Better gap-hamming lower bounds via better round elimination

Authors:
Joshua Brody;Amit Chakrabarti;Oded Regev;Thomas Vidick;Ronald De Wolf
Affiliations:
Department of Computer Science, Dartmouth College, Hanover, NH;Department of Computer Science, Dartmouth College, Hanover, NH;Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel;Department of Computer Science, UC Berkeley;CWI Amsterdam
Venue:
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Year:
2010

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Abstract

Gap Hamming Distance is a well-studied problem in communication complexity, in which Alice and Bob have to decide whether the Hamming distance between their respective n-bit inputs is less than n/2 -√n or greater than n/2 + √n. We show that every k-round bounded-error communication protocol for this problem sends a message of at least Ω(n/(k2 log k)) bits. This lower bound has an exponentially better dependence on the number of rounds than the previous best bound, due to Brody and Chakrabarti. Our communication lower bound implies strong space lower bounds on algorithms for a number of data stream computations, such as approximating the number of distinct elements in a stream.