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

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 Ω(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 αn lg n lg lg n , for some suitable 0 < α < 1. In particular we prove that if space is O(n1− ), then time is Ω(n lg n lg lg n). ∗E-mail: [email protected]. Part of this work was done while the author was visiting University of Toronto. †Basic Research in Computer Science, Centre of the Danish National Research Foundation.