On the Complexity and Approximability of Bounded Access Lempel Ziv Coding

We study the complexity of constructing an optimal parsing \(\varphi \) of a string \(\textbf{s}= s_1 \dots s_n\) under the constraint that given a position p in the original text, and the LZ76 (also known as LZ77 or simply Lempel-Ziv) encoding of T based on \(\varphi \) , it is possible to identify/decompress the character \(s_p\) by performing at most c accesses to the LZ encoding, for a given integer c. We refer to such a parsing \(\varphi \) as a c-bounded access LZ parsing or c-BLZ parsing of \(\textbf{s}.\) We show that for any constant c the problem of computing the optimal c-BLZ parsing of a string, i.e., the one with the minimum number of phrases, is NP-hard and also APX hard, i.e., no PTAS can exist under the standard complexity assumption \(P \ne NP.\) We also study the ratio between the sizes of an optimal c-BLZ parsing of a string \(\textbf{s}\) and an optimal LZ76 parsing of \(\textbf{s}\) (which can be greedily computed in polynomial time).