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Sumanta Guha

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Ohio State University

Ohio State University

Florentin Smarandache

University of New Mexico

The University of Western Australia

Durham University

Riccardo Piergallini

Università degli Studi di Trieste

Nikolaos E Apostolakis

Graduate Center of the City University of New York

University of California, Davis

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Papers by Sumanta Guha

Optimal Algorithms for Curves on Surfaces

We describe an optimal algorithm to decide if one closed curve on a triangulated 2-manifold can b... more 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. Our algorithm runs in O(n+k1+k 2) time and space, where closed curves C1 and C 2 of lengths k1 and k2, resp., on a genus g surface M (g≠2 if M orientable, and g≠3,4 if M is non-orientable) are presented as edge-vertex sequences in a triangulation T of size n of M. This also implies 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 the two classical problems for surfaces posed by the mathematician Max Dehn at the beginning of this century. However, we make novel applications of methods from modern combinatorial group theory for an approach entirely different from previous ones, and much simpler to implement

Topology Preserving Edge Contraction

Transforming Curves on Surfaces

Journal of Computer and System Sciences, 1999

We describe an optimal algorithm to decide if one closed curve on a triangulated 2-manifold can b... more 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. Suppose C 1 and C 2 are two closed curves on a surface M of genus g. Further, suppose T is a triangulation of M of size n such that C 1 and C 2 are represented as edge-vertex sequences of lengths k 1 and k 2 in T , respectively. Then, our algorithm decides if C 1 and C 2 are homotopic in O(n+k 1 +k 2 ) time and space, provided g 6 = 2 if M orientable, and g 6 = 3; 4 if M is non-orientable. This as well implies 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.

Computing homology groups of simplicial complexes in R 3

Journal of The ACM, 1998

Recent developments in analyzing molecular structures and representing solid models using simplic... more 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 R 3 . This paper develops basic techniques required to manipulate and analyze structures of complexes in R 3 .

Algorithms for Manifolds and Simplicial Complexes in Euclidean 3Space

A new approach to analyze simplicial complexes in Euclidean 3-space R 3 is described. First, meth... more A new approach to analyze simplicial complexes in Euclidean 3-space R 3 is described. First, methods from topology are used to analyze triangulated 3manifolds in R 3 . Then it is shown that these methods can in fact be applied to arbitrary simplicial complexes in R 3 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.

Topology preserving edge contraction

On Fault-Tolerant Distributor Communication Architecure

IEEE Transactions on Computers, 1986

On a Graph Partition Problem with Application to VLSI Layout

Information Processing Letters, 1992

$The third constraint, which is a quadratic con- straint, can be rewritten as a linear constraint in the following way: Thus, the number of edges not cut by this parti- tion is given by E,,. = 7L4_, x} Ax;. For the set of nodes V; to be independent sets, the number of edges not cut by the partition must be zero. That is, x; Ax, = 0, for all j, 1 <j <k. Thus if y be the chromatic number of the graph G =(V, E) and C=<{C,,...,C,) be the cost vector, then the optimum cost chromatic partition problem re- duces to the following problem:$

Individual profile graphs for location management in PCS networks

Computer Communications, 2008

A user's profile, for the purpose of location management in a personal communication service (PCS... more A user's profile, for the purpose of location management in a personal communication service (PCS) network, is formalized as a subgraph of the network graph. This subgraph, the so-called individual profile graph (IPG), is determined after a period of observation with the intent of predicting and codifying the user's diurnal routine. The IPG is easily-motivated, robust, straightforwardly computed from observed data, and, under fairly intuitive assumptions, provably predictive of the user's diurnal routine. An IPGbased paging and update strategy is analyzed. It is shown to significantly improve a straight location area (LA) based strategy that ignores user profiles.

Individual profile graphs for location management in PCS networks

A user's profile, for the purpose of location management in a PCS network, is formalized as a sub... more A user's profile, for the purpose of location management in a PCS network, is formalized as a subgraph of the network graph. This subgraph, the so-called individual profile graph (IPG), is determined after a period of observation with the intent of predicting and codifying the user's diurnal routine. The IPG is easily-motivated, robust, straightforwardly computed from observed data, and, under fairly intuitive assumptions, provably predictive of the user's diurnal routine. An IPG-based paging and update strategy is proposed and analyzed.

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

Computer Communications, 2007

The mobility-enabling protocol Mobile IP supports location registration but not paging. However, ... more 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, is proposed. 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 is studied as well. The performance of DIP-MIP is shown to be superior to that of Mobile IP (MIP) in reducing signaling load, managing mobility and supporting a large number of mobile users in IP-based cellular networks.

Random probing to approximate medial axes and plan safe motion

We propose a novel approach to planning the safe motion of a robot - the motion is considered saf... more We propose a novel approach to planning the safe motion of a robot - the motion is considered safe if the robot stays as distant as possible from the nearest obstacle. The approach is based on approximating the medial axis of the free configuration space through randomized probes of the full configuration space. Experiments with 2- and 3-DOF manipulators show that the approach generates longer but significantly safer motion when compared with naive planning to simply finds a shortest motion. In addition, once preprocessing is complete our approach spends much less time planning motion than the naive one. As it is based on probing randomly, rather than at points on a grid, our approach promises to extend effectively to robots with many DOF

A Practical Algorithm for Integer Sorting on a Mesh-connected Computer

International Journal of Parallel, Emergent and Distributed Systems, 1997

This paper presents count-sort, a parallel algorithm for mesh-connected computers to sort integer... more This paper presents count-sort, a parallel algorithm for mesh-connected computers to sort integers where the range of inputs is known. A straightforward counting technique that has not been implemented previously in parallel sorting algorithms is presented. On a mesh-connected computer with N×N processors we are able to sort N integers in the range 1 sN in timewhere cN is very small. For practical values of N, the algorithm is extremely fast. Further, it is possible to expand the range by a factor k to 1 N so that the slowdown is less than k.We produce two implementations of count-sort on the SIMD MasPar MP-I with 8192 processors. The first sorts 8-bit integers, one per processor, significantly faster than the manufacturer's current library routine for sorting 8-bit integers. The second implementation is a fast version that sorts several elements per processor.

Proximity Problems and the Voronoi Diagram an a Rectilinear Plane with Rectangular Obstacles

We consider the following four problems for a set S of k points on a plane, equipped with the rec... more We consider the following four problems for a set S of k points on a plane, equipped with the rectilinear metric and containing a set R of n disjoint rectangular obstacles (so that distance is measured by a shortest rectilinear path avoiding obstacles in R): (a) find a closest pair of points in S, (b) find a nearest neighbor for each point in S, (c) compute the rectilinear Voronoi diagram of S, and (d) compute a rectilinear minimal spanning tree of S. We describe O((n+k)log(n+k)) time sequential algorithms for (a) and (b) based on plane-sweep, and the consideration of geometrically special types of shortest paths, so-called z-first paths. For (c) we present an O((n+k)log(n+k) log n) time sequential algorithm that implements a sophisticated divide-and-conquer scheme with an added extension phase. In the extension phase of this scheme we introduce novel geometric structures, in particular so-called z-diagrams, and techniques associated with the Voronoi diagram. Problem (d) can be reduced to (c) and solved in O((n+k) log(n+k) log n) time as well. All our algorithms are near-optimal, as well as easy to implement.

Optimal Algorithms for Curves on Surfaces

We describe an optimal algorithm to decide if one closed curve on a triangulated 2-manifold can b... more 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. Our algorithm runs in O(n+k1+k 2) time and space, where closed curves C1 and C 2 of lengths k1 and k2, resp., on a genus g surface M (g≠2 if M orientable, and g≠3,4 if M is non-orientable) are presented as edge-vertex sequences in a triangulation T of size n of M. This also implies 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 the two classical problems for surfaces posed by the mathematician Max Dehn at the beginning of this century. However, we make novel applications of methods from modern combinatorial group theory for an approach entirely different from previous ones, and much simpler to implement

Topology Preserving Edge Contraction

Transforming Curves on Surfaces

Journal of Computer and System Sciences, 1999

We describe an optimal algorithm to decide if one closed curve on a triangulated 2-manifold can b... more 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. Suppose C 1 and C 2 are two closed curves on a surface M of genus g. Further, suppose T is a triangulation of M of size n such that C 1 and C 2 are represented as edge-vertex sequences of lengths k 1 and k 2 in T , respectively. Then, our algorithm decides if C 1 and C 2 are homotopic in O(n+k 1 +k 2 ) time and space, provided g 6 = 2 if M orientable, and g 6 = 3; 4 if M is non-orientable. This as well implies 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.

Computing homology groups of simplicial complexes in R 3

Journal of The ACM, 1998

Recent developments in analyzing molecular structures and representing solid models using simplic... more 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 R 3 . This paper develops basic techniques required to manipulate and analyze structures of complexes in R 3 .

Algorithms for Manifolds and Simplicial Complexes in Euclidean 3Space

A new approach to analyze simplicial complexes in Euclidean 3-space R 3 is described. First, meth... more A new approach to analyze simplicial complexes in Euclidean 3-space R 3 is described. First, methods from topology are used to analyze triangulated 3manifolds in R 3 . Then it is shown that these methods can in fact be applied to arbitrary simplicial complexes in R 3 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.

Topology preserving edge contraction

On Fault-Tolerant Distributor Communication Architecure

IEEE Transactions on Computers, 1986

On a Graph Partition Problem with Application to VLSI Layout

Information Processing Letters, 1992

$The third constraint, which is a quadratic con- straint, can be rewritten as a linear constraint in the following way: Thus, the number of edges not cut by this parti- tion is given by E,,. = 7L4_, x} Ax;. For the set of nodes V; to be independent sets, the number of edges not cut by the partition must be zero. That is, x; Ax, = 0, for all j, 1 <j <k. Thus if y be the chromatic number of the graph G =(V, E) and C=<{C,,...,C,) be the cost vector, then the optimum cost chromatic partition problem re- duces to the following problem:$

Individual profile graphs for location management in PCS networks

Computer Communications, 2008

A user's profile, for the purpose of location management in a personal communication service (PCS... more A user's profile, for the purpose of location management in a personal communication service (PCS) network, is formalized as a subgraph of the network graph. This subgraph, the so-called individual profile graph (IPG), is determined after a period of observation with the intent of predicting and codifying the user's diurnal routine. The IPG is easily-motivated, robust, straightforwardly computed from observed data, and, under fairly intuitive assumptions, provably predictive of the user's diurnal routine. An IPGbased paging and update strategy is analyzed. It is shown to significantly improve a straight location area (LA) based strategy that ignores user profiles.

Individual profile graphs for location management in PCS networks

A user's profile, for the purpose of location management in a PCS network, is formalized as a sub... more A user's profile, for the purpose of location management in a PCS network, is formalized as a subgraph of the network graph. This subgraph, the so-called individual profile graph (IPG), is determined after a period of observation with the intent of predicting and codifying the user's diurnal routine. The IPG is easily-motivated, robust, straightforwardly computed from observed data, and, under fairly intuitive assumptions, provably predictive of the user's diurnal routine. An IPG-based paging and update strategy is proposed and analyzed.

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

Computer Communications, 2007

The mobility-enabling protocol Mobile IP supports location registration but not paging. However, ... more 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, is proposed. 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 is studied as well. The performance of DIP-MIP is shown to be superior to that of Mobile IP (MIP) in reducing signaling load, managing mobility and supporting a large number of mobile users in IP-based cellular networks.

Random probing to approximate medial axes and plan safe motion

We propose a novel approach to planning the safe motion of a robot - the motion is considered saf... more We propose a novel approach to planning the safe motion of a robot - the motion is considered safe if the robot stays as distant as possible from the nearest obstacle. The approach is based on approximating the medial axis of the free configuration space through randomized probes of the full configuration space. Experiments with 2- and 3-DOF manipulators show that the approach generates longer but significantly safer motion when compared with naive planning to simply finds a shortest motion. In addition, once preprocessing is complete our approach spends much less time planning motion than the naive one. As it is based on probing randomly, rather than at points on a grid, our approach promises to extend effectively to robots with many DOF

A Practical Algorithm for Integer Sorting on a Mesh-connected Computer

International Journal of Parallel, Emergent and Distributed Systems, 1997

This paper presents count-sort, a parallel algorithm for mesh-connected computers to sort integer... more This paper presents count-sort, a parallel algorithm for mesh-connected computers to sort integers where the range of inputs is known. A straightforward counting technique that has not been implemented previously in parallel sorting algorithms is presented. On a mesh-connected computer with N×N processors we are able to sort N integers in the range 1 sN in timewhere cN is very small. For practical values of N, the algorithm is extremely fast. Further, it is possible to expand the range by a factor k to 1 N so that the slowdown is less than k.We produce two implementations of count-sort on the SIMD MasPar MP-I with 8192 processors. The first sorts 8-bit integers, one per processor, significantly faster than the manufacturer's current library routine for sorting 8-bit integers. The second implementation is a fast version that sorts several elements per processor.

Proximity Problems and the Voronoi Diagram an a Rectilinear Plane with Rectangular Obstacles

We consider the following four problems for a set S of k points on a plane, equipped with the rec... more We consider the following four problems for a set S of k points on a plane, equipped with the rectilinear metric and containing a set R of n disjoint rectangular obstacles (so that distance is measured by a shortest rectilinear path avoiding obstacles in R): (a) find a closest pair of points in S, (b) find a nearest neighbor for each point in S, (c) compute the rectilinear Voronoi diagram of S, and (d) compute a rectilinear minimal spanning tree of S. We describe O((n+k)log(n+k)) time sequential algorithms for (a) and (b) based on plane-sweep, and the consideration of geometrically special types of shortest paths, so-called z-first paths. For (c) we present an O((n+k)log(n+k) log n) time sequential algorithm that implements a sophisticated divide-and-conquer scheme with an added extension phase. In the extension phase of this scheme we introduce novel geometric structures, in particular so-called z-diagrams, and techniques associated with the Voronoi diagram. Problem (d) can be reduced to (c) and solved in O((n+k) log(n+k) log n) time as well. All our algorithms are near-optimal, as well as easy to implement.