EREW Parallel Random Access Machine Research Articles

Accounting for memory bank contention and delay in high-bandwidth multiprocessors

For years, the computation rate of processors has been much faster than the access rate of memory banks, and this divergence in speeds has been constantly increasing in recent years. As a result, several shared-memory multiprocessors consist of more memory banks than processors. The object of this paper is to provide a simple model (with only a few parameters) for the design and analysis of irregular parallel algorithms that will give a reasonable characterization of performance on such machines. For this purpose, we extend Valiant's bulk-synchronous parallel (BSP) model with two parameters: a parameter for memory bank delay, the minimum time for servicing requests at a bank, and a parameter for memory bank expansion, the ratio of the number of banks to the number of processors. We call this model the (d, x)BSP. We show experimentally that the (d, x)-BSP captures the impact of bank contention and delay on the CRAY C90 and J90 for irregular access patterns, without modeling machine-specific details of these machines. The model has clarified the performance characteristics of several unstructured algorithms on the CRAY C90 and J90, and allowed us to explore tradeoffs and optimizations for these algorithms. In addition to modeling individual algorithms directly, we also consider the use of the (d, x)-BSP as a bridging model for emulating a very high-level abstract model, the Parallel Random Access Machine (PRAM). We provide matching upper and lower bounds for emulating the EREW and QRQW PRAMs on the (d, X)-BSP.

2D and 3D Optimal Parallel Image Warping

Spatial image warping is useful for image processing and graphics. In this paper, we present concurrent-read-exclusive-write (CREW) and exclusive-read-exclusive-write (EREW) parallel-random-access-machine (PRAM) algorithms that achieve O(1) asymptotic run time. The significant result is the creative processor assignment that results in an EREW PRAM algorithm. The forward algorithm calculates any nonscaling affine transform including arbitrary skewings, translations, and rotations. The EREW algorithm is the most efficient in practice, and the MasPar MP-1 with 16K processors rotates a 4-million-element image in under a second and a 2-million-element volume in one-half of a second. This high performance allows interactive viewing of volumes from arbitrary viewpoints and illustrates linear speedup. This practical efficiency is analyzed and illustrated by using a bridging model of computation. We develop the mixed cost communication machine (MCCM) to quantify the communication costs and correlate these costs to the MasPar MP-1. The forward algorithm has provable N = 1 congestion on the MCCM, while the backward algorithm has congestion N > 1 which varies with the transform. There are also important quality advantages using our direct warping techniques; empirical measurements are given to provide comparisons to multipass warps.

PARALLEL ALGORITHMS FOR SOME DOMINANCE PROBLEMS BASED ON THE PRAM MODEL

Two parallel geometric algorithms based on the idea of point domination are presented. The first algorithm solves the d-dimensional isothetic rectangles intersection counting problem of input size N/2d, where d>1 and N is a multiple of 2d, in O( log d−1 N) time and O(N log N) space. The second algorithm solves the direct dominance reporting problem for a set of N points in the plane in O( log N+J) time and O(N log N) space, where J denotes the maximum of the number of direct dominances reported by any single point in the set. Both algorithms make use of the EREW PRAM (Exclusive Read Exclusive Write Parallel Random Access Machine) consisting of O(N) processors as the computational model.

PRAM MEMORY ALLOCATION AND INITIALIZATION

Two techniques for managing memory on a parallel random access machine (PRAM) are presented. One is a scheme for an n/log n processor EREW PRAM that dynamically allocates and deallocates up to n records of the same size in O(log n) time. The other is a simulation of a PRAM with initialized memory by one with uninitialized memory. A CREW PRAM variant of the technique justifies the assumption that memory can be assumed to be appropriately initialized with no asymptotic increase in time but a factor of n increase in space. An EREW PRAM solution incurs a factor of O(log n) increase in time but only a constant factor increase in space.

An NC recognition algorithm for cographs

The class of cographs, or complement-reducible graphs, arises naturally in many different areas of applied mathematics and computer science. In this paper we present two new characterizations of the cographs and show how to use them for the purpose of producing a parallel recognition algorithm for this class of graphs. In addition, if the graph under investigation turns out to be a cograph, our algorithm constructs its corresponding cotree. With an arbitrary graph G with n vertices and m edges as input, the proposed algorithm runs in O(log n) time using O(( n 2 + mn)/log n) processors on an EREW parallel random access machine.

A fast parallel algorithm to recognize partitionable graphs

We define a class of graphs which generalizes the well-known class of cographs or complement-reducible graphs and show that the graphs in our class can be recognized fast in parallel. More precisely, with an arbitrary graph G with n vertices and m edges as input, the proposed recognition algorithm runs in O(log n) time using O(( n 2/log n)+ mn) processors on an EREW parallel random access machine.

Parallel binary search

Two arrays of numbers sorted in nondecreasing order are given: an array A of size n and an array B of size m, where n<m. It is required to determine, for every element of A, the smallest element of B (if one exists) that is larger than or equal to it. It is shown how to solve this problem on the EREW PRAM (exclusive-read exclusive-write parallel random-access machine) in O(logm logn/log log m) time using n processors. The solution is then extended to the case in which fewer than n processors are available. This yields an EREW PRAM algorithm for the problem whose cost is O(n log m, which is O(m)) for n<or=m/log m. It is shown how the solution obtained leads to an improved parallel merging algorithm.<<ETX>>