Subir Kumar Ghosh

An output-sensitive algorithm for computing visibility

The visibility graph of a set of nonintersecting polygonal obstacles in the plane is an undirected graph whose vertices are the vertices of the obstacles and whose edges are pairs of vertices (u, v) such that the open line segment between u and v does not intersect any of the obstacles. The visibility graph is an important combinatorial structure in computational geometry and is used in applications such as solving visibility problems and computing shortest paths. An algorithm is presented that computes the visibility graph of s set of obstacles in time O(E + n log n), where E is the number of edges in the visibility graph and n is the total number of vertices in all the obstacles.

An output sensitive algorithm for computing visibility graphs

The visibility graph of a set of nonintersecting polygonal obstacles in the plane is an undirected graph whose vertices are the vertices of the obstacles and whose edges are pairs of vertices (u, v) such that the open line segment between u and v does not intersect any of the obstacles. The visibility graph is an important combinatorial structure in computational geometry and is used in applications such as solving visibility problems and computing shortest paths. An algorithm is presented that computes the visibility graph of s set of obstacles in time O(E + n log n), where E is the number of edges in the visibility graph and n is the total number of vertices in all the obstacles.

Approximation algorithms for art gallery problems in polygons

In this paper, we present approximation algorithms for minimum vertex and edge guard problems for polygons with or without holes with a total of n vertices. For simple polygons, approximation algorithms for both problems run in O(n^4) time and yield solutions that can be at most O(logn) times the optimal solution. For polygons with holes, approximation algorithms for both problems give the same approximation ratio of O(logn), but the running time of the algorithms increases by a factor of n to O(n^5).

Computing the visibility polygon from a convex set and related problems

In this paper, we propose efficient algorithms for computing the complete and weak visibility polygons of a simple polygon P of n vertices from a convex set C inside P. The algorithm for computing the complete visibility polygon of P from C takes O(n + k) time in the worst case, where k is the number of extreme points of the convex set C. Given a triangulation of P - C, the algorithm for computing the weak visibility polygon of P from C takes O(n + k) time in the worst case. We also show that computing the complete and weak visibility polygons of P from a nonconvex set inside P has the same time complexity. The algorithm for computing the complete visibility polygon of P from a convex set inside P leads to efficient algorithms for the following problems: (i) Given a polygon Q of m vertices inside another polygon P of n vertices, construct a minimum nested convex polygon K between P and Q. The algorithm runs in O((n + m)log k) time, where k is the number of vertices of K. This is an improvement over the O((n + m)log(n + m)) time algorithm of Wang and Chan. (ii) Given two points inside a polygon P, compute a minimum link path between them inside P. Given a triangulation of P, the algorithm takes O(n) time. Suri also proposed a linear time algorithm for this problem in a triangulated polygon but our algorithm is simpler.

Unsolved problems in visibility graphs of points, segments, and polygons

In this survey article, we present open problems and conjectures on visibility graphs of points, segments, and polygons along with necessary backgrounds for understanding them.

On recognizing and characterizing visibility graphs of simple polygons

In this paper, we establish three necessary conditions for recognizing visibility graphs of simple polygons and conjecture that these conditions are sufficient. We also show that visibility graphs of simple polygons do not posses the characteristics of several special classes of graphs.

Survey: Online algorithms for searching and exploration in the plane

In this paper, we survey online algorithms in computational geometry that have been designed for mobile robots for searching a target and for exploring a region in the plane.

Optimal on-line algorithms for walking with minimum number of turns in unknown streets

Abstract A simple polygon P with two distinguished vertices s and t is said to be a street if the clockwise and counterclockwise boundary of P from s to t are mutually weakly visible. We consider the problem of traversing a path from s to t in an unknown street P for a mobile robot with on-board vision system such that the number of links in the path is as small as possible. To our knowledge, this problem has not been studied before. We present an algorithm for this problem that requires at most 2m − 1 links to reach from s to t, where m denotes the link distance between s and t in P. Hence the competitive ratio of our algorithm is 2 − 1 m . We also show that any on-line algorithm for the above problem will require 2m − 1 links in the worst case which establishes that our algorithm is optimal. We next consider the above problem for the special case when P is a rectilinear street and the path is required to be a rectilinear path. We propose an algorithm for this problem that requires at most m + 1 links to reach from s to t, where m denotes the rectilinear link distance between s and t in P. Hence the competitive ratio of our algorithm is 1 + 1 m . We also show that any on-line algorithm for this problem will require m + 1 links in the worst case which establishes that our algorithm is optimal.

Triangulating with high connectivity

Abstract We consider the problem of triangulating a given point set, using straight-line edges, so that the resulting graph is “highly connected”. Since the resulting graph is planar, it can be at most 5-connected. Under the nondegeneracy assumption that no three points are collinear, we characterize the point sets with three vertices on the convex hull that admit 4-connected triangulations. More generally, we characterize the planar point sets that admit triangulations having neither chords nor complex (i.e., nonfacial) triangles. We also show that any planar point set can be augmented with at most two extra points to admit a 4-connected triangulation. All our proofs are constructive, and the resulting triangulations can be constructed in O(n log n) time. We conclude by stating several open problems. In particular, it is open whether a polynomial-time algorithm exists for determining whether a point set with no degeneracy restrictions and no restrictions on the number of extreme points admits a 4- or 5-connected triangulation.

Characterizing and recognizing weak visibility polygons

A polygon is said to be a weak visibility polygon if every point of the polygon is visible from some point of an internal segment. In this paper we derive properties of shortest paths in weak visibility polygons and present a characterization of weak visibility polygons in terms of shortest paths between vertices. These properties lead to the following efficient algorithms: (i) an O(E) time algorithm for determining whether a simple polygon P is a weak visibility polygon and for computing a visibility chord if it exist, where E is the size of the visibility graph of P and (ii) an O(n2) time algorithm for computing the maximum hidden vertex set in an n-sided polygon weakly visible from a convex edge.