Greg Aloupis

Decomposition of Multiple Coverings into More Parts

We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any locally finite αk-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth. The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery life.

Coloring geometric range spaces

We study several coloring problems for geometric range-spaces. In addition to their theoretical interest, some of these problems arise in sensor networks. Given a set of points in ℝ2 or ℝ3, we want to color them so that every region of a certain family (e.g.,xa0every disk containing at least a certain number of points) contains points of many (say, k) different colors. In this paper, we think of the number of colors and the number of points as functions of k. Obviously, for a fixed k using k colors, it is not always possible to ensure that every region containing k points has all colors present. Thus, we introduce two types of relaxations: either we allow the number of colors used to increase to c(k), or we require that the number of points in each region increases to p(k).Symmetrically, given a finite set of regions in ℝ2 or ℝ3, we want to color them so that every point covered by a sufficiently large number of regions is contained in regions of k different colors. This requires the number of covering regions or the number of allowed colors to be greater than k.The goal of this paper is to bound these two functions for several types of region families, such as halfplanes, halfspaces, disks, and pseudo-disks. This is related to previous results of Pach, Tardos, and Tóth on decompositions of coverings.

Classic Nintendo Games Are (Computationally) Hard

We prove NP-hardness results for five of Nintendo’s largest video game franchises: Mario, Donkey Kong, Legend of Zelda, Metroid, and Pokemon. Our results apply to generalized versions of Super Mario Bros. 1, 3, Lost Levels, and Super Mario World; Donkey Kong Country 1–3; all Legend of Zelda games; all Metroid games; and all Pokemon role-playing games. In addition, we prove PSPACE-completeness of the Donkey Kong Country games and several Legend of Zelda games.

Reconfiguration of Cube-Style Modular Robots Using O(logn) Parallel Moves

We consider a model of reconfigurable robot, introduced and prototyped by the robotics community. The robot consists of independently manipulable unit-square atoms that can extend/contract arms on each side and attach/detach from neighbors. The optimal worst-case number of sequential moves required to transform one connected configuration to another was shown to be θ(n) at ISAAC 2007. However, in principle, atoms can all move simultaneously. We develop a parallel algorithm for reconfiguration that runs in only O(logn) parallel steps, although the total number of operations increases slightly to θ(n logn). The result is the first (theoretically) almost-instantaneous universally reconfigurable robot built from simple units.

Non-crossing matchings of points with geometric objects

Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a non-crossing matching between point-object pairs. In this paper, we address the algorithmic problem of determining whether a non-crossing matching exists between a given point-object pair. We show that when the objects we match the points to are finite point sets, the problem is NP-complete in general, and polynomial when the objects are on a line or when their size is at most 2. When the objects are line segments, we show that the problem is NP-complete in general, and polynomial when the segments form a convex polygon or are all on a line. Finally, for objects that are straight lines, we show that the problem of finding a min-max non-crossing matching is NP-complete.

Classic Nintendo games are (computationally) hard

We prove NP-hardness results for five of Nintendos largest video game franchises: Mario, Donkey Kong, Legend of Zelda, Metroid, and Pokemon. Our results apply to generalized versions of Super Mario Bros.?1-3, The Lost Levels, and Super Mario World; Donkey Kong Country 1-3; all Legend of Zelda games; all Metroid games; and all Pokemon role-playing games. In addition, we prove PSPACE-completeness of the Donkey Kong Country games and several Legend of Zelda games.

Establishing strong connectivity using optimal radius half-disk antennas

Given a set S of points in the plane representing wireless devices, each point equipped with a directional antenna of radius r and aperture angle @a>=180^o, our goal is to find orientations and a minimum r for these antennas such that the induced communication graph is strongly connected. We show that r=3 if @a@?[180^o,240^o), r=2 if @a@?[240^o,270^o), r=2sin(36^o) if @a@?[270^o,288^o), and r=1 if @a>=288^o suffices to establish strong connectivity, assuming that the longest edge in the Euclidean minimum spanning tree of S is 1. These results are worst-case optimal and match the lower bounds presented in [I. Caragiannis, C. Kaklamanis, E. Kranakis, D. Krizanc, A. Wiese, Communication in wireless networks with directional antennae, in: Proc. of the 20th Symp. on Parallelism in Algorithms and Architectures, 2008, pp. 344-351]. In contrast, r=2 is sometimes necessary when @a<180^o.

Highway hull revisited

A highway H is a line in the plane on which one can travel at a greater speed than in the remaining plane. One can choose to enter and exit H at any point. The highway time distance between a pair of points is the minimum time required to move from one point to the other, with optional use of H. The highway hullH(S,H) of a point set S is the minimal set containing S as well as the shortest paths between all pairs of points in H(S,H), using the highway time distance. We provide a @Q(nlogn) worst-case time algorithm to find the highway hull under the L1 metric, as well as an O(nlog^2n) time algorithm for the L2 metric which improves the best known result of O(n^2) [F. Hurtado, B. Palop, V. Sacristan, Diagramas de Voronoi con distancias temporales, in: Actas de los VIII Encuentros de Geometra Computacional, 1999, pp. 279-288 (in Spanish); B. Palop, Algorithmic problems on proximity and location under metric constraints, PhD thesis, Universitat Politecnica de Catalunya, 2003]. We also define and construct the useful region of the plane: the region that a highway must intersect in order that the shortest path between at least one pair of points uses the highway.

Blocking Colored Point Sets

This paper studies problems related to visibility among points in the plane. A point x blocks two points v and w if x is in the interior of the line segment vw. A set of points P is k-blocked if each point in P is assigned one of k colours, such that distinct points v;w2 P are assigned the same colour if and only if some other point in P blocks v and w. The focus of this paper is the conjecture that each k-blocked set has bounded size (as a function of k). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all setsfn1;n2;n3;n4g such that some 4-blocked set has exactly ni points in the i-th colour class. Amongst other results, for innitely many values of

Efficient reconfiguration of lattice-based modular robots

Abstract Modular robots consist of many identical units (or atoms) that can attach together and perform local motions. By combining such motions, one can achieve a reconfiguration of the global shape of a robot. The term modular comes from the idea of grouping together a fixed number of atoms into a metamodule, which behaves as a larger individual component. Recently, a fair amount of research has focused on algorithms for universal reconfiguration using Crystalline and Telecube metamodules, which use expanding/contracting cubical atoms. From an algorithmic perspective, this work has achieved some of the best asymptotic reconfiguration times under a variety of different physical models. In this paper we show that these results extend to other types of modular robots, thus establishing improved upper bounds on their reconfiguration times. We describe a generic class of modular robots, and we prove that any robot meeting the generic class requirements can simulate the operation of a Crystalline atom by forming a six-arm structure. Previous reconfiguration bounds thus transfer automatically by substituting the six-arm structures for the Crystalline atoms. We also discuss four prototyped robots that satisfy the generic class requirements: M-TRAN, SuperBot, Molecube, and RoomBot.