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Pebbles and Branching Programs for Tree Evaluation

Published:01 January 2012Publication History
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Abstract

We introduce the tree evaluation problem, show that it is in LogDCFL (and hence in P), and study its branching program complexity in the hope of eventually proving a superlogarithmic space lower bound. The input to the problem is a rooted, balanced d-ary tree of height h, whose internal nodes are labeled with d-ary functions on [k] = {1,..., k}, and whose leaves are labeled with elements of [k]. Each node obtains a value in [k] equal to its d-ary function applied to the values of its d children. The output is the value of the root. We show that the standard black pebbling algorithm applied to the binary tree of height h yields a deterministic k-way branching program with O(kh) states solving this problem, and we prove that this upper bound is tight for h = 2 and h = 3. We introduce a simple semantic restriction called thrifty on k-way branching programs solving tree evaluation problems and show that the same state bound of Θ(kh) is tight for all h ≥ 2 for deterministic thrifty programs. We introduce fractional pebbling for trees and show that this yields nondeterministic thrifty programs with Θ(kh/2+1) states solving the Boolean problem “determine whether the root has value 1”, and prove that this bound is tight for h = 2,3,4. We also prove that this same bound is tight for unrestricted nondeterministic k-way branching programs solving the Boolean problem for h = 2,3.

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    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 3, Issue 2
      January 2012
      99 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/2077336
      Issue’s Table of Contents

      Copyright © 2012 ACM

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 1 January 2012
      • Accepted: 1 May 2011
      • Revised: 1 March 2011
      • Received: 1 May 2010
      Published in toct Volume 3, Issue 2

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