Lempel-Ziv Computation in Compressed Space (LZ-CICS)

Abstract

We show that both the Lempel-Ziv-77 and the Lempel-Ziv-78 factorization of a text of length n on an integer alphabet of size σ can be computed in O(n lg lg σ) time (linear time if we allow randomization) using O(n lg σ) bits of working space. Given that a compressed representation of the suffix tree is loaded into RAM, we can compute both factorizations in O(n) time using z lg n + O(n) bits of space, where z is the number of factors.

LZ78 Compression in Low Main Memory Space

“…For example, a simple and classical implementation compresses a text of length n over an alphabet [1..σ] into z integers, where √ n ≤ z = O(n/ lg σ n), in O(n lg σ) deterministic or O(n) randomized time, using O(z lg n) = O(n lg σ) bits of space. A comparable result for LZ77 was obtained only recently [11] and it required sophisticated compressed suffix array construction algorithms.…”

“…They use the compact LZTrie representation described in the previous section. An obstacle to further reducing the space is that they need to build the whole LZTrie before they can build Reference RAM space in bits Compression time Classic [19] O(z lg n) O(n lg σ) O(z lg n) O(n) * Fischer et al [8] (1 + )n lg n + O(n) O(n/ 2 ) Köppl and Sadakane [11] O(n lg σ) O(n lg lg σ) Table 1: Previous and new LZ78 compression algorithms. Times with a star mean expected time of randomized algorithms.…”

“…Recently, Köppl and Sadakane [11] showed how to construct the parsing in O(n lg lg σ) time, using O(n lg σ) bits of working space; note this is Ω(z lg n). Table 1 shows all these previous results, and our contribution in context.…”

“…The fastest deterministic LZ78 compression algorithms require O(n) time, but O(n lg n) bits of main memory [8]. If the main memory is limited to O(n lg σ) bits (i.e., proportional to the input text size), then the time increases to O(n lg lg σ) [11]. Finally, if we limit the main memory to O(z lg n) bits (i.e., proportional to the compressed text size, like the classic scheme), then the compression time becomes O(n(1 + lg σ/ lg lg n)) [1], which improves the classic O(n lg σ) time.…”

Space-Efficient Construction of Compressed Indexes in Deterministic Linear Time

“…Another application is that we can now compute the Lempel-Ziv 77 and 78 parsings [29,46,47] of a string T [0..n − 1] in deterministic linear time using O(n log σ) bits: Köppl and Sadakane [27] recently showed that, if one has a compressed suffix tree on T , then they need only O(n) additional (deterministic) time and O(z log n) bits to produce the parsing, where z is the resulting number of phrases. Since z ≤ n/ log σ n, the space is O(n log σ) bits.…”