Improved Circular k-Mismatch Sketches

The shift distance 𝗌𝗁(S_1,S_2) between two strings S_1 and S_2 of the same length is defined as the minimum Hamming distance between S_1 and any rotation (cyclic shift) of S_2. We study the problem of sketching the shift distance, which is the following communication complexity problem: Strings S_1 and S_2 of length n are given to two identical players (encoders), who independently compute sketches (summaries) 𝚜𝚔(S_1) and 𝚜𝚔(S_2), respectively, so that upon receiving the two sketches, a third player (decoder) is able to compute (or approximate) 𝗌𝗁(S_1,S_2) with high probability. This paper primarily focuses on the more general k-mismatch version of the problem, where the decoder is allowed to declare a failure if 𝗌𝗁(S_1,S_2)>k, where k is a parameter known to all parties. Andoni et al. (STOC'13) introduced exact circular k-mismatch sketches of size O(k+D(n)), where D(n) is the number of divisors of n. Andoni et al. also showed that their sketch size is optimal in the class of linear homomorphic sketches. We circumvent this lower bound by designing a (non-linear) exact circular k-mismatch sketch of size O(k); this size matches communication-complexity lower bounds. We also design (1±ε)-approximate circular k-mismatch sketch of size O(min(ε^-2√(k), ε^-1.5√(n))), which improves upon an O(ε^-2√(n))-size sketch of Crouch and McGregor (APPROX'11).