On Ajtai’s Lower Bound Technique for R-way Branching Programs and the Hamming Distance Problem

Abstract

In this report we study the proof employed by Miklos Ajtai
[Determinism versus Non-Determinism for Linear Time RAMs
with Memory Restrictions, 31st Symposium on Theory of
Computation (STOC), 1999] when proving a non-trivial lower bound
in a general model of computation for the Hamming Distance
problem: given n elements: decide whether any two of them have
"small" Hamming distance. Specifically, Ajtai was able to show
that any R-way branching program deciding this problem using
time O(n) must use space Omega(n lg n).
We generalize Ajtai's original proof allowing us to prove a
time-space trade-off for deciding the Hamming Distance problem
in the R-way branching program model for time between n
and alpha n lg n / lg lg n, for some suitable 0 < alpha < 1. In particular we prove
that if space is O(n^(1−epsilon)), then time is Omega(n lg n / lg lg n).