Roman Kolpakov

Linear-time computation of local periods

We present a linear-time algorithm for computing all local periods of a given word. This subsumes (but is substantially more powerful than) the computation of the (global) period of the word and on the other hand, the computation of a critical factorization, implied by the Critical Factorization Theorem.

Real-time traversal in grammar-based compressed files

Summary form only given. In text compression applications, it is important to be able to process compressed data without requiring (complete) decompression. In this context it is crucial to study compression methods that allow time/space efficient access to any fragment of a compressed file without being forced to perform complete decompression. We study here the real-time recovery of consecutive symbols from compressed files, in the context of grammar-based compression. In this setting, a compressed text is represented as a small (a few Kb) dictionary D (containing a set of code words), and a very long (a few Mb) string based on symbols drawn from the dictionary D. The space efficiency of this kind of compression is comparable with standard compression methods based on the Lempel-Ziv approach. We show, that one can visit consecutive symbols of the original text, moving from one symbol to another in constant time and extra O(|D|) space. This algorithm is an improvement of the on-line linear (amortised) time algorithm presented in (L. Gasieniec et al, Proc. 13th Int. Symp. on Fund. of Comp. Theo., LNCS, vol.2138, p.138-152, 2001).

Real-Time String Matching in Sublinear Space

We study a problem of efficient utilisation of extra memory space in real-time string matching. We propose, for any constant e >0, a real-time string matching algorithm claiming O(m e ) extra space, where m is the size of a pattern. All previously known real-time string matching algorithms use Ω(m) extra space.

Space efficient search for maximal repetitions

We study here a problem of finding all maximal repetitions in a string of length n. We show that the problem can be solved in time O(n log n) in the presence of constant extra space and general (unbounded) alphabets. Subsequently we show that in the model with a constant size alphabet the problem can be solved in time O(n) with a help of o(n) extra space. Previously best known algorithms require linear additional space in both models.

Linear-Time Computation of Local Periods

We present a linear-time algorithm for computing all local periods of a given word. This subsumes (but is substantially more powerful than) the computation of the (global) period of the word and on the other hand, the computation of a critical factorization, implied by the Critical Factorization Theorem.

Classes of Binary Rational Distributions Closed under Discrete Transformations

We study the generation of rational probabilities by Boolean functions. A probability a is generated by a set H of probabilities if a is the probability of f(x 1, ... , x n )=1 for some Boolean function f provided that for any i the probability of x i =1 belongs to H and all the values of x 1, ... , x n are independent. The closure of the set H is the set of all numbers generated by H. A set of probabilities is called closed if it coincides with its closure. We give an explicit characterization of closures for all sets of rational probabilities. Using this result, we describe all closed and all finitely generated closed sets of rational probabilities. Moreover, we determine the structure of the lattice formed of these sets.