Recursive Hashing and One-Pass, One-Hash n-Gram Count Estimation

Many applications use sequences of n consecutive symbols (n-grams). We review n-gram hashing and prove that recursive hash families are pairwise independent at best. We prove that hashing by irreducible polynomials is pairwise independent whereas hashing by cyclic polynomials is quasi-pairwise independent: we make it pairwise independent by discarding n-1 bits. One application of hashing is to estimate the number of distinct n-grams, a view-size estimation problem. While view sizes can be estimated by sampling under statistical assumptions, we desire a statistically unassuming algorithm with universally valid accuracy bounds. Most related work has focused on repeatedly hashing the data, which is prohibitive for large data sources. We prove that a one-pass one-hash algorithm is sufficient for accurate estimates if the hashing is sufficiently independent. For example, we can improve by a factor of 2 the theoretical bounds on estimation accuracy by replacing pairwise independent hashing by 4-wise independent hashing. We show that recursive random hashing is sufficiently independent in practice. Maybe surprisingly, our experiments showed that hashing by cyclic polynomials, which is only quasi-pairwise independent, sometimes outperformed 3-wise independent hashing while being twice as fast. For comparison, we measured the time to obtain exact n-gram counts using suffix arrays and show that, while we used hardly any storage, we were an order of magnitude faster. The experiments used a large collection of English text from Project Gutenberg as well as synthetic data.