Sasanka Roy

Base station placement on boundary of a convex polygon

Let P be a polygonal region which is forbidden for placing a base station in the context of mobile communication. Our objective is to place one base station at any point on the boundary of P and assign a range such that every point in the region is covered by that base station and the range assigned to that base station for covering the region is minimum among all such possible choices of base stations. Here we consider the forbidden region P as convex and base station can be placed on the boundary of the region. We present optimum linear time algorithm for that problem. In addition, we propose a linear time algorithm for placing a pair of base stations on a specified side of the boundary such that the range assigned to those base stations in order to cover the region is minimum among all such possible choices of a pair of base stations on that side.

Approximation algorithms for shortest descending paths in terrains

A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. No efficient algorithm is known to find a shortest descending path (SDP) from s to t in a polyhedral terrain. We present two approximation algorithms that solve the SDP problem on general terrains. We also introduce a generalization of the shortest descending path problem, called the shortest gently descending path (SGDP) problem, where a path descends, but not too steeply. The additional constraint to disallow a very steep descent makes the paths more realistic in practice. We present two approximation algorithms to solve the SGDP problem on general terrains. All of our algorithms are simple, robust and easy to implement.

VARIATIONS OF BASE-STATION PLACEMENT PROBLEM ON THE BOUNDARY OF A CONVEX REGION

Due to the recent growth in the demand of mobile communication services in several typical environments, the development of efficient s ystems for providing specialized services has become an important issue in mobile communication research. An important sub-problem in this area is the base-station placement problem, where the objective is to identify the location for placing the basestations. Mobile terminals communicate with their respective nearest base station, and the base stations communicate with each other over scarce wireless channels in a multi-hop fashion by receiving and transmitting radio signals. Each base station emits signal periodically and all the mobile terminals within its range can identify it as its nearest base station after receiving such radio signal. Here the problem is to position the base stations such that each point in the entire area can communicate with at least one base-station, and total power required for all the base-stations in the network is minimized. A different variation of this problem arises when some portions of the target region is not suitable for placing the base-stations, but the communication inside those regions need to be provided. For example, we may consider the large water bodies or the stiff mountains. In such cases, we need some specialized algorithms for efficiently placing the base-stations on the boundary of the f orbidden zone to provide services inside that region.

Optimal Algorithm for a Special Point-Labeling Problem

We investigate a special class of map labeling problem. Let P = {p1, p2, ..., pn} be a set of point sites distributed on a 2D map. A label associated with each point is a axis-parallel rectangle of a constant height but of variable width.Here height of a label indicates the font size and width indicates the number of characters in that label. For a point pi, its label contains the point pi at its top-left or bottom-left corner, and it does not obscure any other point in P. Width of the label for each point in P is known in advance.The objective is to label the maximum number of points on the map so that the placed labels are mutually nonoverlapping. We first consider a simple model for this problem. Here, for each point pi, the corner specification (i.e., whether the point pi would appear at the top-left or bottom-left corner of the label) is known. We formulate this problem as finding the maximum independent set of a chordal graph, and propose an O(nlogn) time algorithm for producing the optimal solution.If the corner specification of the points in P is not known, our algorithm is a 2-approximation algorithm.Next, we develop a good heuristic algorithm that is observed to produce optimal solutions for most of the randomly generated instances and for all the standard benchmarks available in [13].

Time-Space Tradeoffs for Dynamic Programming Algorithms in Trees and Bounded Treewidth Graphs

The well-known Courcelle’s theorem states that many graph properties (that are expressible in monadic second order logic) can be solved in linear time on graphs of bounded treewidth. Logspace versions of this using automata theoretic framework are also known. In this paper, we develop an alternate methodology using the standard table-based dynamic programming approach to give a space efficient version of Courcelle’s theorem. We assume that the given graph and its tree decomposition are given in a read-only memory. Our algorithms use the recently developed stack-compression machinery and the classical framework of Borie et al. to develop time-space tradeoffs for dynamic programming algorithms that use \({\mathcal {O}}(p \log _p n)\) variables where \(2 \le p \le n\) is a parameter. En route we also generalize the stack compression framework to a broader class of algorithms, which we believe can be of independent interest.

Localized geometric query problems

A new class of geometric query problems are studied in this paper. We are required to preprocess a set of geometric objects P in the plane, so that for any arbitrary query point q, the largest circle that contains q but does not contain any member of P, can be reported efficiently. The geometric sets that we consider are point sets and boundaries of simple polygons.

Constrained minimum enclosing circle with center on a query line segment

In this paper, we will study the problem of locating the center of smallest enclosing circle of a set P of n points, where the center is constrained to lie on a query line segment. The preprocessing time and space complexities of our proposed algorithm are O(n logn) and O(n) respectively; the query time complexity is O(log2n). We will use this method for solving the following problem proposed by Bose and Wang [3] – given r simple polygons with a total of m vertices along with the point set P, compute the smallest enclosing circle of P whose center lies in one of the r polygons. This can be solved in O( nlogn+mlog2n) time using our method in a much simpler way than [3]; the time complexity of the problem is also being improved.

TIGHT ANALYSIS OF SHORTEST PATH CONVERGECAST IN WIRELESS SENSOR NETWORKS

We consider the convergecast problem in wireless sensor networks where readings generated by each sensor node are to reach the sink. Since a sensor reading can usually be encoded in a few bytes, more than one reading can readily fit into a standard transmission packet. We assume that any such packet consumes one unit of energy every time it hops from a node to a neighbor regardless of the total size of the readings in it. Our objective is to minimize the total energy consumed to send all the readings to the sink. Consequently, we ask the question: can we pack the readings in common routes to minimize the number of hops? It is quite elementary to see that this problem is NP-hard when the size of the readings are arbitrary via reductions from bin packing or set partition. We study the simple version with readings normalized to 1 byte in length. However, we make no assumptions on the underlying graph. We show this to be NP-hard by way of a reduction from Set Cover. We study a class SPEP of distributed algorithms that is completely defined by two properties. Firstly, the packets hop along some shortest path to the sink. Secondly, given all the readings that enter into a node, it sends out as many fully packed packets as possible followed by at most one partial packet --- the elementary packing property. We show that any algorithm in this class is (2−3/2k)-approximate where k ≥ 2 is the size of a data packet in bytes. We additionally show that this class is optimal when the underlying sensor network is a tree or grid topology. Our main technical contribution is a lower bound. We show that no algorithm that either follows the shortest path or packs in an elementary manner is a (2 − e)-approximation, for any fixed e > 0.

Shortest monotone descent path problem in polyhedral terrain

Given a polyhedral terrain with n vertices, the shortest monotone descent path problem deals with finding the shortest path between a pair of points, called source (s) and destination (t) such that the path is constrained to lie on the surface of the terrain, and for every pair of points p = (x(p), y(p), z(p)) and q = (x(q), y(q), z(q)) on the path, if dist(s,p) z(q), where dist(s,p) denotes the distance of p from s along the aforesaid path. This is posed as an open problem in [3]. We show that for some restricted classes of polyhedral terrain, the optimal path can be identified in polynomial time. We also propose an elegant method which can return near-optimal path for the general terrain in polynomial time.

Fast Gaussian Process Regression for Big Data

Gaussian Processes are widely used for regression tasks. A known limitation in the application of Gaussian Processes to regression tasks is that the computation of the solution requires performing a matrix inversion. The solution also requires the storage of a large matrix in memory. These factors restrict the application of Gaussian Process regression to small and moderate size data sets. We present an algorithm that combines estimates from models developed using subsets of the data obtained in a manner similar to the bootstrap. The sample size is a critical parameter for this algorithm. Guidelines for reasonable choices of algorithm parameters, based on detailed experimental study, are provided. Various techniques have been proposed to scale Gaussian Processes to large scale regression tasks. The most appropriate choice depends on the problem context. The proposed method is most appropriate for problems where an additive model works well and the response depends on a small number of features. The minimax rate of convergence for such problems is attractive and we can build effective models with a small subset of the data. The Stochastic Variational Gaussian Process and the Sparse Gaussian Process are also appropriate choices for such problems. These methods pick a subset of data based on theoretical considerations. The proposed algorithm uses bagging and random sampling. Results from experiments conducted as part of this study indicate that the algorithm presented in this work can be as effective as these methods. Model stacking can be used to combine the model developed with the proposed method with models from other methods for large scale regression such as Gradient Boosted Trees. This can yield performance gains.