The asymptotic number of prefix normal words

We show that the number of prefix normal binary words of length $n$ is $2^{n-\Theta((\log n)^2)}$. We also show that the maximum number of binary words of length $n$ with a given fixed prefix normal form is $2^{n-O(\sqrt{n\log n})}$.Comment: 9 page

Bubble-Flip---A New Generation Algorithm for Prefix Normal Words

We present a new recursive generation algorithm for prefix normal words. These are binary strings with the property that no substring has more 1s than the prefix of the same length. The new algorithm uses two operations on binary strings, which exploit certain properties of prefix normal words in a smart way. We introduce infinite prefix normal words and show that one of the operations used by the algorithm, if applied repeatedly to extend the string, produces an ultimately periodic infinite word, which is prefix normal. Moreover, based on the original finite word, we can predict both the length and the density of an ultimate period of this infinite word.Comment: 30 pages, 3 figures, accepted in Theoret. Comp. Sc.. This is the journal version of the paper with the same title at LATA 2018 (12th International Conference on Language and Automata Theory and Applications, Tel Aviv, April 9-11, 2018

On Infinite Prefix Normal Words

Prefix normal words are binary words that have no factor with more $1$s than the prefix of the same length. Finite prefix normal words were introduced in [Fici and Lipt\'ak, DLT 2011]. In this paper, we study infinite prefix normal words and explore their relationship to some known classes of infinite binary words. In particular, we establish a connection between prefix normal words and Sturmian words, between prefix normal words and abelian complexity, and between prefix normality and lexicographic order.Comment: 20 pages, 4 figures, accepted at SOFSEM 2019 (45th International Conference on Current Trends in Theory and Practice of Computer Science, Nov\'y Smokovec, Slovakia, January 27-30, 2019

Algorithms and Data Structures for Coding, Indexing, and Mining of Sequential Data

In recent years, the production of sequential data has been rapidly increasing. This requires solving challenging problems about how to represent information, how to retrieve information, and how to extract knowledge, from sequential data. These questions belong to the areas of coding, indexing, and mining, respectively. In this thesis, we investigate problems from those three areas. Coding refers to the way in which information is represented. Coding aims at generating optimal codes, that are codes having a minimum expected length. Codes can be generated for different purposes, from data compression to error detection/correction. The Lempel-Ziv 77 parsing produces an asymptotically optimal code in terms of compression. We study algorithms to efficiently decompress strings from the Lempel-Ziv 77 parsing, using memory proportional to the size of the parsing itself. We provide the first implementation of an algorithm by Bille et al., the only work we are aware of on this problem. We present a practical evaluation of this approach and several optimizations which improve the performance on all datasets we tested. Through the Ulam-R{'e}nyi game, it is possible to provide optimal adaptive error-correcting codes. The game consists of discovering an unknown $m$-bit number by asking membership questions the answers to which can be erroneous. Questions are formulated knowing the answers to all previous ones. We want to find an optimal strategy, i.e., a strategy that can identify any $m$-bit number using the theoretical minimum number of questions. We studied the case where questions are a union of up to a fixed number of intervals, and up to three answers can be erroneous. We first show that for any sufficiently large $m$, there exists a strategy to identify an initially unknown $m$-bit number which uses at most four intervals per question. We further refine our main tool to turn the above asymptotic result into a complete characterization of those instances of the Ulam-R{'e}nyi game that admit optimal strategies. Indexing refers to the way in which information is retrieved. An index for texts permits finding all occurrences of any substring, without traversing the whole text. Many applications require to look for approximate substrings. One of these is the problem of jumbled pattern matching, where two strings match if one is a permutation of the other. We study combinatorial aspects of prefix normal words, a class of binary words introduced in this context. These words can be used as indices for the Indexed Binary Jumbled Pattern Matching problem. We present a new recursive generation algorithm for prefix normal words that is competitive with the previous one but allows to list all prefix normal words sharing the same prefix. This sheds lights on novel insights that may help solving the problem of counting the number of prefix normal words of a given length. We then introduce infinite prefix normal words, and we show that one of the operations used by the algorithm, when repeatedly applied to extend a word, produces an infinite prefix normal word. This motivates the seeking for other operations that produce infinite prefix normal words. We found that one of these operations establishes a connection between prefix normal words and Sturmian words. We also explored the relationship between prefix normal words and Abelian complexity, as well as between prefix normal words and lexicographic order. Mining refers to the way in which information is converted into knowledge. The process of knowledge discovery covers several processing steps, including knowledge extraction. We analyze the problem of mining assertions for an embedded system from its simulation traces. This problem can be modeled as a pattern discovery problem on colored strings. We present two problems of pattern discovery on colored strings: patterns for one color only, or for all colors at the same time. We present two suffix tree-based algorithms. The first algorithm solves both the one color problem and the all colors problem. We then, introduce modifications which improve performance of the algorithm both on synthetic and on real data. We implemented and evaluated the proposed approaches, highlighting time trade-offs that can be obtained. A different way of knowledge extraction is based on the information-theoretic perspective of Pearl's model of causality. It has been postulated that the true causality direction between two phenomena A and B is related to the problem of finding the minimum entropy joint distribution between A and B. This problem is known to be NP-hard, and greedy algorithms have recently been proposed. We provide a novel analysis of one of the proposed heuristic showing that this algorithm guarantees an additive approximation of 1 bit. We then, provide a general criterion for guaranteeing an additive approximation factor of 1. This criterion may be of independent interest in other contexts where couplings are used

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum