Binay K. Bhattacharya

Clustering algorithms based on minimum and maximum spanning trees

We consider clustering problems under two different optimization criteria. One is to minimize the maximum intracluster distance (diameter), and the other is to maximize the minimum intercluster distance. In particular, we present an algorithm which partitions a set <italic>S</italic> of <italic>n</italic> points in the plane into two subsets so that their larger diameter is minimized in time <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic>) and space <italic>&Ogr;</italic>(<italic>n</italic>). Another algorithm with the same bounds computes a <italic>k</italic>-partition of <italic>S</italic> for any <italic>k</italic> so that the minimum intercluster distance is maximized. In both instances it is first shown that an optimal parition is determined by either a maximum or minimum spanning tree of <italic>S</italic>.

Optimal movement of mobile sensors for barrier coverage of a planar region

Intrusion detection, area coverage and border surveillance are important applications of wireless sensor networks today. They can be (and are being) used to monitor large unprotected areas so as to detect intruders as they cross a border or as they penetrate a protected area. We consider the problem of how to optimally move mobile sensors to the fence (perimeter) of a region delimited by a simple polygon in order to detect intruders from either entering its interior or exiting from it. We discuss several related issues and problems, propose two models, provide algorithms and analyze their optimal mobility behavior.

Joint cluster analysis of attribute data and relationship data: The connected k -center problem, algorithms and applications

Attribute data and relationship data are two principal types of data, representing the intrinsic and extrinsic properties of entities. While attribute data have been the main source of data for cluster analysis, relationship data such as social networks or metabolic networks are becoming increasingly available. It is also common to observe both data types carry complementary information such as in market segmentation and community identification, which calls for a joint cluster analysis of both data types so as to achieve better results. In this article, we introduce the novel Connected k-Center (CkC) problem, a clustering model taking into account attribute data as well as relationship data. We analyze the complexity of the problem and prove its NP-hardness. Therefore, we analyze the approximability of the problem and also present a constant factor approximation algorithm. For the special case of the CkC problem where the relationship data form a tree structure, we propose a dynamic programming method giving an optimal solution in polynomial time. We further present NetScan, a heuristic algorithm that is efficient and effective for large real databases. Our extensive experimental evaluation on real datasets demonstrates the meaningfulness and accuracy of the NetScan results.

On Geometric Algorithms that use the Furthest-Point Voronoi Diagram*

In this paper it is shown that the diameter D( P ) of a set of n points P on the plane is not necessarily an edge in the dual of the furthest-point Voronoi diagram (FPVD) of P , as previously claimed in [1] and [2]. It is also proved that if P is contained in the disk determined by D( P ) then the above property does hold. Furthermore, it is shown that an edge e in the dual of the FPVD( P ) intersects its corresponding edge in the FPVD( P ) if, and only if, P is contained in the disk determined by e. These results invalidate several algorithms for solving the diameter, all-furthest-neighbor, and maximal spanning tree problems proposed in [1] and [2]. A proof of correctness is given for the minimum spanning circle algorithm proposed in [2] and [3]. Finally new O(n log n) algorithms are offered for the minimum spanning circle and all-furthest-neighbor problems.

On the multimodality of distances in convex polygons

Abstract Examples are given of n vertex convex polygons for which the distances between a fixed vertex and the remaining vertices, visited in order, form a multi-modal function. We show that this function may have as many as n /2 modes, or local maxima. Further examples are given of n vertex convex polygons in which n 2 /8 pairs of vertices are local maxima of their corresponding distance functions. These results are used to construct an example that shows that a general algorithm of Dobkin and Snyder may not, in fact, be used to find the diameter of a convex polygon.

Efficient algorithms for center problems in cactus networks

Efficient algorithms for solving the center problems in weighted cactus networks are presented. In particular, we have proposed the following algorithms for the weighted cactus networks of size n: an O(nlogn) time algorithm to solve the 1-center problem, and an O(nlog^3n) time algorithm to solve the weighted continuous 2-center problem. We have also provided improved solutions to the general p-center problems in cactus networks. The developed ideas are then applied to solve the obnoxious 1-center problem in weighted cactus networks.

Optimal Algorithms for Two-Guard Walkability of Simple Polygons

A polygon P admits a walk from a boundary point s to another boundary point t if two guards can simultaneously walk along the two boundary chains of P from s to t such that they are always visible to each other. A walk is called a straight walk if no backtracking is required during the walk. A straight walk is discrete if only one guard is allowed to move at a time, while the other guard waits at a vertex. We present simple, optimal O(n) time algorithms to determine all pairs of points of P which admit walks, straight walks and discrete straight walks. The chief merits of the algorithms are that these require simple data structures and do not assume a triangulation of P. Furthermore, the previous algorithms for the straight walk and the discrete straight walk versions ran in O(n log n) time even after assuming a triangulation.

Mobile facility location (extended abstract)

In this paper we investigate the location of mobile facilities (in <italic>L</italic><subscrpt>∞</subscrpt> and <italic>L</italic><subscrpt>2</subscrpt> metric) under the motion of clients. In particular, we present lower bounds and efficient algorithms for exact and approximate maintenance of 1-center and 1-median for a set of moving points in the plane. Our algorithms are based on the <italic>kinetic</italic> framework introduced by Basch et. al [5].

Optimal Movement of Mobile Sensors for Barrier Coverage of a Planar Region

Intrusion detection, area coverage and border surveillance are important applications of wireless sensor networks today. They can be (and are being) used to monitor large unprotected areas so as to detect intruders as they cross a border or as they penetrate a protected area. We consider the problem of how to optimally move mobile sensors to the fence (perimeter) of a region delimited by a simple polygon in order to detect intruders from either entering its interior or exiting from it. We discuss several related issues and problems, propose two models, provide algorithms and analyze their optimal mobility behavior.

On a Simple, Practical, Optimal, Output-Sensitive Randomized Planar Convex Hull Algorithm

In this paper we present a truly practical and provably optimalO(nlogh) time output-sensitive algorithm for the planar convex hull problem. The basic algorithm is similar to the algorithm presented by2, where the median-finding step is replaced by an approximate median. We analyze two such schemes and show that for both methods, the algorithm runs in expectedO(nlogh) time. We further show that the probability of deviation from expected running time approaches 0 rapidly with increasing values ofnandhfor any input. Our experiments suggest that this algorithm is a practical alternative to the worst-caseO(nlogn) algorithms such as Grahams and is especially faster for small output-sizes. Our approach bears some resemblance to a recent algorithm of13, but our analysis is substantially different.