Faster Compact On-Line Lempel-Ziv Factorization

We present a new on-line algorithm for computing the Lempel-Ziv factorization of a string that runs in O(N log N) time and uses only O(N log sigma) bits of working space, where N is the length of the string and sigma is the size of the alphabet. This is a notable improvement compared to the performance of previous on-line algorithms using the same order of working space but running in either O(N log(3) N) time (Okanohara & Sadakane 2009) or O(N log(2) N) time (Starikovskaya 2012). The key to our new algorithm is in the utilization of an elegant but less popular index structure called Directed Acyclic Word Graphs, or DAWGs (Blumer et al. 1985). We also present an opportunistic variant of our algorithm, which, given the run length encoding of size m of a string of length N, computes the Lempel-Ziv factorization of the string on-line, in O (m . min {(log log m)(log log N)/log log log N, root lon m/log log m}) time and O(m log N) bits of space.