Simple Low-Overhead Communication-Efficient String Reconciliation and Edit Distance

Suppose two parties, Alice and Bob, hold long character strings, $X$ and $Y$, respectively, and they are interested in determining how similar $X$ and $Y$ are. {Moreover, they want to exchange the strings with cost proportional to their degree of dissimilarity.} Such problems arise, for example, in database and file system synchronization operations, as well as in DNA sequence comparisons. Since the strings are long, we are interested in methods that are communication-efficient and have low overhead in terms of the computations that Alice and Bob must perform, when the strings are similar enough. In this paper, we provide simple low-overhead communication-efficient algorithms for such string reconciliation and edit distance problems. In the general case, %where the only assumption we make is that we have an upper bound, $k$, on the edit distance between $X$ and $Y$, we show how to determine the edit distance $k$ between $X$ and~$Y$ using only $O(k^2\log n)$ bits of communication and optimal $O(n)$ time overhead, with high probability. For specialized cases, such as typical English text or DNA sequences, where we can make additional well-justified assumptions about the distribution of the input strings, we show how to achieve possibly better bounds, such as $O(k\log^3 n)$ bits of communication.