Algorithmica

, Volume 36, Issue 1, pp 41–57 | Cite as

The Solution of Linear Probabilistic Recurrence Relations

  • Bazzi
  • Mitter
Article

Abstract

Linear probabilistic divide-and-conquer recurrence relations arise when analyzing the running time of divide-and-conquer randomized algorithms. We consider first the problem of finding the expected value of the random process T(x) , described as the output of a randomized recursive algorithm T . On input x , T generates a sample (h1,···,hk) from a given probability distribution on [0,1]k and recurses by returning g(x) + Σi=1kciT(hi x) until a constant is returned when x becomes less than a given number. Under some minor assumptions on the problem parameters, we present a closed-form asymptotic solution of the expected value of T(x) . We show that E[T(x)] = Θ( xp + xp∈t1x(g(u)/ up+1 ) du) , where p is the nonnegative unique solution of the equation Σi=1kciE[hip] = 1 . This generalizes the result in [1] where we considered the deterministic version of the recurrence. Then, following [2], we argue that the solution holds under a broad class of perturbations including floors and ceilings that usually accompany the recurrences that arise when analyzing randomized divide-and-conquer algorithms.

Keywords

Randomized algorithms, Divide-and-conquer algorithms, Recurrence relations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag New York Inc. 2003

Authors and Affiliations

  • Bazzi
    • 1
  • Mitter
    • 1
  1. 1.Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA 02139, louay@lids.mit.edu, mitter@lids.mit.edu.USA

Personalised recommendations