Data Structure Lower Bounds on Random Access to Grammar-Compressed Strings

Abstract

In this paper we investigate the problem of building a static data structure that represents a string s using space close to its compressed size, and allows fast access to individual characters of s. This type of data structures was investigated by the recent paper of Bille et al. [3]. Let n be the size of a context-free grammar that derives a unique string s of length L. (Note that L might be exponential in n.) Bille et al. showed a data structure that uses space O(n) and allows to query for the i-th character of s using running time O(logL). Their data structure works on a word RAM with a word size of logL bits.

Here we prove that for such data structures, if the space is poly(n), the query time must be at least $(\log L)^{1-\varepsilon }/\log \mathcal{S}$ where $\mathcal{S}$ is the space used, for any constant ε > 0. As a function of n, our lower bound is Ω(n ^{1/2 − ε}). Our proof holds in the cell-probe model with a word size of logL bits, so in particular it holds in the word RAM model. We show that no lower bound significantly better than n ^{1/2 − ε} can be achieved in the cell-probe model, since there is a data structure in the cell-probe model that uses O(n) space and achieves $O(\sqrt{n \log n})$ query time. The “bad” setting of parameters occurs roughly when $L=2^{\sqrt{n}}$. We also prove a lower bound for the case of not-as-compressible strings, where, say, L = n ^1 + ε. For this case, we prove that if the space is O(n ·polylog(n)), the query time must be at least Ω(logn / loglogn).

The proof works by reduction from communication complexity, namely to the LSD (Lopsided Set Disjointness) problem, recently employed by Pǎtraşcu and others. We prove lower bounds also for the case of LZ-compression. All of our lower bounds hold even when the strings are over an alphabet of size 2 and hold even for randomized data structures with 2-sided error.

The authors acknowledge support from the Danish National Research Foundation and The National Science Foundation of China (under the grant 61061130540) for the Sino-Danish Center for the Theory of Interactive Computation, within which part of this work was performed. Part of the work was done while the authors were working in IIIS, Tsinghua University in China.