although it uses about half as many links per node-a fact that has positive ramifications on the implementation of HCN-connected systems. >

Computational Aspects of Vlsi

Citations

A logarithmic time sort for linear size networks

Group action graphs and parallel architectures

Tight bounds on the complexity of parallel sorting

Communication Complexity and Parallel Computing

Hierarchical cubic networks

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