Block Heavy Hitters

We study a natural generalization of the heavy hitters problem in the streaming context. We term this generalization block heavy hitters and define it as follows. We are to stream over a matrix A, and report all rows that are heavy, where a row is heavy if its `1-norm is at least φ fraction of the `1 norm of the entire matrix A. In comparison, in the standard heavy hitters problem, we are required to report the matrix entries that are heavy. As is common in streaming, we solve the problem approximately: we return all rows with weight at least φ, but also possibly some other rows that have weight no less than (1− )φ. To solve the block heavy hitters problem, we show how to construct a linear sketch of A from which we can recover the heavy rows of A. The block heavy hitters problem has already found applications for other streaming problems. In particular, it is a crucial building block in a streaming algorithm of [AIK08] that constructs a small-size sketch for the Ulam metric, a metric on non-repetitive strings under the edit (Levenshtein) distance. We prove the following theorem. Let Mn,m be the set of real matrices A of size n by m, with entries from E = 1 nm · {0, 1, . . . nm}. For a matrix A, let Ai denote its i th row. Theorem 0.1. Fix some > 0, and n,m ≥ 1, and φ ∈ [0, 1]. There exists a randomized linear map (sketch) μ : Mn,m → {0, 1}s, where s = O( 1 5φ2 log n), such that the following holds. For a matrix A ∈ Mn,m, it is possible, given μ(A), to find a set W ⊂ [n] of rows such that, with probability at least 1− 1/n, we have: • for any i ∈W , ‖Ai‖1 ‖A‖1 ≥ (1− )φ and • if ‖Ai‖1 ‖A‖1 ≥ φ, then i ∈W . Moreover, μ can be of the form μ(A) = μ(ρ(A1), ρ(A2), . . . ρ(An)), where ρ : Em → Rk and μ′ : Rkn → {0, 1}s are randomized linear mappings. That is, the sketch μ is obtained by first sketching the rows of A (using the same function ρ) and then sketching those sketches. Our construction is inspired by the CountMin sketch of [CM05], and may be seen as a CountMin sketch on the projections of the rows of A. Proof. Construction of the sketch. We define the function ρ as an `1 projection into a space with k = O( 1 2 log n) dimensions, achieved through a standard Cauchy distribution projection.