Computing the rotation distance between two binary trees with $n$ internal nodes efficiently (in $poly(n)$ time) is a long standing open question in the study of height balancing in tree data structures. In this paper, we initiate the study of this problem bounding the rank of the trees given at the input (defined by Ehrenfeucht and Haussler (1989) in the context of decision trees). We define the rank-bounded rotation distance between two given binary trees $T_1$ and $T_2$ (with $n$ internal nodes) of rank at most $r$, denoted by $d_r(T_1,T_2)$, as the length of the shortest sequence of rotations that transforms $T_1$ to $T_2$ with the restriction that the intermediate trees must be of rank at most $r$. We show that the rotation distance problem reduces in polynomial time to the rank bounded rotation distance problem. This motivates the study of the problem in the combinatorial and algorithmic frontiers. Observing that trees with rank $1$ coincide exactly with skew trees (binary trees where every internal node has at least one leaf as a child), we show the following results in this frontier : We present an $O(n^2)$ time algorithm for computing $d_1(T_1,T_2)$. That is, when the given trees are skew trees (we call this variant as skew rotation distance problem) - where the intermediate trees are restricted to be skew as well. In particular, our techniques imply that for any two skew trees $d(T_1,T_2) \le n^2$. We show the following upper bound : for any two trees $T_1$ and $T_2$ of rank at most $r_1$ and $r_2$ respectively, we have that: $d_r(T_1,T_2) \le n^2 (1+(2n+1)(r_1+r_2-2))$ where $r = max\{r_1,r_2\}$. This bound is asymptotically tight for $r=1$. En route our proof of the above theorems, we associate binary trees to permutations and bivariate polynomials, and prove several characterizations in the case of skew trees.

We introduce a game where players selfishly choose a resource and endure a cost depending on the number of players choosing nearby resources. We model the influences among resources by a weighted graph, directed or not. These games are generalizations of well-known games like Wardrop and congestion games. We study the conditions of equilibria existence and their efficiency if they exist. We conclude with studies of games whose influences among resources can be modelled by simple graphs.

This investigation is firstly focused into showing that two metric parameters represent the same object in graph theory. That is, we prove that the multiset resolving sets and the ID-colorings of graphs are the same thing. We also consider some computational and combinatorial problems of the multiset dimension, or equivalently, the ID-number of graphs. We prove that the decision problem concerning finding the multiset dimension of graphs is NP-complete. We consider the multiset dimension of king grids and prove that it is bounded above by 4. We also give a characterization of the strong product graphs with one factor being a complete graph, and whose multiset dimension is not infinite.

We apply automata theory and Karp's minimum mean weight cycle algorithm to minimum density problems in coding theory. Using this method, we find the new upper bound $53/126 \approx 0.4206$ for the minimum density of an identifying code on the infinite hexagonal grid, down from the previous record of $3/7 \approx 0.4286$.

We introduce two new classes of covering codes in graphs for every positive integer $r$. These new codes are called local $r$-identifying and local $r$-locating-dominating codes and they are derived from $r$-identifying and $r$-locating-dominating codes, respectively. We study the sizes of optimal local 1-identifying codes in binary hypercubes. We obtain lower and upper bounds that are asymptotically tight. Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible. For some small $n$ optimal constructions are obtained. Moreover, the upper bound is obtained by a linear code construction. Also, we study the densities of optimal local 1-identifying codes and local 1-locating-dominating codes in the infinite square grid, the hexagonal grid, the triangular grid, and the king grid. We prove that seven out of eight of our constructions have optimal densities.

We study decision problems of the form: given a regular or linear context-free language $L$, is there a word of a given fixed form in $L$, where given fixed forms are based on word operations copy, marked copy, shuffle and their combinations.

A morphism $g$ from the free monoid $X^*$ into itself is called upper triangular if the matrix of $g$ is upper triangular. We characterize all upper triangular binary morphisms $g_1$ and $g_2$ such that $g_1g_2=g_2g_1$.

In the Fully Leafed Induced Subtrees, one is given a graph $G$ and two integers $a$ and $b$ and the question is to find an induced subtree of $G$ with $a$ vertices and at least $b$ leaves. This problem is known to be NP-complete even when the input graph is $4$-regular. Polynomial algorithms are known when the input graph is restricted to be a tree or series-parallel. In this paper we generalize these results by providing an FPT algorithm parameterized by treewidth. We also provide a polynomial algorithm when the input graph is restricted to be a chordal graph.

This paper studies how to use relation algebras, which are useful for high-level specification and verification, for proving the correctness of lower-level array-based implementations of algorithms. We give a simple relation-algebraic semantics of read and write operations on associative arrays. The array operations seamlessly integrate with assignments in computation models supporting while-programs. As a result, relation algebras can be used for verifying programs with associative arrays. We verify the correctness of an array-based implementation of disjoint-set forests using the union-by-rank strategy and find operations with path compression, path splitting and path halving. All results are formally proved in Isabelle/HOL. This paper is an extended version of [1].

Given a set of disks in the plane, the goal of the problem studied in this paper is to choose a subset of these disks such that none of its members contains the centre of any other. Each disk not in this subset must be merged with one of its nearby disks that is, increasing the latter's radius. This problem has applications in labelling rotating maps and in visualizing the distribution of entities in static maps. We prove that this problem is NP-hard. We also present an ILP formulation for this problem, and a polynomial-time algorithm for the special case in which the centres of all disks are on a line.

The state complexity, respectively, nondeterministic state complexity of a regular language $L$ is the number of states of the minimal deterministic, respectively, of a minimal nondeterministic finite automaton for $L$. Some of the most studied state complexity questions deal with size comparisons of nondeterministic finite automata of differing degree of ambiguity. More generally, if for a regular language we compare the size of description by a finite automaton and by a more powerful language definition mechanism, such as a context-free grammar, we encounter non-recursive trade-offs. Operational state complexity studies the state complexity of the language resulting from a regularity preserving operation as a function of the complexity of the argument languages. Determining the state complexity of combined operations is generally challenging and for general combinations of operations that include intersection and marked concatenation it is uncomputable.

Foucaud et al. recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Given a graph $G=(V(G), E(G))$, a set $M \subseteq V(G)$ is a distance-edge-monitoring set if for every edge $e \in E(G)$, there is a vertex $x \in M$ and a vertex $y \in V(G)$ such that the edge $e$ belongs to all shortest paths between $x$ and $y$. The smallest size of such a set in $G$ is denoted by $\operatorname{dem}(G)$. Denoted by $G-e$ (resp. $G \backslash u$) the subgraph of $G$ obtained by removing the edge $e$ from $G$ (resp. a vertex $u$ together with all its incident edges from $G$). In this paper, we first show that $\operatorname{dem}(G-e)- \operatorname{dem}(G)\leq 2$ for any graph $G$ and edge $e \in E(G)$. Moreover, the bound is sharp. Next, we construct two graphs $G$ and $H$ to show that $\operatorname{dem}(G)-\operatorname{dem}(G\setminus u)$ and $\operatorname{dem}(H\setminus v)-\operatorname{dem}(H)$ can be arbitrarily large, where $u \in V(G)$ and $v \in V(H)$. We also study the relation between $\operatorname{dem}(H)$ and $\operatorname{dem}(G)$, where $H$ is a subgraph of $G$. In the end, we give an algorithm to judge whether the distance-edge monitoring set still remain in the resulting graph when any edge of the graph $G$ is deleted.

This note is a complement to the paper by Eberlein, Kabanov, and Schmidt on the asymptotic of the ruin probability in a Sparre Andersen non-life insurance model with investments a risky asset whose price follows a geometric Lévy process. Using the techniques of semi-Markov processes we extend the result of the mentioned paper to the case of annuities and models with two-sided jumps.

We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.

In process discovery, the goal is to find, for a given event log, the model describing the underlying process. While process models can be represented in a variety of ways, Petri nets form a theoretically well-explored description language and are therefore often used. In this paper, we extend the eST-Miner process discovery algorithm. The eST-Miner computes a set of Petri net places which are considered to be fitting with respect to a certain fraction of the behavior described by the given event log as indicated by a given noise threshold. It evaluates all possible candidate places using token-based replay. The set of replayable traces is determined for each place in isolation, i.e., these sets do not need to be consistent. This allows the algorithm to abstract from infrequent behavioral patterns occurring only in some traces. However, when combining places into a Petri net by connecting them to the corresponding uniquely labeled transitions, the resulting net can replay exactly those traces from the event log that are allowed by the combination of all inserted places. Thus, inserting places one-by-one without considering their combined effect may result in deadlocks and low fitness of the Petri net. In this paper, we explore adaptions of the eST-Miner, that aim to select a subset of places such that the resulting Petri net guarantees a definable minimal fitness while maintaining high precision with respect to the input event log. Furthermore, current place evaluation techniques tend to block the execution of infrequent activity labels. Thus, a refined place fitness metric is introduced and thoroughly investigated. In our experiments we use real and artificial event logs to evaluate and compare the impact of the various place selection strategies and place fitness evaluation metrics on the returned Petri net.

A model of an information system describes its processes and how resources are involved in these processes to manipulate data objects. This paper presents an extension to the Petri nets formalism suitable for describing information systems in which states refer to object instances of predefined types and resources are identified as instances of special object types. Several correctness criteria for resource- and object-aware information systems models are proposed, supplemented with discussions on their decidability for interesting classes of systems. These new correctness criteria can be seen as generalizations of the classical soundness property of workflow models concerned with process control flow correctness.

Coverings of undirected graphs are used in distributed computing, and unfoldings of directed graphs in semantics of programs. We study these two notions from a graph theoretical point of view so as to highlight their similarities, as they are both defined in terms of surjective graph homomorphisms. In particular, universal coverings and complete unfoldings are infinite trees that are regular if the initial graphs are finite. Regularity means that a tree has finitely many subtrees up to isomorphism. Two important theorems have been established by Leighton and Norris for coverings. We prove similar statements for unfoldings. Our study of the difficult proof of Leighton's Theorem lead us to generalize coverings and similarly, unfoldings, by attaching finite or infinite weights to edges of the covered or unfolded graphs. This generalization yields a canonical factorization of the universal covering of any finite graph, that (provably) does not exist without using weights. Introducing infinite weights provides us with finite descriptions of regular trees having nodes of countably infinite degree. We also generalize to weighted graphs and their coverings a classical factorization theorem of their characteristic polynomials.

The study of strings is an important combinatorial field that precedes the digital computer. Strings can be very long, trillions of letters, so it is important to find compact representations. Here we first survey various forms of one potential compaction methodology, the cover of a given string x, initially proposed in a simple form in 1990, but increasingly of interest as more sophisticated variants have been discovered. We then consider covering by a seed; that is, a cover of a superstring of x. We conclude with many proposals for research directions that could make significant contributions to string processing in future.

In time Petri nets (TPNs), time and control are tightly connected: time measurement for a transition starts only when all resources needed to fire it are available. Further, upper bounds on duration of enabledness can force transitions to fire (this is called urgency). For many systems, one wants to decouple control and time, i.e. start measuring time as soon as a part of the preset of a transition is filled, and fire it after some delay \underline{and} when all needed resources are available. This paper considers an extension of TPN called waiting nets that dissociates time measurement and control. Their semantics allows time measurement to start with incomplete presets, and can ignore urgency when upper bounds of intervals are reached but all resources needed to fire are not yet available. Firing of a transition is then allowed as soon as missing resources are available. It is known that extending bounded TPNs with stopwatches leads to undecidability. Our extension is weaker, and we show how to compute a finite state class graph for bounded waiting nets, yielding decidability of reachability and coverability. We then compare expressiveness of waiting nets with that of other models w.r.t. timed language equivalence, and show that they are strictly more expressive than TPNs.

We study minimization problems for deterministic $ω$-automata in the presence of don't care words. We prove that the number of priorities in deterministic parity automata can be efficiently minimized under an arbitrary set of don't care words. We derive that from a more general result from which one also obtains an efficient minimization algorithm for deterministic parity automata with informative right-congruence (without don't care words). We then analyze languages of don't care words with a trivial right-congruence. For such sets of don't care words it is known that weak deterministic Büchi automata (WDBA) have a unique minimal automaton that can be efficiently computed from a given WDBA (Eisinger, Klaedtke 2006). We give a congruence-based characterization of the corresponding minimal WDBA, and show that the don't care minimization results for WDBA do not extend to deterministic $ω$-automata with informative right-congruence: for this class there is no unique minimal automaton for a given don't care set with trivial right congruence, and the minimization problem is NP-hard. Finally, we extend an active learning algorithm for WDBA (Maler, Pnueli 1995) to the setting with an additional set of don't care words with trivial right-congruence.

We consider a class of problems of Discrete Tomography which has been deeply investigated in the past: the reconstruction of convex lattice sets from their horizontal and/or vertical X-rays, i.e. from the number of points in a sequence of consecutive horizontal and vertical lines. The reconstruction of the HV-convex polyominoes works usually in two steps, first the filling step consisting in filling operations, second the convex aggregation of the switching components. We prove three results about the convex aggregation step: (1) The convex aggregation step used for the reconstruction of HV-convex polyominoes does not always provide a solution. The example yielding to this result is called \textit{the bad guy} and disproves a conjecture of the domain. (2) The reconstruction of a digital convex lattice set from only one X-ray can be performed in polynomial time. We prove it by encoding the convex aggregation problem in a Directed Acyclic Graph. (3) With the same strategy, we prove that the reconstruction of fat digital convex sets from their horizontal and vertical X-rays can be solved in polynomial time. Fatness is a property of the digital convex sets regarding the relative position of the left, right, top and bottom points of the set. The complexity of the reconstruction of the lattice sets which are not fat remains an open question.

We prove that quasi-isometries of horospherical products of hyperbolic spaces are geometrically rigid in the sense that they are uniformly close to product maps, this is a generalisation of the result obtained by Eskin, Fisher and Whyte in [7]. Our work covers the case of solvable Lie groups of the form R ___ (N 1 x N 2), where N 1 and N 2 are nilpotent Lie groups, and where the action on R contracts the metric on N 1 while extending it on N 2. We obtain new quasi-isometric invariants and classi cations for these spaces.

We establish a Myhill-Nerode type theorem for higher-dimensional automata (HDAs), stating that a language is regular if and only if it has finite prefix quotient. HDAs extend standard automata with additional structure, making it possible to distinguish between interleavings and concurrency. We also introduce deterministic HDAs and show that not all HDAs are determinizable, that is, there exist regular languages that cannot be recognised by a deterministic HDA. Using our theorem, we develop an internal characterisation of deterministic languages. Lastly, we develop analogues of the Myhill-Nerode construction and of determinacy for HDAs with interfaces.

We introduce a method to construct closed rigid associative submanifolds in twisted connected sum $G_2$-manifolds. More precisely, we prove a gluing theorem of asymptotically cylindrical (ACyl) associative submanifolds in ACyl $G_2$-manifolds under a transverse intersection hypothesis. This is analogous to the gluing theorem for $G_2$-instantons introduced in [SW15]. We rephrase the hypothesis in the special cases where the ACyl associative submanifolds are obtained from holomorphic curves or special Lagrangians in ACyl Calabi-Yau $3$-folds, in terms of algebraic-geometric conditions and topological conditions, respectively. This yields many rigid associative submanifolds with new topological types $S^3$, $\mathbf R\mathbf P^3$ or $\mathbf R\mathbf P^3\#\mathbf R\mathbf P^3$.

We simulate the transport of colloidal particles driven by a static and homogeneous drift force, and subject to the optical potential created by two lattices of optical tweezers. The lattices of optical tweezers are parallel to each other, shifted, and rotated by a twist angle. Due to a negative interference between the potential of the two lattices, flat channels appear in the total optical potential. At specific twist angles, known as magic-angles, the flat channels percolate the entire system and the colloidal particles can then be transported using a weak external drift force. We characterize the transport in both square and hexagonal lattices of twisted optical tweezers

Since the 1980s, it has been known that essential surfaces in alternating link complements can be isotoped to be transverse to the link diagram almost everywhere, with the exception of some well-understood intersections, and described combinatorially as a result. This was called standard position for surfaces and has had numerous applications. However, the original techniques only apply to classical alternating links projected onto the 2-sphere inside the 3-sphere. In this paper, we prove that standard position for surfaces can be extended to a broader class, namely weakly generalized alternating links. Such links include all classical prime non-split alternating links in the 3-sphere, and also many links that are alternating on higher genus surfaces, or lie in manifolds besides the 3-sphere. As an application, we show that all such links are prime, and that under mild restrictions, essential Conway spheres for such links interact with the diagram exactly as in the classical alternating setting.

Discrete tomography focuses on the reconstruction of functions $f: A \to \mathbb{R}$ from their line sums in a finite number $d$ of directions, where $A$ is a finite subset of $\mathbb{Z}^2$. Consequently, the techniques of discrete tomography often find application in areas where only a small number of projections are available. In 1978 M.B. Katz gave a necessary and sufficient condition for the uniqueness of the solution. Since then, several reconstruction methods have been introduced. Recently Pagani and Tijdeman developed a fast method to reconstruct $f$ if it is uniquely determined. Subsequently Ceko, Pagani and Tijdeman extended the method to the reconstruction of a function with the same line sums of $f$ in the general case. Up to here we assumed that the line sums are exact. In this paper we investigate the case where a small number of line sums are incorrect as may happen when discrete tomography is applied for data storage or transmission. We show how less than $d/2$ errors can be corrected and that this bound is the best possible.

Graph embeddings play a significant role in the design and analysis of parallel algorithms. It is a mapping of the topological structure of a guest graph G into a host graph H, which is represented as a one-to-one mapping from the vertex set of the guest graph to the vertex set of the host graph. In multiprocessing systems the interconnection networks enhance the efficient communication between the components in the system. Obtaining minimum wirelength in embedding problems is significant in the designing of network and simulating one architecture by another. In this paper, we determine the wirelength of embedding 3-ary n-cubes into cylinders and certain trees.

Colored Petri nets offer a compact and user friendly representation of the traditional P/T nets and colored nets with finite color ranges can be unfolded into the underlying P/T nets, however, at the expense of an exponential explosion in size. We present two novel techniques based on static analysis in order to reduce the size of unfolded colored nets. The first method identifies colors that behave equivalently and groups them into equivalence classes, potentially reducing the number of used colors. The second method overapproximates the sets of colors that can appear in places and excludes colors that can never be present in a given place. Both methods are complementary and the combined approach allows us to significantly reduce the size of multiple colored Petri nets from the Model Checking Contest benchmark. We compare the performance of our unfolder with state-of-the-art techniques implemented in the tools MCC, Spike and ITS-Tools, and while our approach is competitive w.r.t. unfolding time, it also outperforms the existing approaches both in the size of unfolded nets as well as in the number of answered model checking queries from the 2021 Model Checking Contest.

An integral quadratic form q is usually identified with a bilinear form b such that its Gram matrix with respect to the canonical basis is upper triangular. Two integral quadratic forms are called strongly (resp. weakly) Gram congruent if their corresponding upper triangular bilinear forms (resp. their symmetrizations) are equivalent. If q is unitary, such upper triangular bilinear form is unimodular, and one considers the associated Coxeter transformation and its characteristic polynomial, the so-called Coxeter polynomial of q with this identification. Two strongly Gram congruent quadratic unit forms are weakly Gram congruent and have the same Coxeter polynomial. Here we show that the converse of this statement holds for the connected non-negative case of Dynkin type A_r and arbitrary corank, and use this characterization to complete a combinatorial classification of such quadratic forms started in [Fundamenta Informaticae 184(1):49-82, 2021] and [Fundamenta Informaticae 185(3):221-246, 2022].