  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 $\tilde O(k^3)$.
