The shift distance $mathsf{sh}(S_1,S_2)$ 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) $mathtt{sk}(S_1)$ and $mathtt{sk}(S_2)$, respectively, so that upon receiving the two sketches, a third player (decoder) is able to compute (or approximate) $mathsf{sh}(S_1,S_2)$ 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 $mathsf{sh}(S_1,S_2)>k$, where $k$ is a parameter known to all parties. Andoni et al. (STOC13) introduced exact circular $k$-mismatch sketches of size $widetilde{O}(k+D(n))$, 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 $widetilde{O}(k)$; this size matches communication-complexity lower bounds. We also design $(1pm varepsilon)$-approximate circular $k$-mismatch sketch of size $widetilde{O}(min(varepsilon^{-2}sqrt{k}, varepsilon^{-1.5}sqrt{n}))$, which improves upon an $widetilde{O}(varepsilon^{-2}sqrt{n})$-size sketch of Crouch and McGregor (APPROX11).
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