We show that in the document exchange problem, where Alice holds xβ{0,1}n and Bob holds yβ{0,1}n, Alice can send Bob a message of
size O(K(log2K+logn)) bits such that Bob can recover x using the
message and his input y if the edit distance between x and y is no more
than K, and output "error" otherwise. Both the encoding and decoding can be
done in time O~(n+poly(K)). This result significantly
improves the previous communication bounds under polynomial encoding/decoding
time. We also show that in the referee model, where Alice and Bob hold x and
y respectively, they can compute sketches of x and y of sizes
poly(Klogn) bits (the encoding), and send to the referee, who can
then compute the edit distance between x and y together with all the edit
operations if the edit distance is no more than K, and output "error"
otherwise (the decoding). To the best of our knowledge, this is the first
result for sketching edit distance using poly(Klogn) bits.
Moreover, the encoding phase of our sketching algorithm can be performed by
scanning the input string in one pass. Thus our sketching algorithm also
implies the first streaming algorithm for computing edit distance and all the
edits exactly using poly(Klogn) bits of space.Comment: Full version of an article to be presented at the 57th Annual IEEE
Symposium on Foundations of Computer Science (FOCS 2016