Asish Mukhopadhyay

Computing a centerpoint of a finite planar set of points in linear time

The notion of a centerpoint of a finite set of points in two and higher dimensions is a generalization of the concept of the median of a set of reals. In this paper we present a linear-time algorithm for computing a centerpoint of a set ofn points in the plane, which is optimal compared with theO(n log3n) complexity of the previously best-known algorithm. We use suitable modifications of the hamsandwich cut algorithm in [Me2] and the prune-and-search technique of Megiddo [Me1] to achieve this improvement.

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.

On intersecting a set of parallel line segments with a convex polygon of minimum area

Let S = {l1, l2, l3, . . . , ln} be a set of n vertical line segments in the plane. Though not essential, to simplify proofs we assume that no two li ’s are on the same vertical line. A convex polygon weakly intersects S if it contains a point of each line segment on its boundary or interior. In this paper, we propose an O(n logn) algorithm for the problem of finding a minimum area convex polygon that weakly intersects S. The principal motivation behind this paper is the open problem proposed by Tamir [5] at the fourth Computational Geometry day at NYU to decide if there exists a convex polygon whose boundary intersects a set of arbitrarily oriented line segments.

Voronoi diagrams for direction-sensitive distances

Most computational geometry research on planar problems assumes that the underlying plane is perfectly ‘flat’, in the sense that movement between any two points cm the plane always takes the same cost as long as the Euclidean distance between the two points is the same. In real environments, distances may depend on the direction one moves along [10], or even may be influenced by local properties of the plane [8]. These situations sometimes can be modeled by considering a piecewiselinear surface as the underlying ‘plane’, and measuring distances therein; see e.g. [7]. In fact, many distance problems on non-flat planes are hard to deal with from the computational geometry point of view. We study distance problems for the basic case of a ‘tilted’ plane in three-space. In this model, the cost of moving depends not only on the Euclidean distance but also on how much upwards or downwards the movement has to travel, simulating the situation when driving a vehicle on the tilted plane. Direction-sensitive distances and, in particular, their induced Voronoi di-

An Optimal Algorithm for the Intersection Radius of a Set of Convex Polygons

The intersection radius of a finite collection of geometrical objects in the plane is the radius of the smallest closed disk that intersects all the objects in the collection. Bhattacharyaet al.showed how the intersection radius can be found in linear time for a collection of line segments in the plane by combining the prune-and-search strategy of Megiddo (J. Assoc. Comput. Mach.31(1) (1984), 114?127) with the strategy of replacing line segments by lines or points (B. K. Bhattacharya, S. Jadhav, A. Mukhopadhyay, and J. M. Robert,in“Proceedings of the Seventh Annual ACM Symposium on Computational Geometry,” pp. 81?88, 1991). In this paper, we enlarge the scope of this technique by showing that it can also be used to find the intersection radius of a collection of convex polygons inO(n) time, wherenis the total number of polygon vertices.

On the Minimum Perimeter Triangle Enclosing a Convex Polygon

We consider the problem of computing a minimum perimeter triangle enclosing a convex polygon. This problem defied a linear-time solution due to the absence of a property called the interspersing property. This property was crucial in the linear-time solution for the minimum area triangle enclosing a convex polygon. We have discovered a non-trivial interspersing property for the minimum perimeter problem. This resulted in an optimal solution to the minimum perimeter triangle problem.

A linear time algorithm for computing the shortest line segment from which a polygon is weakly externally visible

A simple polygon P is said to be weakly externally visible from a line segment if the line segment is outside P and if for every point x on the boundary of P there is a point y on the line segment such that the interior of the line segment xy does not intersect the interior of P. In this paper a linear time algorithm is proposed for computing the shortest line segment from which a simple polygon is weakly externally visible. This is done by a suitable generalization of a linear time algorithm which solves the same problem for a convex polygon.

A new necessary condition for the vertex visibility graphs of simple polygons

A new necessary condition conjectured by Everett [2], which is essentially a stronger version of a necessary condition by Ghosh [3], for a graph to be the vertex visibility graph of a simple polygon is established.

Using simplicial partitions to determine a closest point to a query line

In this note we show that simplicial partitions can be used to answer the following queries efficiently: given a point set in the plane, determine a point that is closest to an arbitrary query line. We also propose an alternate scheme that is very practical and easy to implement.

On Computing a Largest Empty Arbitrarily Oriented Rectangle

Given a set P of n points within a rectangle R, we present an O(n3) time algorithm for computing an arbitrarily oriented empty rectangle of largest area in R that is bounded by a point of P on each of its four sides. We assume that R is large enough to contain such a rectangle.