On the Multi-Interval Ulam-R\'enyi Game: for 3 lies 4 intervals suffice

We study the problem of identifying an initially unknown $m$-bit number by using yes-no questions when up to a fixed number $e$ of the answers can be erroneous. In the variant we consider here questions are restricted to be the union of up to a fixed number of intervals. For any $e \geq 1$ let $k_e$ be the minimum $k$ such that for all sufficiently large $m$, there exists a strategy matching the information theoretic lower bound and only using $k$-interval questions. It is known that $k_e = O(e^2)$. However, it has been conjectured that the $k_e = \Theta(e).$ This linearity conjecture is supported by the known results for small values of $e$. For $e\leq2$ we have $k_e = e.$ We extend these results to the case $e=3$. We show $k_3 \leq 4$ improving upon the previously known bound $k_3 \leq 10.$Comment: 31 pages, 5 figures, extension of the result to non-asymptotic strategie

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 Constrained Intersection Representations of Graphs and Digraphs

We study the problem of determining minimal directed intersection representations of DAGs in a model introduced by [Kostochka, Liu, Machado, and Milenkovic, ISIT2019]: vertices are assigned color sets, two vertices are connected by an arc if and only if they share at least one color and the tail vertex has a strictly smaller color set than the head, and the goal is to minimize the total number of colors. We show that the problem is polynomially solvable in the class of triangle-free and Hamiltonian DAGs and also disclose the relationship of this problem with several other models of intersection representations of graphs and digraphs

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