Shripad Thite

Strong edge coloring for channel assignment in wireless radio networks

We give efficient sequential and distributed approximation algorithms for strong edge coloring graphs modeling wireless networks. Strong edge coloring is equivalent to computing a conflict-free assignment of channels or frequencies to pairwise links between transceivers in the network

Capturing a Convex Object With Three Discs

This paper addresses the problem of capturing an arbitrary convex object P in the plane with three congruent disc-shaped robots. Given two stationary robots in contact with P, we characterize the set of positions of a third robot, the so-called capture region, that prevent P from escaping to infinity via continuous rigid motion. We show that the computation of the capture region reduces to a visibility problem. We present two algorithms for solving this problem, and for computing the capture region when P is a polygon and the robots are points (zero-radius discs). The first algorithm is exact and has polynomial time complexity. The second one uses simple hidden surface removal techniques from computer graphics to output an arbitrarily accurate approximation of the capture region; it has been implemented, and examples are presented.

Spacetime meshing with adaptive refinement and coarsening

We propose a new algorithm for constructing finite-element meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain Ω and a target time value T, our method constructs a tetrahedral mesh of the spacetime domain Ω X [0,T] in constant running time per tetrahedron in ℝ3 using an advancing front method. Elements are added to the evolving mesh in small patches by moving a vertex of the front forward in time. Spacetime discontinuous Galerkin methods allow the numerical solution within each patch to be computed as soon as the patch is created. Our algorithm employs new mechanisms for adaptively coarsening and refining the front in response to a posteriori error estimates returned by the numerical code. A change in the front induces a corresponding refinement or coarsening of future elements in the spacetime mesh. Our algorithm adapts the duration of each element to the local quality, feature size, and degree of refinement of the underlying space mesh. We directly exploit the ability of discontinuous Galerkin methods to accommodate discontinuities in the solution fields across element boundaries.

An h-adaptive spacetime-discontinuous Galerkin method for linear elastodynamics

We present an h-adaptive version of the spacetime-discontinuous Galerkin (SDG) finite element method for linearized elastodynamics (Abedi et al., 2006). The adaptive version inherits key properties of the basic SDG formulation, including element-wise balance of linear and angular momentum, complexity that is linear in the number of elements and oscillationfree shock capturing. Unstructured spacetime grids allow simultaneous adaptation in space and time. A localized patch-by-patch solution process limits the cost of reanalysis when the error indicator calls for more refinement. Numerical examples demonstrate the method’s performance and shock-capturing capabilities.

Capturing a convex object with three discs

This paper addresses the problem of capturing an arbitrary convex object P in the plane with three congruent disc-shaped robots. Given two stationary robots in contact with P, we characterize the set of positions of a third robot that prevent P from escaping to infinity and show that the computation of this so-called capture region reduces to the resolution of a visibility problem. We present two algorithms for solving this problem and computing the capture region when P is a polygon and the robots are points (zero-radius discs). The first algorithm is exact and has polynomial-time complexity. The second one uses simple hidden-surface removal techniques from computer graphics to output an arbitrarily accurate approximation of the capture region; it has been implemented and examples are presented.

Adaptive spacetime meshing for discontinuous Galerkin methods

Spacetime-discontinuous Galerkin (SDG) finite element methods are used to solve hyperbolic spacetime partial differential equations (PDEs) to accurately model wave propagation phenomena arising in important applications in science and engineering. Tent Pitcher is a specialized algorithm, invented by Ungor and Sheffer (2000) and extended by Erickson et al. (2005) to construct an unstructured simplicial (d+1)-dimensional spacetime mesh over an arbitrary d-dimensional space domain. Tent Pitcher is an advancing front algorithm that incrementally adds groups of elements to the evolving spacetime mesh. It supports an accurate, local, and parallelizable solution strategy by interleaving mesh generation with an SDG solver. When solving nonlinear PDEs, previous versions of Tent Pitcher must make conservative worst-case assumptions about the physical parameters which limit the duration of spacetime elements. Thus, these algorithms create a mesh with many more elements than necessary. In this paper, we extend Tent Pitcher to give the first spacetime meshing algorithm suitable for efficient simulation of nonlinear phenomena using SDG methods. We adapt the duration of spacetime elements to changing physical parameters due to nonlinear response. Given a triangulated 2-dimensional Euclidean space domain M corresponding to time t=0 and initial and boundary conditions of the underlying hyperbolic spacetime PDE, we construct an unstructured tetrahedral mesh in the spacetime domain E^2xR. For every target time T>=0, our algorithm meshes the spacetime volume Mx[0,T] with a bounded number of non-degenerate tetrahedra. A recent extension of Tent Pitcher due to Abedi et al. (2004) adapts the spatial size of spacetime elements in 2Dxtime to a posteriori estimates of numerical error. Our extension of Tent Pitcher retains the ability to perform adaptive refinement and coarsening of the mesh. We thus obtain the first adaptive nonlinear Tent Pitcher algorithm to build spacetime meshes in 2Dxtime.

The Complexity of Bisectors and Voronoi Diagrams on Realistic Terrains

We prove tight bounds on the complexity of bisectors and Voronoi diagrams on so-called realistic terrains, under the geodesic distance. In particular, if ndenotes the number of triangles in the terrain, we show the following two results. (i) If the triangles of the terrain have bounded slope and the projection of the set of triangles onto the xy-plane has low density, then the worst-case complexity of a bisector is i¾?(n). (ii) If, in addition, the triangles have similar sizes and the domain of the terrain is a rectangle of bounded aspect ratio, then the worst-case complexity of the Voronoi diagram of mpoint sites is

I/O-efficient map overlay and point location in low-density subdivisions

\Theta(n+m\sqrt{n})

Cache-oblivious selection in sorted X+Y matrices

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The distance-2 matching problem and its relationship to the MAC-Layer capacity of ad hoc wireless networks

We present improved and simplified I/O-efficient algorithms for two problems on planar low-density subdivisions, namely map overlay and point location. More precisely, we show how to preprocess a low-density subdivision with n edges in O(sort (n)) I/Os into a compressed linear quadtree such that one can: (i) compute the overlay of two preprocessed subdivisions in O(scan(n)) I/Os, where n is the total number of edges in the two subdivisions, (ii) answer a single point location query in O(logB n) I/Os and k batched point location queries in O(scan(n) + sort(k)) I/Os. For the special case where the subdivision is a fat triangulation, we show how to obtain the same bounds with an ordinary (uncompressed) quadtree, and we show how to make the structure fully dynamic using O(logB n) I/Os per update. Our algorithms and data structures improve on the previous best known bounds for general subdivisions both in the number of I/Os and storage usage, they are significantly simpler, and several of our algorithms are cache-oblivious.