Sumanta Guha

On a graph partition problem with application to VLSI layout

We discuss a graph partition problem with application to VLSI layout. It is not difficult to show that the general partition problem is NPComplete, so we restrict our attention to some special classes of graphs. A graph is called a circZe graph if the nodes of the graph represent the chords of a circle and there is an edge between the two nodes if the corresponding chords intersect. Supowit in [17] had posed the partition problem of circle graphs as an open problem. However, it is also not too difficult to show that the partition problem on circle graphs remains NP-Complete. We give an integer linear programming formulation for the general graph par-

Computing homology groups of simplicial complexes in R 3

Recent developments in analyzing molecular structures and representing solid models using simplicial complexes have further enhanced the need for computing structural information about simplicial complexes in <bold>R</bold><supscrpt>3</supscrpt>. This paper develops basic techniques required to manipulate and analyze structures of complexes in <bold>R</bold><supscrpt>3</supscrpt>. A new approach to analyze simplicial complexes in Euclidean 3-space <bold>R</bold><supscrpt>3</supscrpt> is described. First, methods from topology are used to analyze triangulated 3-manifolds in <bold>R</bold><supscrpt>3</supscrpt>. Then, it is shown that these methods can, in fact, be applied to arbitrary simplicial complexes in <bold>R</bold><supscrpt>3</supscrpt> after (simulating) the process of thickening a complex to a 3-manifold homotopic to it. As a consequence considerable structural information about the complex can be determined and certain discrete problems solved as well. For example, it is shown how to determine homology groups, as well as concrete representations of their generators, for a given complex in <bold>R</bold><supscrpt>3</supscrpt>

Transforming Curves on Surfaces

We describe an optimal algorithm to decide if one closed curve on a triangulated 2-manifold can be continuously transformed to another, i.e., if they are homotopic. SupposeC1andC2are two closed curves on a surfaceMof genusg. Further, supposeTis a triangulation ofMof sizensuch thatC1andC2are represented as edge?vertex sequences of lengthsk1andk2inT, respectively. Then, our algorithm decides ifC1andC2are homotopic inO(n+k1+k2) time and space, providedg?2 ifMis orientable, andg?3, 4 ifMis nonorientable. This implies as well an optimal algorithm to decide if a closed curve on a surface can be continuously contracted to a point. Except for three low genus cases, our algorithm completes an investigation into the computational complexity of two classical problems for surfaces posed by the mathematician Max Dehn at the beginning of this century. The novelty of our approach is in the application of methods from modern combinatorial group theory.

Parallel methods for visibility and shortest-path problems in simple polygons

In this paper we give efficient parallel algorithms for solving a number of visibility and shortest-path problems for simple polygons. Our algorithms all run inO(logn) time and are based on the use of a new data structure for implicitly representing all shortest paths in a simple polygonP, which we call thestratified decomposition tree. We use this approach to derive efficient parallel methods for computing the visibility ofP from an edge, constructing the visibility graph of the vertices ofP (using an output-sensitive number of processors), constructing the shortest-path tree from a vertex ofP, and determining all-farthest neighbors for the vertices inP. The computational model we use is the CREW PRAM.

Parallel methods for visibility and shortest path problems in simple polygons (preliminary version)

In this paper we give efficient parallel algorithms for solving a number of visibility and shortest path problems for simple polygons. Our algorithms all run in <italic>&Ogr;</italic>(log <italic>n</italic>) time and are based on the use of a new data structure for implicitly representing all shortest paths in a simple polygon <italic>P</italic>, which we call the <italic>stratified decomposition tree</italic>. We use this approach to derive efficient parallel methods for computing the visibility of <italic>P</italic> from an edge, constructing the visibility graph of the vertices of <italic>P</italic> (using an output-sensitive number of processors), constructing the shortest path tree from a vertex of <italic>P</italic>, and determining all-farthest neighbors for the vertices in <italic>P</italic>. The computational model we use is the CREW PRAM.

Proximity Problems for Points on a Rectilinear Plane with Rectangular Obstacles

We consider the following four problems for a setS ofk points on a plane, equipped with the rectilinear metric and containing a setR ofn disjoint rectangular obstacles (so that distance is measured by a shortest rectilinear path avoiding obstacles inR): (a) find aclosest pair of points inS, (b) find anearest neighbor for each point inS, (c) compute the rectilinearVoronoi diagram ofS, and (d) compute a rectilinearminimal spanning tree ofS. We describeO ((n+k) log(n+k))-time sequential algorithms for (a) and (b) based onplane-sweep, and the consideration of geometrically special types of shortest paths, so-calledz-first paths. For (c) we present anO ((n+k) log(n+k) logn)-time sequential algorithm that implements a sophisticateddivide-and-conquer scheme with an addedextension phase. In the extension phase of this scheme we introduce novel geometric structures, in particular so-calledz-diagrams, and techniques associated with the Voronoi diagram. Problem (d) can be reduced to (c) and solved inO ((n+k) log(n+k) logn) time as well. All our algorithms arenear-optimal, as well as easy to implement.

DIP-MIP: distributed individual paging extension for mobile IP in IP-based cellular networks

The mobility-enabling protocol mobile IP supports location registration but not paging. However, current cellular networks use registration as well as paging procedures to minimize signaling cost. Accordingly, an extension to mobile IP using distributed individual paging, the so-called DIP-MIP protocol, is proposed for managing mobility in IP-based cellular networks. In DIP-MIP, each mobile host derives its own paging area size by optimizing a signaling cost function based on its individual mobility pattern. The cost function itself may use either of two mobility models - fluid flow and random walk and the performance of DIP-MIP is analyzed for both. The impact of various parameters on the DIP-MIP signaling cost and the optimal size of the paging area is studied as well.

An optimal mesh computer algorithm for constrained Delaunay triangulation

We present an optimal parallel algorithm that runs in O(/spl radic/n) time on a /spl radic/n/spl timesspl radic/n mesh to compute the constrained Delaunay triangulation of a planar straight line graph G whose vertices lie in an n-element set S. Implications of our result also include an efficient PRAM algorithm for the same problem, a new optimal mesh algorithm to compute a planar Voronoi diagram, as well as a partial solution to the problem of the geodesic Voronoi diagram of a point set inside a simple polygon.<<ETX>>

Mining movies to extract song sequences

This paper proposes a system to automatically locate and extract songs from digitized movies. We focus on the genre of Bollywood movies. A song grammar particularly applicable to this genre is proposed and used subsequently to construct a probabilistic timed automaton to differentiate songs. The proposed system has been implemented and test results indicate both high precision and recall. Songs being a major driver in the success of Bollywood movies, a potentially significant application of the proposed system lies in automatically mining the vast Bollywood movie archives.

Reconstructing Curves without Delaunay Computation

A new non-Delaunay-based approach is presented to reconstruct a curve, lying in 2- or 3-space, from a sampling of points. The underlying theory is based on bounding curvature to determine monotone pieces of the curve. Theoretical guarantees are established. The implemented algorithm, based heuristically on the theory, proceeds by iteratively partitioning the sample points using an octree data structure. The strengths of the approach are (a) simple implementation, (b) efficiency-experimental performance compares favorably with Delaunay-based algorithms, (c) robustness-curves with multiple components and sharp corners are reconstructed satisfactorily, and (d) potential extension to surface reconstruction.