Longest Alignment with Edits in Data Streams

Analyzing patterns in data streams generated by network traffic, sensor networks, or satellite feeds is a challenge for systems in which the available storage is limited. In addition, real data is noisy, which makes designing data stream algorithms even more challenging. Motivated by such challenges, we study algorithms for detecting the similarity of two data streams that can be read in sync. Two strings $S, T\in \Sigma^n$ form a $d$-near-alignment if the distance between them in some given metric is at most $d$. We study the problem of identifying a longest substring of $S$ and $T$ that forms a $d$-near-alignment under the edit distance, in the simultaneous streaming model. In this model, symbols of strings $S$ and $T$ are streamed at the same time, and the amount of available processing space is sublinear in the length of the strings. We give several algorithms, including an exact one-pass algorithm that uses $\mathcal{O}(d^2+d\log n)$ bits of space. We couple these results with comparable lower bounds.