The communication complexity of the Hamming distance problem

Authors:
Wei Huang;Yaoyun Shi;Shengyu Zhang;Yufan Zhu
Affiliations:
Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI;Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI;Computer Science Department, Princeton University, NJ;Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI
Venue:
Information Processing Letters
Year:
2006

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Abstract

We investigate the randomized and quantum communication complexity of the HAMMING DISTANCE problem, which is to determine if the Hamming distance between two n-bit strings is no less than a threshold d. We prove a quantum lower bound of Ω(d) qubits in the general interactive model with shared prior entanglement. We also construct a classical protocol of O(d log d) bits in the restricted Simultaneous Message Passing model with public random coins, improving previous protocols of O(d2) bits [A.C.-C. Yao, On the power of quantum fingerprinting, in: Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 77-81], and O(d log n) bits [D. Gavinsky, J. Kempe, R. de Wolf, Quantum communication cannot simulate a public coin, quant-ph/0411051, 2004].