Linear-Space Substring Range Counting over Polylogarithmic Alphabets

Bille and Gørtz (2011) recently introduced the problem of substring range counting, for which we are asked to store compactly a string S of n characters with integer labels in [0, u], such that later, given an interval [a, b] and a pattern P of length m, we can quickly count the occurrences of P whose first characters’ labels are in [a, b]. They showed how to store S in O(n logn/ log logn) space and answer queries in O(m + log log u) time. We show that, if S is over an alphabet of size polylog(n), then we can achieve optimal linear space. Moreover, if u = npolylog(n), then we can also reduce the time to O(m). Our results give linear space and time bounds for position-restricted substring counting and the counting versions of indexing substrings with intervals, indexing substrings with gaps and aligned