Bounded branching process and and/or tree evaluation

  • Authors:

  • Richard M. Karp;Yanjun Zhang

  • Affiliations:

  • Computer Science Division, University of California, Berkeley, CA 94720 and International Computer Science Institute, Berkeley, CA 94704;Department of Computer Science and Engineering, Southern Methodist University, Dallas, TX 75275

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 1995

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Abstract

We study the tail distribution of supercritical branching processes for which the number of offspring of an element is bounded. Given a supercritical branching process {Zn} ***image*** with a bounded offspring distribution, we derive a tight bound, decaying super‐exponentially fast as c increases, on the probability Pr[Zn cE(Zn)], and a similar bound on the probability Pr[Zn ≤ E(Zn)/c] under the assumption that each element generates at least two offspring. As an application, we observe that the execution of a canonical algorithm for evaluating uniform AND/OR trees in certain probabilistic models can be viewed as a two‐type supercritical branching process with bounded offspring, and show that the execution time of this algorithm is likely to concentrate around its expectation, with a standard deviation of the same order as the expectation. © 1995 Wiley Periodicals, Inc.