A tight analysis and near-optimal instances of the algorithm of Anderson and Woll

  • Authors:
  • Grzegorz Malewicz

  • Affiliations:
  • Department of Computer Science, University of Alabama, 116 Houser Hall, Tuscaloosa, AL

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2004

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Abstract

This paper shows an asymptotically tight analysis of the Certified Write-All algorithm called AWT that was introduced by Anderson and Woll, SIAM J. Comput. 26 (1997) 1277, and a method for creating near-optimal instances of the algorithm. This algorithm is the best known deterministic algorithm that can be used to simulate n synchronous parallel processors on n asynchronous processors. The algorithm is instantiated with q permutations on {1,...,q}, where q can be chosen from a wide range of values. When implementing a simulation on a specific parallel system with n processors, one would like to select the best possible value of q and the best possible q permutations, in order to maximize the efficiency of the simulation.This paper shows that work complexity of any instance of AWT is Θ (q2/C ċ n1+logq(C/q)), where q is the number of permutations selected, and C is a value related to their combinatorial properties. The choice of q turns out to be critical for obtaining an instance of the AWT algorithm with near-optimal work. For any ε 0, and any large enough n, work of any instance of the algorithm must be at least n1+(1+n)√2 ln ln n/ln n. Under certain conditions, however, that q is about e√1/2 ln n ln ln n and for infinitely many large enough n, this lower bound can be nearly attained by instances of the algorithm that use certain q permutations and have work at most n1+(1+n)√2 ln ln n/ln n. The paper also shows a penalty for not selecting q well. When q is significantly away from e√1/2 ln n ln ln n, then work of any instance of the algorithm with this displaced q must be considerably higher than otherwise.