The Communication Complexity of the Hamming Distance Problem

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, improving previous protocols of O(d^2) bits (A. C.-C. Yao, Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pp. 77-81, 2003), and O(d\log n) bits (D. Gavinsky, J. Kempe, and R. de Wolf, quant-ph/0411051, 2004).

8 pages, v3, updated reference. to appear in Information Processing Letters, 2006

References in corpus (2)

• Quantum Communication Cannot Simulate a Public Coin
• Lower bounds on the Deterministic and Quantum Communication Complexity of Hamming Distance

Cited by in corpus (13)

• Tight bounds on the randomized communication complexity of symmetric XOR functions in one-way and SMP models
• Communication Complexities of XOR functions
• Quantum sketching protocols for Hamming distance and beyond
• Approximate Hamming distance in a stream
• Quantum communication complexity of block-composed functions
• Near log-convexity of measured heat in (discrete) time and consequences
• The k-mismatch problem revisited
• Space Lower Bounds for Online Pattern Matching
• Linear Sketching over ${\mathbb{F}}_2$
• Approximate ${\mathbb{F}}_2$-Sketching of Valuation Functions
• Time Bounds for Streaming Problems
• On The Communication Complexity of Linear Algebraic Problems in the Message Passing Model
• Streaming algorithms for recognizing nearly well-parenthesized expressions