Linear-Space Substring Range Counting over Polylogarithmic Alphabets

Bille and G{\o}rtz (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 $\text{Oh}n\log n\mathrm{/}\log \log n$ space and answer queries in $\Oh{m + \log \log u}$time\mathrm{.}Weshowthat,if$S$ is over an alphabet of size (\polylog (n)), then we can achieve optimal linear space. Moreover, if (u = n \polylog (n)), then we can also reduce the time to $\text{Oh}m$. 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 pattern matching.