Constrained generalized Delaunay graphs are plane spanners

We look at generalized Delaunay graphs in the constrained setting by introducing line segments which the edges of the graph are not allowed to cross. Given an arbitrary convex shape C, a constrained Delaunay graph is constructed by adding an edge between two vertices p and q if and only if there exists a homothet of C with p and q on its boundary that does not contain any other vertices visible to p and q. We show that, regardless of the convex shape C used to construct the constrained Delaunay graph, there exists a constant t (that depends on C) such that it is a plane t-spanner of the visibility graph

Pay secrecy and perceptions of fairness in a university environment

Le texte intégral de ce document de travail n'est pas disponible en ligne. Une copie papier est disponible à l'Annexe de la bibliothéque. Effectuez une recherche par titre dans le catalogue pour réserver le document. // The full text of this working paper is not available online. A print copy is available in the Library Annex. Search by title in the catalogue to request the paper

Point location in well-shaped meshes using jump-and-walk

We present results on executing point location queries in well-shaped meshes in R2 and R3 using the Jump- And-Walk paradigm. If the jump step is performed on a nearest-neighbour search structure built on the vertices of the mesh, we demonstrate that the walk step can be performed in guaranteed constant time. Constant time for the walk step holds even if the jump step starts with an approximate nearest neighbour

Constrained generalized Delaunay are plane spanners

We look at generalized Delaunay graphs in the constrained setting by introducing line segments which the edges of the graph are not allowed to cross. Given an arbitrary convex shape C, a constrained Delaunay graph is constructed by adding an edge between two vertices p and q if and only if there exists a homothet of C with p and q on its boundary that does not contain any other vertices visible to p and q. We show that, regardless of the convex shape C used to construct the constrained Delaunay graph, there exists a constant t (that depends on C) such that it is a plane t-spanner of the visibility graph. Furthermore, we reduce the upper bound on the spanning ratio for the special case where the empty convex shape is an arbitrary rectangle to 2⋅(2l/s+1), where l and s are the length of the long and short side of the rectangle

Voronoi games and epsilon nets

Competitive facility location is concerned with the strategic placement of facilities by competing market players. In the Discrete Voronoi Game V G(k; l), two players P1 and P2, respectively, strive to attract as many of n users as possible. Initially, P1 first chooses a set F of k locations in the plane to place its facilities. Then, P2 chooses a set S of l locations in the plane to place its facilities, where S ∩ F = Ø. Finally, the users choose the facilities based on the nearest-neighbour rule. The goal for each player is to maximize the number of users served by its set of facilities. By establishing a connection between V G(2; 1) and ε-nets, we provide an algorithm running in O(n log4 n) time to find a 7/4 -factor approximation of the optimal strategy of P1 in V G(2; 1). We also prove that for any real number 0 < α < 1, there exists a placement of 42/α facilities by P1 such that P2 can serve at most αn users by placing one facility

Discrete Voronoi games and ε-nets, in two and three dimensions

The one-round discrete Voronoi game, with respect to an n-point user set U, consists of two players Player 1 (P1) and Player 2 (P2). At first, P1 chooses a set of facilities F1 following which P2 chooses another set of facilities F2, disjoint from F1. The payoff of P2 is defined as the cardinality of the set of points in U which are closer to a facility in F2 than to every facility in F1, and the payoff of P1 is the difference between the number of users in U and the payoff of P2. The objective of both the players in the game is to maximize their respective payoffs. In this paper we study the one-round discrete Voronoi game where P1 places k facilities and P2 places one facility. We denote this game as VG(k,1). Although the optimal solution of this game can be found in polynomial time, the polynomial has a very high degree. In this paper, we focus on achieving approximate solutions to VG(k,1) with significantly better running times. We provide a constant-factor approximate solution to the optimal strategy of P1 in VG(k,1) by establishing a connection between VG(k,1) and weak ε-nets. To the best of our knowledge, this is the first time that Voronoi games are studied from the point of view of ε-nets

Essential constraints of edge-constrained proximity graphs

Given a plane forest F = (V, E) of |V | = n points, we find the minimum set S ⊆ E of edges such that the edge-constrained minimum spanning tree over the set V of vertices and the set S of constraints contains F . We present an O(n log n)-time algorithm that solves this problem. We generalize this to other proximity graphs in the constraint setting, such as the relative neighbourhood graph, Gabriel graph, β-skeleton and Delaunay triangulation. We present an algorithm that identifies the minimum set S ⊆ E of edges of a given plane graph I = (V, E) such that I ⊆ CGβ (V, S) for 1 ≤ β ≤ 2, where CGβ (V, S) is the constraint β-skeleton over the set V of vertices and the set S of constraints. The running time of our algorithm is O(n), provided that the constrained Delaunay triangulation of I is given

Optimal Art Gallery Localization is NP-hard

Art Gallery Localization (AGL) is the problem of placing a set T of broadcast towers in a simple polygon P in order for a point to locate itself in the interior. For any point p∈P: for each tower t∈T∩V(p) (where V(p) denotes the visibility polygon of p) the point p receives the coordinates of t and the Euclidean distance between t and p. From this information p can determine its coordinates. We study the computational complexity of AGL problem. We show that the problem of determining the minimum number of broadcast towers that can localize a point anywhere in a simple polygon P is NP-hard. We show a reduction from Boolean Three Satisfiability problem to our problem and give a proof that the reduction takes polynomial time

A note on the unsolvability of the weighted region shortest path problem

Let S be a subdivision of the plane into polygonal regions, where each region has an associated positive weight. The weighted region shortest path problem is to determine a shortest path in S between two points s,tεR2, where the distances are measured according to the weighted Euclidean metric - the length of a path is defined to be the weighted sum of (Euclidean) lengths of the sub-paths within each region. We show that this problem cannot be solved in the Algebraic Computation Model over the Rational Numbers (ACMQ). In the ACMQ, one can compute exactly any number that can be obtained from the rationals Q by applying a finite number of operations from +, -, ×, ÷, √k, for any integer k≥2. Our proof uses Galois theory and is based on Bajaj's technique

Similarity of polygonal curves in the presence of outliers

The Fréchet distance is a well studied and commonly used measure to capture the similarity of polygonal curves. Unfortunately, it exhibits a high sensitivity to the presence of outliers. Since the presence of outliers is a frequently occurring phenomenon in practice, a robust variant of Fréchet distance is required which absorbs outliers. We study such a variant here. In this modified variant, our objective is to minimize the length of subcurves of two polygonal curves that need to be ignored (MinEx problem), or alternately, maximize the length of subcurves that are preserved (MaxIn problem), to achieve a given Fréchet distance. An exact solution to one problem would imply an exact solution t