Abstract Huemer et al. (Discrete Mathematics, 2019) proved that for any two point sets R and B with $$|R|=|B|$$ | R | = | B | , the perfect matching that matches points of R with points of B, and maximizes the total squared Euclidean distance of the matched pairs, has the property that all the disks induced by the matching have a common point. Each pair of matched points $$p\in R$$ p ∈ R and $$q\in B$$ q ∈ B induces the disk of smallest diameter that covers p and q. Following this research line, in this paper we consider the perfect matching that maximizes the total Euclidean distance. First, we prove that this new matching for R and B does not always ensure the common intersection property of the disks. Second, we extend the study of this new matching for sets of 2n uncolored points in the plane, where a matching is just a partition of the points into n pairs. As the main result, we prove that in this case all disks of the matching do have a common point.

We give improved lower bounds for M(n, d), for various positive integers d and n with d <; n, where M(n, d) is the largest number of permutations on n symbols with pairwise Hamming distance at least d. Permutation arrays are used for constructing error correcting permutation codes, which have been proposed for power-line communications. We describe two techniques, which use a modified Kronecker product and a tiling operation, called doubling. Our techniques improve the size of permutation arrays, and improve lower bounds on M(n, d), for infinitely many n and d, d <; n.

Monadic predicates play a prominent role in many decidable cases, including decision procedures for symbolic automata. We are here interested in discovering whether a formula can be rewritten into a Boolean combination of monadic predicates. Our setting is quantifier-free formulas whose satisfiability is decidable, such as linear arithmetic. Here we develop a semidecision procedure for extracting a monadic decomposition of a formula when it exists.

This paper addresses a synchronization problem that arises when a team of robots needs to communicate while repeatedly performing assigned tasks in a cooperative scenario. Each robot has a limited communication range and moves along a previously defined closed trajectory. When two robots are close enough, a communication link may be established, allowing the robots to exchange information. The goal is to schedule the motions such that the entire system can be synchronized for maximum information exchange; that is, every pair of neighbors always visit the feasible communication link at the same time. An algorithm for scheduling the team of robots in this scenario is proposed and a robust framework that assures the synchronization of a large team of robots is presented. Simulations, experiments, and computational results demonstrate the applicability of the algorithm. The approach allows the design of fault-tolerant systems that can be used for multiple tasks, such as surveillance, area exploration, and searching for targets in hazardous environments, among others.

Consider a finite sequence of permutations of the elements $1,\ldots,n$ with the property that each element changes its position by at most 1 from any permutation to the next. We call such a sequence a tangle, and we define a move of element $i$ to be a maximal subsequence of at least two consecutive permutations during which its positions form an arithmetic progression of common difference +1 or -1. We prove that for any initial and final permutations, there is a tangle connecting them in which each element makes at most 5 moves, and another in which the total number of moves is at most 4n. On the other hand, there exist permutations that require at least 3 moves for some element, and at least 2n-2 moves in total. If we further require that every pair of elements exchange positions at most once, then any two permutations can be connected by a tangle with at most $O(\log n)$ moves per element, but we do not know whether this can be reduced to O(1) per element, or to O(n) in total. A key tool is the introduction of certain restricted classes of tangle that perform pattern-avoiding permutations.

This paper addresses a synchronization problem that arises when a team of aerial robots (ARs) need to communicate while performing assigned tasks in a cooperative scenario. Each robot has a limited communication range and flies within a previously assigned closed path. When two robots are close enough, a communication link may be established allowing the robots to share information. The goal is to schedule the flights such that the entire system can be synchronized for maximum information exchange, that is, every pair of neighbors are on the feasible communication link at the same time. We propose an algorithm for scheduling a team of robots in this scenario and propose a robust framework where the synchronization of a large team of robots is assured. The approach allows us to design a fault-tolerant system that can be used for multiple tasks such as surveillance, area exploration, searching for targets in a hazardous environment, and assembly and structure construction, to name a few.

Consider an orthogonal art gallery partitioned into n rectangular rooms. If two rooms are adjacent, there is a door connecting them and a guard positioned at this door will see both rooms. In Czyzowicz et al. [(1994) Guarding rectangular art galleries. Discrete Appl. Math., 50, 149–157], it is shown that any rectangular gallery can be guarded with ⌈n/2⌉ guards. We prove that the same bound holds for L-shape polygons. We extend it to staircases and prove that an orthogonal staircase with n rooms and r reflex vertices can be guarded with ⌈(n+⌊ r/2⌋)/2⌉ guards. Then we prove an upper bound on the number of guards for arbitrary orthogonal polygon with orthogonal holes. This result improves the previous bound by Czyzowicz et al. [(1994) Guarding rectangular art galleries. Discrete Appl. Math., 50, 149–157] (even in the case of polygon without holes).

We address the problem of realizing a given distance matrix by a planar phylogenetic network with a minimum number of faces. With the help of the popular software SplitsTree4, we start by approximating the distance matrix with a distance metric that is a linear combination of circular splits. The main results of this paper are the necessary and sufficient conditions for the existence of a network with a single face. We show how such a network can be constructed, and we present a heuristic for constructing a network with few faces using the first algorithm as the base case. Experimental results on biological data show that this heuristic algorithm can produce phylogenetic networks with far fewer faces than the ones computed by SplitsTree4, without affecting the approximation of the distance matrix.

The ham-sandwich theorem states that, given d ≥ 2 measures in ℝd, it is possible to divide all of them in half with a single (d – 1)-dimensional hyperplane. We study an orthogonal version of the ham-sandwich theorem and define an orthogonal cut using at most d hyperplanes orthogonal to coordinate axes. For example, a hyperplane orthogonal to a coordinate axis and the boundary of an orthant are orthogonal cuts. We prove that any three measures in ℝ3 can be divided in half each with a single orthogonal cut. Applied to point measures, it implies that any three finite sets of points in ℝ3 can be simultaneously bisected by an orthogonal cut. We present an algorithm for computing an orthogonal ham-sandwich cut in O(n log n) time.

Network-wide broadcast (simply broadcast) is a frequently used operation in wireless ad hoc networks (WANETs). One promising practical approach for energy-efficient broadcast is to use localized algorithms to minimize the number of nodes involved in the propagation of the broadcast messages. In this context, the minimum forwarding set problem (MFSP) (also known as multipoint relay (MPR) problem) has received a considerable attention in the research community. Even though the general form of the problem is shown to be NP-complete, the complexity of the problem has not been known under the practical application context of ad hoc networks. In this paper, we present a polynomial time algorithm to solve the MFSP for wireless network under unit disk coverage model. We prove the existence of some geometrical properties for the problem and then propose a polynomial time algorithm to build an optimal solution based on these properties. To the best of our knowledge, our algorithm is the first polynomial time solution to the MFSP under the unit disk coverage model. We believe that the work presented in this paper will have an impact on the design and development of new algorithms for several wireless network applications including energy-efficient multicast, broadcast, and topology control protocols for WANETs and sensor networks.

Given a finite set of points S in Rd, consider visiting thepoints in S with a polygonal path that makes a minimum number ofturns, or equivalently, has the the minimum number of segments(links). We call this minimization problem the minimum linkspanning path problem. This natural problem has appeared severaltimes in the literature under different variants. The simplest oneis where the allowed paths are axis-aligned. Let L(S) be theminimum number of links of an axis-aligned path for S denote by Gdn the d-dimensional grid of size n. Kranakis, Krizanc andMeertens (Ars Combinatoria, vol. 38, pp. 177--192, 1994)showed that in 2-dimensions L(G2n)=2n-1 and in three dimensions 4/3 n2-O(n)< L(G3n) < 3/2 n2+O(n). Kranakiset al. conjectured that, for all d ≥ 3, L(Gdn)= d/d-1 nd-1 ± O(nd-2). We prove theconjecture for d=3 by showing that L(G3n) ≥ 3/2 n2 -O(n). For d=4, we prove that 4/3 n3 -O(n2) ≤ L(G4n) ≤ 4/3 n3 +O(n5/2).For general d, we give new estimates on L(Gdn), that bring usvery close to the conjectured value. The new lower bound of (1+ 1/d)nd-1-O(nd-2) improves previous result byCollins and Moret (Information Processing Letters, vol. 68,pp. 317--319, 1998), while the new upper bound of (1+ 1/d-1)nd-1+O(nd-3/2) differs from the conjecturedvalue only in the lower order terms. For arbitrary point sets, we give an exact bound on the minimumnumber of links needed in an axis-aligned path traversing any planar n-point set. We obtain similar tight estimates (within 1) in anynumber of dimensions d. For the general problem of traversing anarbitrary set of points in Rd with an axis-aligned spanning pathhaving a minimum number of links, we present a constant ratio(depending on the dimension d) approximation algorithm.

A classical result in phylogenetic trees is that a binary phylogenetic tree adhering to the molecular clock hypothesis exists if and only if the matrix of distances between taxa is ultrametric. The ultrametric condition is very restrictive. In this paper we study phylogenetic networks that can be constructed assuming the molecular clock hypothesis. We characterize distance matrices that admit such networks for 3 and 4 taxa. We also design two algorithms for constructing networks optimizing the least-squares fit.

We present a new method of constructing a phylogenetic network from a given phylogenetic tree. It is based on a procedure that locally improves the tree. The procedure is quite general and can be applied to phylogenetic networks. By repeating local improvements user can introduce a given number of recombination cycles. A sequence of networks with decreasing distance deviation can be generated. The algorithm is efficient and shows a good performance on an example with plants. This is due to the fact that the update in every step is local and optimal.

A classical result in phylogenetic trees is that a binary phylogenetic tree adhering to the molecular clock hypothesis exists if and only if the matrix of distances between taxa is ultrametric. The ultrametric condition is very restrictive. In this paper we study phylogenetic networks that can be constructed assuming the molecular clock hypothesis. We characterize distance matrices that admit such networks for 3 and 4 taxa. We design an efficient algorithm for a special class of phylogenetic networks that can detect the existence of a network and constructs it.

Given a polyhedral terrain T with n vertices, the two-watchtower problem for T calls for finding two vertical segments, called watchtowers, of smallest common height, whose bottom endpoints (bases) lie on T, and whose top endpoints guard T, in the sense that each point on T is visible from at least one of them. In this paper we present the following results for the two-watchtower problem in R2 and R3: (1) We show that the discrete two-watchtowers problem in R2, where the bases are constrained to lie at vertices of T, can be solved in O(n2 log4n) time, significantly improving previous solutions. The algorithm works, without increasing its asymptotic running time, even if, one of the towers is allowed to be placed anywhere on T. (2) We show that the continuous two-watchtower problem in R2, where the bases can lie anywhere on T, can be solved in O(n3α(n)log3n) time, again significantly improving previous results. (3) Still in R2, we show that the continuous version of the problem of guarding a finite set P ⊂ T of m points by two watchtowers of smallest height can be solved in O(mn log4n) time. (4) The discrete version of the two-watchtower problem in R3 can be solved in O(n11/3 polylog(n)) time; this is the first nontrivial result for this problem in R3.

We study minimally rigid graphs in the plane or plane isostatic graphs. These graphs (also called Laman graphs) admit characterizations based on decomposition into trees (Crapo's theorem and Récski's theorem). Tree partitions can be viewed as certificates of plane isostatic graphs. Unfortunately, they require Ω(n2) time to verify their validity where n is the number of vertices in the graph. We present a new construction (which can be viewed as a hierarchical decomposition of the graph) called red-black hierarchy that (i) is a certificate for plane isostatic graphs, and (ii) can be verified in linear time. We also show that it can be computed in O(n2) time.A classical result in Rigidity Theory by Henneberg [9] states that the plane isostatic graphs can be constructed incrementally by special vertex insertions. We study the following computational problem: given a Laman graph G, compute a sequence of Henneberg insertions that yields G. We show that the red-bl ack hierarchy can be used to compute a Henneberg construction in O(n2) time. Applied to planar graphs our algorithm can speed up a recent algorithm by Haas et al. [8] for embedding a planar Laman graph as a pointed pseudo-triangulation by a factor of O(n).

Abstract Given a pair of start and target configurations, each consisting of n pairwise disjoint disks in theplane, what is the minimum number of moves that suffice for transforming the start configuration into the target configuration? In one move a disk is lifted from the plane and placed back in the plane atanother location, without intersecting any other disk. We discuss efficient algorithms for this task and estimate their number of moves under different assumptions on disk radii. We then extend our results forarbitrary disks to systems of pseudodisks, in particular to sets of homothetic copies of a convex object. 1 Introduction Consider a set (system) of n pairwise disjoint objects in the plane that need to be brought from a givenstart (initial) configuration S into a desired goal (target) configuration T. The motion planning problemfor such a system is that of computing a sequence of object motions (schedule) that achieves this task. If

We consider the existence and efficient construction of bounded curvature paths traversing constant-width regions of the plane, called corridors. We make explicit a width threshold τ with the property that (a) all corridors of width at least τ admit a unit-curvature traversal and (b) for any width w < τ there exist corridors of width w with no such traversal. Applications to the design of short, but not necessarily shortest, and high clearance, but not necessarily maximum clearance, curvature-bounded paths in general polygonal domains, are also discussed.

Abstract Thep‐center problem is to locatepfacilities on a network so as to minimize the largest distance from a demand point to its nearest facility. Thep‐median problem is to locatepfacilities on a network so as to minimize the average distance from a demand point to its closest facility. We consider these problems when the network can be modeled by an interval or circular‐arc graph whose edges have unit lengths. We provide, given the interval model of annvertex interval graph, anO(n) time algorithm for the 1‐median problem on the interval graph. We also show how to solve thep‐median problem, for arbitraryp, on an interval graph inO(pnlogn) time and on a circular‐arc graph inO(pn2logn) time. We introduce a spring representation of the objective function and show how to solve thep‐center problem on a circular‐arc graph inO(pn) time, assuming that the arc endpoints are sorted. © 2002 Wiley Periodicals, Inc.

Article Generalizing ham sandwich cuts to equitable subdivisions Share on Authors: Sergei Bespamyatnikh Department of Computer Science, University of British Columbia Department of Computer Science, University of British ColumbiaView Profile , David Kirkpatrick Department of Computer Science, University of British Columbia Department of Computer Science, University of British ColumbiaView Profile , Jack Snoeyink Department of Computer Science, University of British Columbia Department of Computer Science, University of British ColumbiaView Profile Authors Info & Claims SCG '99: Proceedings of the fifteenth annual symposium on Computational geometryJune 1999 Pages 49–58https://doi.org/10.1145/304893.304909Online:13 June 1999Publication History 8citation363DownloadsMetricsTotal Citations8Total Downloads363Last 12 Months7Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Given a set S of n points in i&dimensional space, and an Lt metric, the dynamic clos-