Pebbles and branching programs for tree evaluation

Stephen Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, Rahul Santhanam

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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(k h) 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.

Original languageEnglish (US)
Article number4
JournalACM Transactions on Computation Theory
Issue number2
StatePublished - Jan 2012
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computational Theory and Mathematics


  • Branching programs
  • Log space
  • LogDCFL
  • Lower bounds


Dive into the research topics of 'Pebbles and branching programs for tree evaluation'. Together they form a unique fingerprint.

Cite this