Re-pairing brackets

Consider the following one-player game. Take a well-formed sequence of opening and closing brackets (a Dyck word). As a move, the player can pair any opening bracket with any closing bracket to its right, erasing them. The goal is to re-pair (erase) the entire sequence, and the cost of a strategy is measured by its width: the maximum number of nonempty segments of symbols (separated by blank space) seen during the play. For various initial sequences, we prove upper and lower bounds on the minimum width sufficient for re-pairing. (In particular, the sequence associated with the complete binary tree of height n admits a strategy of width sub-exponential in log n.) Our two key contributions are (1) lower bounds on the width and (2) their application in automata theory: quasi-polynomial lower bounds on the translation from one-counter automata to Parikh-equivalent nondeterministic finite automata. The latter result answers a question by Atig et al. (2016)

Automata Equipped with Auxiliary Data Structures and Regular Realizability Problems

We consider general computational models: one-way and two-way finite automata, and logarithmic space Turing machines, all equipped with an auxiliary data structure (ADS). The definition of an ADS is based on the language of protocols of work with the ADS. We describe the connection of automata-based models with ``Balloon automata'' that are another general formalization of automata equipped with an ADS presented by Hopcroft and Ullman in 1967. This definition establishes the connection between the non-emptiness problem for one-way automata with ADS, languages recognizable by nondeterministic log-space Turing machines equipped with the same ADS, and a regular realizability problem (NRR) for the language of ADS' protocols. The NRR problem is to verify whether the regular language on the input has a non-empty intersection with the language of protocols. The computational complexity of these problems (and languages) is the same up to log-space reductions.Comment: 25 pages. An extended version of the conference paper (DCFS 2021), submitted to International Journal of Foundations of Computer Scienc

Avoidability beyond paths

The concept of avoidable paths in graphs was introduced by Beisegel, Chudnovsky, Gurvich, Milani\v{c}, and Servatius in 2019 as a common generalization of avoidable vertices and simplicial paths. In 2020, Bonamy, Defrain, Hatzel, and Thiebaut proved that every graph containing an induced path of order $k$ also contains an avoidable induced path of the same order. They also asked whether one could generalize this result to other avoidable structures, leaving the notion of avoidability up to interpretation. In this paper we address this question: we specify the concept of avoidability for arbitrary graphs equipped with two terminal vertices. We provide both positive and negative results, some of which appear to be related to the recent work by Chudnovsky, Norin, Seymour, and Turcotte [arXiv:2301.13175]

Characterizing (quasi-)ultrametric finite spaces in terms of (directed) graphs

AbstractLet D=(V,A) be a complete directed graph (digraph) with a positive real weight function d:A→{d1,…,dk}⊆R+ such that 0<d1<⋯<dk. For every i∈[k]={1,…,k}, let us set Ai={(u,w)∈A∣d(u,w)≤di} and assume that each subgraph Di=(V,Ai),i∈[k], in the obtained nested family is transitive, that is, (u,w)∈Ai whenever (u,v),(v,w)∈Ai for some v∈V. This assumption implies that the considered weighted digraph (D,d) defines a quasi-ultrametric finite space (QUMFS) and, conversely, each QUMFS is uniquely (up to an isometry) is realized by a nested family of transitive digraphs.These simple observations imply important corollaries. For example, each QUMFS can be realized by a multi-pole flow network. Furthermore, k≤(n2)+n−1=12(n−1)(n+2), where n=|V|, and this upper bound for the number k of pairwise distinct distances is precise. Moreover, we characterize all QUMFSes for which the equality holds.In the symmetric case, d(u,w)=d(w,u), we obtain a canonical representation of an ultrametric finite space (UMFS) together with the well-known bound k≤n−1. Interestingly, due to this representation, a UMFS can be viewed as a positional game structure of k players {1,…,k} such that, in every play, they make moves in a monotone strictly decreasing order