An Improved Sketching Bound for Edit Distance
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We provide improved upper bounds for the simultaneous sketching complexity of edit distance. Consider two parties, Alice with input x∈Σ^n and Bob with input y∈Σ^n, that share public randomness and are given a promise that the edit distance 𝖾𝖽(x,y) between their two strings is at most some given value k. Alice must send a message sx and Bob must send sy to a third party Charlie, who does not know the inputs but shares the same public randomness and also knows k. Charlie must output 𝖾𝖽(x,y) precisely as well as a sequence of 𝖾𝖽(x,y) edits required to transform x into y. The goal is to minimize the lengths |sx|, |sy| of the messages sent. The protocol of Belazzougui and Zhang (FOCS 2016), building upon the random walk method of Chakraborty, Goldenberg, and Koucký (STOC 2016), achieves a maximum message length of Õ(k^8) bits, where Õ(·) hides poly(log n) factors. In this work we build upon Belazzougui and Zhang's protocol and provide an improved analysis demonstrating that a slight modification of their construction achieves a bound of Õ(k^3).
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