We provide improved upper bounds for the simultaneous sketching complexity of edit distance. Consider two parties, Alice with input $x\in\Sigma^n$ and Bob with input $y\in\Sigma^n$, that share public randomness and are given a promise that the edit distance $\mathsf{ed}(x,y)$ 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 $\mathsf{ed}(x,y)$ precisely as well as a sequence of $\mathsf{ed}(x,y)$ 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 Kouck\'y (STOC 2016), achieves a maximum message length of $\tilde O(k^8)$ bits, where $\tilde O(\cdot)$ hides $\mathrm{poly}(\log n)$ 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 $\tilde O(k^3)$.
翻译:我们为同时完成编辑距离的复杂度提供更好的上限值。 爱丽丝必须给第三方查理发送信息 $xx$, 鲍勃必须给第三方发送 $sx$, 他不知道输入但共享相同的公共随机性, 并且知道$k$。 Charlie必须准确输出$\mathsf{Sigma}( y) 美元, 并承诺将两个字符串之间的编辑距离 $\ mathsfsf{ed} (x,y) 最多某些给定值 $。 爱丽丝必须给第三方查理发送信息$sx$sx$, 鲍勃必须给第三方发送$sy$sy, 他不知道输入但共享相同的公共随机性, 也知道$k$。 Charlie必须准确输出$s\masfesf{( x,y) 以及将美元xxxxxxxxxxxxxxxxxx。 目标是将发送的信息的长度最小化 。 Belazzougui和Zh( FOC 2016) 的随机行走法方法将显示 $xlexlexlexlexxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx