Top Tree Compression of Tries

We present a compressed representation of tries based on top tree compression [ICALP 2013] that works on a standard, comparison-based, pointer machine model of computation and supports efficient prefix search queries. Namely, we show how to preprocess a set of strings of total length n over an alphabet of size $$\sigma$$ into a compressed data structure of worst-case optimal size $$O(n/\log _\sigma n)$$ that given a pattern string P of length m determines if P is a prefix of one of the strings in time $$O(\min (m\log \sigma ,m + \log n))$$ . We show that this query time is in fact optimal regardless of the size of the data structure. Existing solutions either use $$\Omega (n)$$ space or rely on word RAM techniques, such as tabulation, hashing, address arithmetic, or word-level parallelism, and hence do not work on a pointer machine. Our result is the first solution on a pointer machine that achieves worst-case o(n) space. Along the way, we develop several interesting data structures that work on a pointer machine and are of independent interest. These include an optimal data structures for random access to a grammar-compressed string and an optimal data structure for a variant of the level ancestor problem.