A Multi-Round Communication Lower Bound for Gap Hamming and Some Consequences

—The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their n-bit input strings is large (i.e., at least n/2 + √ n) or small (i.e., at most n/2 − √ n); they do not care if it is neither large nor small. This Θ( √ n) gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an Ω(n) lower bound on this problem was known only in the one-way setting. We prove an Ω(n) lower bound for randomized protocols that use any constant number of rounds. As a consequence we conclude, for instance, that εapproximately counting the number of distinct elements in a data stream requires Ω(1/ε2 ) space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a...

• Type: Conference
• Year: 2009
• Where: COCO
• Authors: Joshua Brody, Amit Chakrabarti
• Keywords: COCO 2009, Data Stream, Lower Bound, Space Lower Bounds, Theoretical Computer Science