David Rappaport

Algorithms for Computing Geometric Measures of Melodic Similarity

Greg Aloupis, Thomas Fevens, Stefan Langerman, Tomomi Matsui, Antonio Mesa, Yurai Nunez, David Rappaport, and Godfried Toussaint *School of Computer Science, McGill University 3480 University Street Montreal, Quebec, Canada H3A 2A7 {athens,godfried}@cs.mcgill.ca †Department of Computer Science and Software Engineering, Concordia University 1455 de Maisonneuve Boulevard West Montreal, Quebec, Canada H3G 1M8 fevens@cs.concordia.ca ‡ Chercheur Qualifie du FNRS, Departement d’Informatique, Universite Libre de Bruxelles CP212 Boulevard du Triomphe, 1050 Bruxelles, Belgium Stefan.Langerman@ulb.ac.be §Department of Mathematical Informatics Graduate School of Information Science and Technology, University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656 Japan tomomi@misojiro.t.u-tokyo.ac.jp ¶Departamento Ciencias de la Computacion Facultad de Matematica y Computacion, Universidad de La Habana San Lazaro y L, Vedado 10400 Ciudad de La Habana, Cuba tonymesa@matcom.uh.cu, yurainr@yahoo.com **School of Computing, Queen’s University Kingston, Ontario, Canada K7L 3N6 daver@cs.queensu.ca Algorithms for Computing Geometric Measures of Melodic Similarity

On point-sets that support planar graphs

A universal point-set supports a crossing-free drawing of any planar graph. For a planar graph with n vertices, if bends on edges of the drawing are permitted, universal point-sets of size n are known, but only if the bend points are in arbitrary positions. If the locations of the bend points must also be specified as part of the point-set, we prove that any planar graph with n vertices can be drawn on a universal set S of O(n^2/logn) points with at most one bend per edge and with the vertices and the bend points in S. If two bends per edge are allowed, we show that O(nlogn) points are sufficient, and if three bends per edge are allowed, O(n) points are sufficient. When no bends on edges are permitted, no universal point-set of size o(n^2) is known for the class of planar graphs. We show that a set of n points in balanced biconvex position supports the class of maximum-degree-3 series-parallel lattices.

THE LARGEST EMPTY ANNULUS PROBLEM

Given a set of n points S in the Euclidean plane, we address the problem of computing an annulus A, (open region between two concentric circles) of largest width, that partitions S into a subset of points inside and a subset of points outside the circles, such that no point p∈S lies in the interior of A. This problem can be considered as a maximin facility location problem for n points such that the facility is a circumference. We give a characterization of the centres of annuli which are locally optimal and we show that the problem can be solved in O(n3logn) time and O(n) space. We also consider the case in which the number of points in the inner circle is a fixed value k. When k∈O(n) our algorithm runs in O(n3logn) time and O(n) space, furthermore, we can simultaneously optimize for all values of k within the same time bound. When k is small, that is a fixed constant, we can solve the problem in O(n logn) time and O(n) space.

On representing graphs by touching cuboids

We consider contact representations of graphs where vertices are represented by cuboids, i.e. interior-disjoint axis-aligned boxes in 3D space. Edges are represented by a proper contact between the cuboids representing their endvertices. Two cuboids make a proper contact if they intersect and their intersection is a non-zero area rectangle contained in the boundary of both. We study representations where all cuboids are unit cubes, where they are cubes of different sizes, and where they are axis-aligned 3D boxes. We prove that it is NP-complete to decide whether a graph admits a proper contact representation by unit cubes. We also describe algorithms that compute proper contact representations of varying size cubes for relevant graph families. Finally, we give two new simple proofs of a theorem by Thomassen stating that all planar graphs have a proper contact representation by touching cuboids.

Minimum convex partition of a constrained point set

A convex partition with respect to a point set S is a planar subdivision whose vertices are the points of S, where the boundary of the unbounded outer face is the boundary of the convex hull of S, and every bounded interior face is a convex polygon. A minimum convex partition with respect to S is a convex partition of S such that the number of convex polygons is minimised. In this paper, we will present a polynomial time algorithm to find a minimum convex partition with respect to a point set S where S is constrained to lie on the boundaries of a fixed number of nested convex hulls.

Approximation Algorithms for Finding a Minimum Perimeter Polygon Intersecting a Set of Line Segments

Let S denote a set of line segments in the plane. We say that a polygon P intersects S if every segment in S has a non-empty intersection with the interior or boundary of P . Currently, the best known algorithm finding a minimum perimeter polygon intersecting a set of line segments has a worst case exponential running time. It is also still unknown whether this problem is NP-hard. In this note we explore several approximation algorithms. We present efficient approximation algorithms that yield good empirical results, but can perform very poorly on pathological examples. We also present an O(n logn ) algorithm with a guaranteed worst case performance bound that is at most *** /2 times that of the optimum.

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.

Super Generalized 4PCS for 3D Registration

The 4-Points Congruent Sets (4PCS) Algorithm is an established approach to registering two overlapping 3D point sets with partial overlap and arbitrary initial poses. 4PCS performs the registration efficiently using a special set of 4 points, also known as a base, formed by two co-planar pairs of points within a RANSAC framework. The SUPER 4PCS algorithm uses intelligent indexing to reduce the complexity of the original 4PCS algorithm. Although SUPER 4PCS is efficient, we show in this work that one can gain significant practical improvements in runtime by reducing the number of congruent 4-point bases across the two 3D point sets. We accomplish this by using a generalized 4-point base which considers non-coplanar 4-point bases as well as planar ones. We show through experimentation that the number of 4-point bases decreases, sometimes exponentially, with a non-coplanar base. Using this property, we propose the Super Generalized 4PCS algorithm which can exhibit a significant speed-up of up to 6.5x over the Super 4PCS algorithm as demonstrated experimentally.

Generalized 4-Points Congruent Sets for 3D Registration

The 4-Points Congruent Sets (4PCS) algorithm is a state-of-the-art RANSAC-based algorithm for registering two partially overlapping 3D point sets using raw points. Unlike other RANSAC-based algorithms, which try to achieve registration by searching for matching 3-point bases, it uses a base of two coplanar pairs of points to reduce the search space matching bases. In this work, we first generalize the algorithm by allowing the two pairs to fall on two different planes which have an arbitrary distance, i.e. Degree of separation, between them. Furthermore, we show that increasing the degree of separation exponentially decreases the search space of matching bases. Using this property, we show that using the new generalized base allows for more efficient registration than the original 4PCS base type. We achieve a maximum run-time improvement of 83.10% for 3D registration.

Biclique edge cover graphs and confluent drawings

Confluent drawing is a technique that allows some non-planar graphs to be visualized in a planar way. This approach merges edges together, drawing groups of them as single tracks, similar to train tracks. In the general case, producing confluent drawings automatically has proven quite difficult. We introduce the biclique edge cover graph that represents a graph G as an interconnected set of cliques and bicliques. We do this in such a way as to permit a straightforward transformation to a confluent drawing of G. Our result is a new sufficient condition for confluent planarity and an additional algorithmic approach for generating confluent drawings. We give some experimental results gauging the performance of existing confluent drawing heuristics.