Dense Matrix Algorithms -- Chapter 8 Introduction

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4 ScalabilityAnalysis • The parallel overhead is: T o = pT p – W • That is:, T o = t s p log p + t w np • To determine the isoefficiency function, we use the equation W = KT o where K = E/(1–E) and E is constant • We try to bound and express W in terms of p in terms of constants E, t w and t s – Use the larger term t w np in T o to show that W = K 2 t w2 p 2 or W = Θ(p 2 ) – Using the degree of concurrency: » Maximum number of processors p is n: that is, p = O(n) » That is, n = Ω(p), n 2 = Ω(p 2 ), and thus W = Ω(p 2 ) » This places a lower bound on the isoefficiency function which is the same as the upper bound • Thus, the isoefficiency function for 1-D row partition matrix-vector multiply is Θ(p 2 ) 5/6/2003 densematrix 7 2-D Partitioned Algorithm For <strong>Matrix</strong>-Vector Multiplication • We begin by assuming n 2 processors arranged in an n×n 2-d grid • A is assumed to be distributed with one element per processor • x is assumed distributed as one element per processor in the last column processors of the grid • Algorithm: • Distribute the elements of x from the last processor in each row to the diagonal processor in each row • Perform a simultaneous one-to-all broadcast from the diagonal processors to the processors in each column • Each processor performs one element of A by one element of x multiplication (the elements each processor owns) • Finally, a sum reduction along the rows to the last processor in the row is performed to compute each component of y 5/6/2003 densematrix 8
5 Diagram Of The Computational And Communication Process For 2-D Row Partition <strong>Matrix</strong>-Vector Multiplication 5/6/2003 densematrix 9 Parallel Runtime Of 2-D Partition Algorithm • The three communication steps • The first (from last column to diagonal) takes Θ(1) • The second and third (broadcast and reduction) each take Θ(log n) -- see Table 4.1 with m = 1 • The computation takes Θ(1) • Thus, the parallel runtime is: Θ(log n) • The cost (process-time product) is: Θ(n 2 log n) • Thus, the algorithm is not cost optimal 5/6/2003 densematrix 10
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Page 7 and 8: 7 Determining The Cost-Optimal Cons
Page 9 and 10: 9 Performance And Analysis for the
Page 11 and 12: 11 The DNS Algorithm • Partitions
Page 13: 13 The DNS Algorithm Continued 5/6/

Dense Matrix Algorithms -- Chapter 8 Introduction

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