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Bille and G{\o}rtz (2011) recently introduced the problem of substring rangecounting, 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 occurrencesof $P$ whose first characters' labels are in ([a, b]). They showed how to store$S$ in $\Oh{n \log n / \log \log n}$ space and answer queries in $\Oh{m + \log\log u}$ time. We show that, if $S$ is over an alphabet of size (\polylog (n)),then we can achieve optimal linear space. Moreover, if (u = n \polylog (n)),then we can also reduce the time to $\Oh{m}$. Our results give linear space andtime bounds for position-restricted substring counting and the countingversions of indexing substrings with intervals, indexing substrings with gapsand aligned pattern matching.

Linear-Space Substring Range Counting over Polylogarithmic Alphabets