Sue Whitesides

Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems

AbstractnWe obtain faster algorithms for problems such as r-dimensional matching and r-set packing when the size k of the solution is considered a parameter. We first establish a general framework for finding and exploiting small problem kernels (of size polynomial in k). This technique lets us combine Alon, Yuster and Zwick’s color-coding technique with dynamic programming to obtain faster fixed-parameter algorithms for these problems. Our algorithms run in time O(n+2O(k)), an improvement over previous algorithms for some of these problems running in time O(n+kO(k)). The flexibility of our approach allows tuning of algorithms to obtain smaller constants in the exponent.n

On the Parameterized Complexity of Layered Graph Drawing

AbstractnWe consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case either the total span or the maximum span of edges can be minimized. In contrast to the Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.n

Depth potential function for folding pattern representation, registration and analysis.

Some surfaces present folding patterns formed by juxtapositions of ridges and valleys as, for example, the cortical surface of the human brain. The fundamental problem with ridges is to find a correspondence among and analyze the variability among them. Many techniques to achieve these goals exist but use scalar functions. Depth maps are used to efficiently project the geometry of folds into a scalar function in the case where a natural projection plane exists. However, in most cases of curved surfaces, there is no natural projection plane to represent folding patterns. This paper studies the problem of shape matching and analysis of folding patterns by extending the notion of depth maps when no natural projection plane exists. The novel depth measure is called a depth potential function. The depth potential function integrates the information known from the curvature of the surface into a point-of-view invariant representation. The main advantage of the depth potential function is that it is computed by solving a time independent Poisson equation. The Poisson equation endows our surface representation with a significant computational advantage that makes it orders of magnitude faster to compute compared with other available surface representations. The method described in this paper was validated using both synthetic surfaces and cortical surfaces of human brain acquired by magnetic resonance imaging. On average, the improvement in shape matching when using the depth potential was of 11%, which is considerable.

Parameterized Complexity of Geometric Problems

This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixed-parameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixed-parameter intractability results are surveyed as well. Finally, we give some directions for future research.

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.

Computing k-regret minimizing sets

Regret minimizing sets are a recent approach to representing a dataset D by a small subset R of size r of representative data points. The set R is chosen such that executing any top-1 query on R rather than D is minimally perceptible to any user. However, such a subset R may not exist, even for modest sizes, r. In this paper, we introduce the relaxation to k-regret minimizing sets, whereby a top-1 query on R returns a result imperceptibly close to the top-k on D. n nWe show that, in general, with or without the relaxation, this problem is NP-hard. For the specific case of two dimensions, we give an efficient dynamic programming, plane sweep algorithm based on geometric duality to find an optimal solution. For arbitrary dimension, we give an empirically effective, greedy, randomized algorithm based on linear programming. With these algorithms, we can find subsets R of much smaller size that better summarize D, using small values of k larger than 1.

Indexing Reverse Top-k Queries in Two Dimensions

We consider the recently introduced monochromatic reverse top−k query which asks for, given a (possibly new) tuple q and a dataset (mathcal{D}), all possible top−k queries on (mathcal{D}cup{q}) for which q is in the result. Towards this problem, we introduce the first query-agnostic approach, which leads to an efficient index. We present the novel insight that by representing the dataset as an arrangement of lines, a critical k-polygon can be identified and can singularly answer reverse top−k queries.

Sampled medial loci for 3D shape representation

The medial axis transform is valuable for shape representation as it is complete and captures part structure. However, its exact computation for arbitrary 3D models is not feasible. We introduce a novel algorithm to approximate the medial axis of a polyhedron with a dense set of medial points, with a guarantee that each medial point is within a specified tolerance from the medial axis. Given this discrete approximation to the medial axis, we use Damons work on radial geometry (Damon, 2005 [1]) to design a numerical method that recovers surface curvature of the object boundary from the medial axis transform alone. We also show that the number of medial sheets comprising this representation may be significantly reduced without substantially compromising the quality of the reconstruction, to create a more useful part-based representation.

Embeddability Problems for Upward Planar Digraphs

We study two embedding problems for upward planar digraphs. Both problems arise in the context of drawing sequences of upward planar digraphs having the same set of vertices, where the location of each vertex is to remain the same for all the drawings of the graphs. We develop a method, based on the notion of book embedding, that gives characterization results for embeddability as well as testing and drawing algorithms.

Sampled medial loci and boundary differential geometry

We introduce a novel algorithm to compute a dense sample of points on the medial locus of a polyhedral object, with a guarantee that each medial point is within a specified tolerance ε from the medial surface. Motivated by Damons work on the relationship between the differential geometry of the smooth boundary of an object and its medial surface [8], we then develop a computational method by which boundary differential geometry can be recovered directly from this dense medial point cloud. Experimental results on models of varying complexity demonstrate the validity of the approach, with principal curvature values that are consistent with those provided by an alternative method that works directly on the boundary. As such, we demonstrate the richness of a dense medial point cloud as a shape descriptor for 3D data processing.