A fully compressed pattern matching algorithm for simple collage systems

We study the fully compressed pattern matching problem (FCPM problem): Given ${\mathcal T}$ and ${\mathcal P}$ which are descriptions of text T and pattern P respectively, find the occurrences of P in Twithout decompressing${\mathcal T}$or${\mathcal P}$. This problem is rather challenging since patterns are also given in a compressed form. In this paper we present an FCPM algorithm for simple collage systems. Collage systems are a general framework representing various kinds of dictionary-based compressions in a uniform way, and simple collage systems are a subclass that includes LZW and LZ78 compressions. Collage systems are of the form $\langle {\mathcal D}, {\mathcal S} \rangle$, where ${\mathcal D}$ is a dictionary and ${\mathcal S}$ is a sequence of variables from ${\mathcal D}$. Our FCPM algorithm performs in $O(\|{\mathcal D}\|^2 + mn \log |{\mathcal S}|)$ time, where $n = |{\mathcal T}| = \|{\mathcal D}\| + |{\mathcal S}|$ and $m = |{\mathcal P}|$. This is faster than the previous best result of O...