Guilherme Dias da Fonseca

The stable marriage problem with restricted pairs

A stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching. In an instance of the STABLE MARRIAGE problem, each of the n men and n women ranks the members of the opposite sex in order of preference. It is well known that at least one stable matching exists for every STABLE MARRIAGE problem instance. We consider extensions of the STABLE MARRIAGE problem obtained by forcing and by forbidding sets of pairs. We present a characterization for the existence of a solution for the STABLE MARRIAGE WITH FORCED AND FORBIDDEN PAIRS problem. In addition, we describe a reduction of the STABLE MARRIAGE WITH FORCED AND FORBIDDEN PAIRS problem to the STABLE MARRIAGE WITH FORBIDDEN PAIRS problem. Finally, we also present algorithms for finding a stable matching, all stable pairs and all stable matchings for this extension. The complexities of the proposed algorithms are the same as the best known algorithms for the unrestricted version of the problem.

Efficient sub-5 approximations for minimum dominating sets in unit disk graphs

Abstract A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their applicability in wireless ad-hoc networks. Because the minimum dominating set problem for unit disk graphs is NP-hard, numerous approximation algorithms have been proposed in the literature, including some PTASs. However, since the proposal of a linear-time 5-approximation algorithm in 1995, the lack of efficient algorithms attaining better approximation factors has aroused attention. We introduce an O ( n + m ) algorithm that takes the usual adjacency representation of the graph as input and outputs a 44/9-approximation. This approximation factor is also attained by a second algorithm, which takes the geometric representation of the graph as input and runs in O ( n log n ) time regardless of the number of edges. Additionally, we propose a 43/9-approximation which can be obtained in O ( n 2 m ) time given only the graphs adjacency representation. It is noteworthy that the dominating sets obtained by our algorithms are also independent sets.

Optimal area-sensitive bounds for polytope approximation

Approximating convex bodies is a fundamental question in geometry and has applications to a wide variety of optimization problems. Given a convex body K in REd for fixed d, the objective is to minimize the number of vertices or facets of an approximating polytope for a given Hausdorff error ε. The best known uniform bound, due to Dudley (1974), shows that O((diam(K)/ε)(d-1)/2) facets suffice. While this bound is optimal in the case of a Euclidean ball, it is far from optimal for skinny convex bodies. We show that, under the assumption that the width of the body in any direction is at least ε, it is possible to approximate a convex body using O(√area(K)/ε(d-1)/2) facets, where area(K) is the surface area of the body. This bound is never worse than the previous bound and may be significantly better for skinny bodies. This bound is provably optimal in the worst case and improves upon our earlier result (which appeared in SODA 2012). Our improved bound arises from a novel approach to sampling points on the boundary of a convex body in order to stab all (dual) caps of a given width. This approach involves the application of an elegant concept from the theory of convex bodies, called Macbeath regions. While Macbeath regions are defined in terms of volume considerations, we show that by applying them to both the original body and its dual, and then combining this with known bounds on the Mahler volume, it is possible to achieve the desired width-based sampling.

Kinetic heap-ordered trees: tight analysis and improved algorithms

The most natural kinetic data structure for maintaining the maximum of a collection of continuously changing numbers is the kinetic heap. Basch, Guibas, and Ramkumar proved that the maximum number of events processed by a kinetic heap with n numbers changing as linear functions of time is O(n log2 n) and Ω (n log n). We prove that this number is actually Θ(n log n). In the kinetic heap, a linear number of events are stored in a priority queue, consequently, it takes O(log n) time to determine the next event at each iteration. We also present a modified version of the kinetic heap that processes O(n log n/log log n) events, with the same O(log n) time complexity to determine the next event.

A unified approach to approximate proximity searching

The inability to answer proximity queries efficiently for spaces of dimension d > 2 has led to the study of approximation to proximity problems. Several techniques have been proposed to address different approximate proximity problems. In this paper, we present a new and unified approach to proximity searching, which provides efficient solutions for several problems: spherical range queries, idempotent spherical range queries, spherical emptiness queries, and nearest neighbor queries. In contrast to previous data structures, our approach is simple and easy to analyze, providing a clear picture of how to exploit the particular characteristics of each of these problems. As applications of our approach, we provide simple and practical data structures that match the best previous results up to logarithmic factors, as well as advanced data structures that improve over the best previous results for all aforementioned proximity problems.

Linear Time Approximation for Dominating Sets and Independent Dominating Sets in Unit Disk Graphs

A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their application in wireless ad-hoc networks. Since the minimum dominating set problem for unit disk graphs is NP-hard, several approximation algorithms with different merits have been proposed in the literature. On one extreme, there is a linear time 5-approximation algorithm. On another extreme, there are two PTAS whose running times are polynomials of very high degree. We introduce a linear time approximation algorithm that takes the usual adjacency representation of the graph as input and attains a 44/9 approximation factor. This approximation factor is also attained by a second algorithm we present, which takes the geometric representation of the graph as input and runs in O(n logn) time, regardless of the number of edges. The analysis of the approximation factor of the algorithms, both of which are based on local improvements, exploits an assortment of results from discrete geometry to prove that certain graphs cannot be unit disk graphs. It is noteworthy that the dominating sets obtained by our algorithms are also independent sets.

Tradeoffs in Approximate Range Searching Made Simpler

Range searching is a fundamental problem in computational geometry. The problem involves preprocessing a set of n points in R^d into a data structure, so that it is possible to determine the subset of points lying within a given query range. In approximate range searching, a parameter eps epsiv > 0 is given, and for a given query range R the points lying within distance eps diam(R) of the ranges boundary may be counted or not. In this paper we present three results related to the issue of tradeoffs in approximate range searching. First, we introduce the range sketching problem. Next, we present a space-time tradeoff for smooth convex ranges, which generalize spherical ranges. Finally, we show how to modify the previous data structure to obtain a space-time tradeoff for simplex ranges. In contrast to existing results, which are based on relatively complex data structures, all three of our results are based on simple, practical data structures.

Optimal approximate polytope membership

In the polytope membership problem, a convex polytope K in ℝd is given, and the objective is to preprocess K into a data structure so that, given a query point q ∈ ℝd, it is possible to determine efficiently whether q ∈ K. We consider this problem in an approximate setting and assume that d is a constant. Given an approximation parameter e > 0, the query can be answered either way if the distance from q to Ks boundary is at most e times Ks diameter. Previous solutions to the problem were on the form of a space-time trade-off, where logarithmic query time demands O(1/ed−1) storage, whereas storage O(1/e(d−1)/2) admits roughly O(1/e(d−1)/8) query time. In this paper, we present a data structure that achieves logarithmic query time with storage of only O(1/e(d−1)/2), which matches the worst-case lower bound on the complexity of any e-approximating polytope. Our data structure is based on a new technique, a hierarchy of ellipsoids defined as approximations to Macbeath regions. As an application, we obtain major improvements to approximate Euclidean nearest neighbor searching. Notably, the storage needed to answer e-approximate nearest neighbor queries for a set of n points in O(log n/e) time is reduced to O(n/ed/2). This halves the exponent in the e-dependency of the existing space bound of roughly O(n/ed), which has stood for 15 years (Har-Peled, 2001).

On the recognition of unit disk graphs and the Distance Geometry Problem with Ranges

We introduce a method to decide whether a graph G admits a realization on the plane in which two vertices lie within unitary distance from one another exactly if they are neighbors in G . Such graphs are called unit disk graphs, and their recognition is a known NP-hard problem. By iteratively discretizing the plane, we build a sequence of geometrically defined trigraphs-graphs with mandatory, forbidden and optional adjacencies-until either we find a realization of G or the interruption of such a sequence certifies that no realization exists. Additionally, we consider the proposed method in the scope of the more general Distance Geometry Problem with Ranges, where arbitrary intervals of pairwise distances are allowed.

approximate range searching: the absolute model

Range searching is a well known problem in the area of geometric data structures. We consider this problem in the context of approximation, where an approximation parameter e > 0 is provided. Most prior work on this problem has focused on the case of relative errors, where each range shape R is bounded, and points within distance e ċ diam (R) of the ranges boundary may or may not be included. We consider a different approximation model, called the absolute model, in which points within distance e of the ranges boundary may or may not be included, regardless of the diameter of the range. We consider range spaces consisting of halfspaces, Euclidean balls, simplices, axis-aligned rectangles, and general convex bodies. We consider a variety of problem formulations, including range searching under general commutative semigroups, idempotent semigroups, groups, and range emptiness. We show how idempotence can be used to improve not only approximate, but also exact halfspace range searching. Our data structures are much simpler than both their exact and relative model counterparts, and so are amenable to efficient implementation.