Abstract
In this paper we provide a new locally consistent decomposition of strings. Each string x is decomposed into blocks that can be described by grammars of size O(k) (using some amount of randomness).
If we take two strings x and y of edit distance at most k then their block decomposition uses the same number of grammars and the i-th grammar of x is the same as the i-th grammar of y except for at most k indexes i. The edit distance of x and y equals to the sum of edit distances of pairs of blocks where x and y differ.
Our decomposition can be used to design a sketch of size O(k^2) for edit distance, and also a rolling sketch for edit distance of size O(k^2). The rolling sketch allows to update the sketched string by appending a symbol or removing a symbol from the beginning of the string.