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Given a text and a pattern over an alphabet, the classic exact matching problem searches for all occurrences of the pattern in the text. Unlike exact matching, order-preserving pattern matching (OPPM) considers the relative order of elements, rather than their exact values. In this paper, we propose efficient algorithms for the OPPM problem using the “duel-and-sweep” paradigm. For a pattern of length m and a text of length n, our serial algorithm runs in (O(n + mlog m)) time, and our parallel algorithm runs in (O(log ^2 m)) time and (O(n log ^2 m)) work with (O(log m)) time and (O(m log m)) work pattern preprocessing on the Priority Concurrent Read Concurrent Write Parallel Random-Access Machines (P-CRCW PRAM).

The linear-size suffix tries (LSTries) (Crochemore et al. in Theor Comput Sci 638:171–178, 2016) are a version of suffix trees in which the edge labels are single characters, yet are able to perform pattern matching queries in optimal time. Instead of explicitly storing the input text, LSTries have some extra non-branching internal nodes called type-2 nodes. The extended techniques are then used in the linear-size compact directed acyclic word graphs (LCDAWGs) (Takagi et al., in: SPIRE 2017, pp. 304–316, 2017), which can be stored with (O(textsf{el}(T)+textsf{er}(T))) space (i.e. without the text), where (textsf{el}(T)) and (textsf{er}(T)) are the numbers of left- and right-extensions of the maximal repeats in the input text string T, respectively. In this paper, we present simpler alternatives to the aforementioned indexing structures, called the simplified LSTries (simLSTries) and the simplified LCDAWGs (simLCDAWGs), in which most of the type-2 nodes are removed. In particular, our simLCDAWGs require only (O(textsf{er}(T))) space and work in a weaker model of computation (i.e. the pointer machine model). This contrasts the (O(textsf{er}(T)))-space CDAWG representation of Belazzougui and Cunial (in: Proceedings of the 24th international symposium on string processing and information retrieval, pp. 161–175, 2017), which works on the word RAM model.

The Kemeny method is one of the popular tools for rank aggregation. However, computing an optimal Kemeny ranking is (textsf{NP})-hard. Consequently, the computational task of finding a Kemeny ranking has been studied under the lens of parameterized complexity with respect to many parameters. We study the parameterized complexity of the problem of computing all distinct Kemeny rankings. We consider the target Kemeny score, number of candidates, average distance of input rankings, maximum range of any candidate, and unanimity width as our parameters. For all these parameters, we already have (textsf{FPT}) algorithms. We find that any desirable number of Kemeny rankings can also be found without substantial increase in running time. We also present (textsf{FPT}) approximation algorithms for Kemeny rank aggregation with respect to these parameters.

Word-representable graphs were introduced in 2008 by Kitaev and Pyatkin in the context of semigroup theory. Graphs are called word-representable if there exists a word with the graph’s nodes as letters such that the letters in the word alternate iff there is an edge between them in the graph. Until today numerous works investigated the word-representability of graphs but mostly from the graph perspective. In this work, we change the perspective to the words, i.e., we take classes of words and investigate the represented graphs. Our first subject of interest are the conjugates of words: we determine exactly which graphs are represented if we rotate the word. Afterwards, we look at k-local words introduced by Day et al. (FSTTCS LIPIcs, 2017) in order to gain more insights into this class of words. Here, we investigate especially which graphs are represented by 1-local words. Lastly, we prove that the language of all words representing a graph is regular. We were also able to characterise k-representable graphs, solving an open problem.

A bipartite graph G(X, Y) is called a star-convex bipartite graph with convexity on X if there is an associated star T(X, F), such that for each vertex in Y, its neighborhood in X induces a subtree in T. A graph G is said to be a split graph if G can be partitioned into a clique (K) and an independent set (I). The objective of this study is twofold: (i) to strengthen the complexity results presented in Chen et al. (J Comb Optim 32(1):95–110, 2016) for the Hamiltonian cycle (HCYCLE), the Hamiltonian path (HPATH), and the Domination (DS) problems on star-convex bipartite graphs (ii) to reinforce the results of Müller (Discret Math 156(1–3):291–298, 1996) for HCYCLE, and HPATH on split graphs by introducing a convex ordering on one of the partitions (K or I). As part of our fine-grained analysis study with the diameter being the parameter, we first show that the diameter of star-convex bipartite graphs is at most six. Next, we observe that the reduction instances of Chen et al. (J Comb Optim 32(1):95–110, 2016) are star-convex bipartite graphs with at most diameter 4, and hence HCYCLE and HPATH are NP-complete on star-convex bipartite graphs with at most diameter 4. We strengthen this result and establish the following results on star-convex bipartite graphs: (i) HCYCLE is NP-complete for diameter 3, and polynomial-time solvable for diameters 2, 5, and 6 (a transformation in complexity: P to NPC to P) (ii) HPATH is polynomial-time solvable for diameter 2, and NP-Complete, otherwise (a dichotomy). Further, with convexity being the parameter, for split graphs with convexity on K (resp. I), we show that HCYCLE and HPATH are NP-complete on star-convex (resp. comb) split graphs with convexity on K (resp. I). Further, we show that HCYCLE is NP-complete on (k_{1,r})-free star-convex split graphs with convexity on I, (rge 6). On the positive side, we show that for (K_{1,5})-free star-convex split graphs with convexity on I, HCYCLE is polynomial-time solvable. Thus, we establish a dichotomy for HCYCLE on star-convex split graphs with convexity on I. We further show that the dominating set problem (DS) and its variants (resp. Connected, Total, Outer-Connected, and Dominating biclique) are NP-complete on star-convex bipartite graphs with diameter 3 (resp. diameter 5, and diameter 6). On the parameterized complexity front, we prove that the parameterized version of the domination problem and its variants, with the parameter being the solution size, is not fixed-parameter tractable for star-convex bipartite graphs with diameter 3 (resp. diameter 5, and diameter 6), whereas it is fixed-parameter tractable when the parameter is the number of leaves in the associated star. Further, we show that for star-convex bipartite graphs with diameters 5, and 6, the domination

Parametric timed automata (PTA) extend timed automata (TA) with parameters instead of fixed timing constraints, providing the flexibility to accommodate uncertainties during the design phase. Once a parametric model is obtained, the next step is finding the optimal parameters such that the resulting TA satisfies the specifications. This paper introduces a new algorithm for determining parameters from safety specifications for PTA with bounded integer parameters and no nested cycles. The algorithm searches for unsafe paths through a depth-first search and generates parameter constraints. In particular, the realizability of simple and cyclic paths are encoded via mixed integer linear programming and non-linear programming problems. Then, the parameter constraints rendering the path unrealizable are derived via quantifier elimination. The accumulated constraints through the depth-first search guarantee that a parameter valuation satisfying these constraints solves the synthesis problem. The results are illustrated over benchmarks.

Motivated by computing duplication patterns in sequences, a new problem called the longest letter-duplicated subsequence (LLDS) is proposed. Given a sequence S of length n, a letter-duplicated subsequence is a subsequence of S in the form of (x_1^{d_1}x_2^{d_2}ldots x_k^{d_k}) with (x_iin Sigma ), (x_jne x_{j+1}) and (d_ige 2) for all i in [k] and j in ([k-1]). A linear time algorithm for computing a longest letter-duplicated subsequence (LLDS) of S can be easily obtained. In this paper, we focus on two variants of this problem: (1) ‘all-appearance’ version, i.e., all letters in (Sigma ) must appear in the solution, and (2) the weighted version. For the former, we obtain dichotomous results: We prove that, when each letter appears in S at least 4 times, the problem and a relaxed version on feasibility testing (FT) are both NP-hard. The reduction is from ((3^+,1,2^-))-SAT, where all 3-clauses (i.e., containing 3 lals) are monotone (i.e., containing only positive literals) and all 2-clauses contain only negative literals. We then show that when each letter appears in S at most 3 times, then the problem admits an O(n) time algorithm. Finally, we consider the weighted version, where the weight of a block (x_i^{d_i} (d_ige 2)) could be any positive function which might not grow with (d_i). We give a non-trivial (O(n^2)) time dynamic programming algorithm for this version, i.e., computing an LD-subsequence of S whose weight is maximized.

Invariant relations are used to analyze while loops; while their primary application is to derive the function of a loop, they can also be used to derive loop invariants, weakest preconditions, strongest postconditions, sufficient conditions of correctness, necessary conditions of correctness, and termination conditions of loops. In this paper we present two generic invariant relations that capture the semantics of loops whose loop body applies affine transformations on numeric variables.

In this paper, we propose a decision procedure of reachability for a linear system (xi '=Axi +u), where the matrix (A's) eigenvalues can be arbitrary algebraic number and the input u is a vector of trigonometric-exponential polynomials. If the initial set contains only one point, the reachability problem under consideration is reduced to the decidability of the sign of trigonometric-exponential polynomial and then achieved by being reduced to verification of a series of univariate polynomial inequalities through Taylor expansions of the related exponential functions and trigonometric functions. If the initial set is open semi-algebraic, we will propose a decision procedure based on OpenCAD and an algorithm of real root isolation derived from the sign-deciding procedure for the trigonometric-exponential polynomials. The experimental results indicate the efficiency of our approach. Under the assumption of Schanuel’s Conjecture, the above procedures are complete for bounded time except for several cases.

Red–black (RB) trees are one of the most efficient variants of balanced binary search trees. However, they have often been criticized for being too complicated, hard to explain, and unsuitable for pedagogical purposes, particularly their delete operation. Sedgewick (in: Dagstuhl Workshop on Data Structures, 2008. https://sedgewick.io/wp-content/themes/sedgewick/papers/2008LLRB.pdf) identified the length of code as the root of the problems and introduced left-leaning red–black (LLRB) trees. The delete operation of LLRB trees has a compact recursive code. Unfortunately, it may perform many unnecessary operations. The crux of the deletion algorithm is dealing with a “deficient” subtree, that is one whose black-height has become one less than that of its sibling subtree. In this paper, we revisit 2–3 red–black trees and propose a parity-seeking delete algorithm with the basic idea of making a deficient subtree on a par with its sibling: either by fixing the deficient subtree or by turning the sibling deficient as well, ascending deficiency to the parent node. Interestingly, the proposed parity-seeking delete algorithm also works for 2–3–4 RB trees. Our experiments show that the proposed parity-seeking delete algorithm is as efficient as the best preceding RB trees. The proposed parity-seeking delete algorithm is easily understandable and suitable for pedagogical and practical purposes.