R. Chandrasekaran

An

Many known algorithms are based on selection in a set whose cardinality is superlinear in terms of the input length. It is desirable in these cases to have selection algorithms that run in sublinear time in terms of the cardinality of the set. This paper presents a successful development in this direction. The methods developed here are applied to improve the previously known upper bounds for the time complexity of various location problems.

O(n\log ^2 n)

Cognitive radios (CR) have the ability to dynamically adapt to local spectrum availability. In a multi-hop wireless network comprised of CR-enabled devices, medium access control (MAC) layer scheduling for data communication involves assignment of timeslots and channels to either links or nodes in the network. The number of channels available and the channel identities vary from one node to another within the CR network. This is in contrast to the existing use of multiple channels where all the nodes have the same set of channels available (for example in IEEE 802.11 networks). In this paper, we present an integer linear programming (ILP) formulation for the MAC-layer scheduling problem and find an optimal schedule. We also propose a simple and efficient distributed heuristic for MAC-layer scheduling. Simulation results indicate that the proposed distributed heuristic provides near optimal schedule

Algorithm for the kth Longest Path in a Tree with Applications to Location Problems

The Fermat—Weber location problem is to find a point in ℝn that minimizes the sum of the weighted Euclidean distances fromm given points in ℝn. A popular iterative solution method for this problem was first introduced by Weiszfeld in 1937. In 1973 Kuhn claimed that if them given points are not collinear then for all but a denumerable number of starting points the sequence of iterates generated by Weiszfelds scheme converges to the unique optimal solution. We demonstrate that Kuhns convergence theorem is not always correct. We then conjecture that if this algorithm is initiated at the affine subspace spanned by them given points, the convergence is ensured for all but a denumerable number of starting points.

MAC-layer scheduling in cognitive radio based multi-hop wireless networks

Given an undirected graph G: (N;E) with a node set N and an edge set E and numbers Ce and De, e ϵ E, we provide a polynominally bounded algorithm to solve the problem: Find a spanning tree T such that the ratio is minimized. An extension to finding bases in matroids that minimize such ratio functions is immediate. It is shown that an algorithm that is “greedy,” in the sense of Edmonds [2], will not work for this problem.

Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem

The Fermat-Weber location problem is to find a point in ℝn that minimizes the sum of the (weighted) Euclidean distances fromm given points in ℝn. In this work we discuss some relevant complexity and algorithmic issues. First, using Tarskis theory on solvability over real closed fields we argue that there is an infinite scheme to solve the problem, where the rate of convergence is equal to the rate of the best method to locate a real algebraic root of a one-dimensional polynomial. Secondly, we exhibit an explicit solution to the strong separation problem associated with the Fermat-Weber model. This separation result shows that anε-approximation solution can be constructed in polynomial time using the standard Ellipsoid Method.

Minimal ratio spanning trees

We consider a wireless sensor network made of sensor nodes capable of sensing and communication, relay nodes capable of communication, and base stations responsible for collecting data generated by sensor nodes, to be deployed in sensor field. We address the problem of placing the sensor nodes, relay nodes and base stations in the sensor field such that (i) each point of interest in the sensor field is covered by a subset of sensors of desired cardinality (ii) the resulting sensor network is connected and (iii) the sensor network has sufficient bandwidth. We propose several deployment strategies to determine optimal placements of sensor nodes, relay nodes and base stations for guaranteed coverage, connectivity, bandwidth and robustness. We study several different objectives such as minimizing the number of sensor nodes deployed, minimizing the total cost, minimizing the energy consumption, maximizing the network lifetime and maximizing the network utilization. The placement problems for reliable as well as unreliable/probabilistic detection models are formulated as integer linear programs (ILPs). The practicality, effectiveness and performance of the proposed strategies are illustrated through simulations.

Algebraic optimization: the Fermat-Weber location problem

We discuss several forms of thep-center location problems on an undirected tree network. Our approach is based on utilizing results for rigid circuit graphs to obtain polynomial algorithms for solving the model. Duality theory on perfect graphs is used to define and solve the dual location model.

Energy efficient sensor, relay and base station placements for coverage, connectivity and routing

A robotic cell-a manufacturing system widely used in industry-contains two or more robot-served machines, repetitively producing a number of part types. In this paper, we consider scheduling of operations in a bufferless dual-gripper robotic cell processing multiple part types. The processing constraints specify the cell to be a flowshop. The objective is to determine the robot move sequence and the sequence in which parts are to be processed so as to maximize the long-run average throughput rate for repetitive production of parts. We provide a framework to study the problem, and address the issues of problem complexity and solvability. Focusing on a particular class of robot move sequences, we identify all potentially optimal robot move sequences for the part-sequencing problem in a two-machine dual-gripper robot cell. In the case when the gripper switching time is sufficiently small, we specify the best robot move sequence in the class. We prove the problem of finding an optimal part sequence to be strongly NP-hard, even when the robot move sequence is specified. We provide a heuristic approach to solve the general two-machine problem and evaluate its performance on the set of randomly generated problem instances. We perform computations to estimate the productivity gain of using a dual-gripper robot in place of a single-gripper robot. Finally, we extend our results for the two-machine cell to solve an m -machine problem.

Polynomially bounded algorithms for locatingp-centers on a tree

Cognitive radios (CR) have the ability to dynamically adapt to local spectrum availability. In a network comprised of CR-enabled devices, layer-2 auto-configuration involves determining a common set of channels to facilitate communication among participating nodes. This is a unique challenge as nodes in the CR network may be unaware of (a) their neighbors and (b) the channels on which they can communicate with a neighbor. In this paper, we propose a time-efficient distributed algorithm for layer-2 auto-configuration for a CR network. Our algorithm finds the globally common channel set in 2MN+O(DN) timeslots, where each node is assigned a unique identifier from the range [1,...,N], M is the maximum number of channels available for communication, and D is the diameter of the network. All nodes know M and N. We present both diameter-aware and diameter-unaware versions of the algorithm. We then show that the proposed algorithms are efficient by proving a matching lower bound. Finally, we investigate a special case when nodes have more knowledge available at their disposal and discuss how the time-complexity of our algorithm can be improved under this case.

Scheduling Multiple Parts in a Robotic Cell Served by a Dual-Gripper Robot

A recently developed data separation/classification method, called isotonic separation, is applied to breast cancer prediction. Two breast cancer data sets, one with clean and sufficient data and the other with insufficient data, are used for the study and the results are compared against those of decision tree induction methods, linear programming discrimination methods, learning vector quantization, support vector machines, adaptive boosting, and other methods. The experiment results show that isotonic separation is a viable and useful tool for data classification in the medical domain. 2006 Elsevier B.V. All rights reserved.