Partha P. Goswami

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.

Smallest k -point enclosing rectangle and square of arbitrary orientation

Given a set of n points in 2D, the problem of identifying the smallest rectangle of arbitrary orientation, and containing exactly k (≤ n) points is studied in this paper. The worst case time and space complexities of the proposed algorithm are O(n2 logn + nk(n - k)(n - k + logk)) and O(n), respectively. The algorithm is then used to identify the smallest square of arbitrary orientation, and containing exactly k points in O(n2 logn + kn(n - k)2 logn) time.

SMALLEST COLOR-SPANNING OBJECT REVISITED

Given a set of n colored points in IR2 with a total of m (3 ≤ m ≤ n) colors, the problem of identifying the smallest color-spanning object of some predefined shape is studied in this paper. We shall consider two different shapes: (i) corridor and (ii) rectangle of arbitrary orientation. Our proposed algorithm for identifying the smallest color-spanning corridor is simple and runs in O(n2log n) time using O(n) space. A dynamic version of the problem is also studied, where new points may be added, and the narrowest color-spanning corridor at any instance can be reported in O(mn(α(n))2log m) time. Our algorithm for identifying the smallest color-spanning rectangle of arbitrary orientation runs in O(n3log m) time and O(n) space.

An efficient k nearest neighbors searching algorithm for a query line

We present an algorithm for finding k nearest neighbors of a given query line among a set of n points distributed arbitrarily on a two-dimensional plane. Our algorithm requires O(n2) time and O(n2/log n) space to preprocess the given set of points, and it answers the query for a given line in O(k + log n) time, where k may also be an input at the query time. Almost a similar technique works for finding k farthest neighbors of a query line, keeping the time and space complexities invariant. We also show that if k is known at the time of preprocessing, the time and space complexities for the preprocessing can be reduced keeping the query times unchanged.

k-enclosing axis-parallel square

Let P be a set of n points in the plane. Here we present an efficient algorithm to compute the smallest square containing at least k points of P for large values of k. The worst case time complexity of the algorithm is O(n + (n - k) log2(n - k)) using O(n) space which is the best known bound for worst case time complexity.

Triangular range counting query in 2D and its application in finding k nearest neighbors of a line segment

We present an efficient algorithm for finding k nearest neighbors of a query line segment among a set of points distributed arbitrarily on a two dimensional plane. Along the way to finding this algorithm, we have obtained an improved triangular range searching technique in 2D. Given a set of n points, we preprocess them to create a data structure using O(n2) time and space, such that given a triangular query region Δ, the number of points inside Δ can be reported in O(logn) time. Finally, this triangular range counting technique is used for finding k nearest neighbors of a query straight line segment in O(log2 n + k) time.

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].

Computing the maximum clique in the visibility graph of a simple polygon

In this paper, we present an algorithm for computing the maximum clique in the visibility graph G of a simple polygon P in O(n^2e) time, where n and e are number of vertices and edges of G respectively. We also present an O(ne) time algorithm for computing the maximum hidden vertex set in the visibility graph G of a convex fan P. We assume in both algorithms that the Hamiltonian cycle in G that corresponds to the boundary of P is given as an input along with G.

Optimal algorithm for a special point-labeling problem

A special class of map labeling problem is studied. Let P = {p1, p2,..., pn} be a set of point sites distributed on a 2D map. A label associated with each point pi is an axis-parallel rectangle ri of specified width. The height of all ri, i = 1, 2, ..., n are same. The placement of ri must contain pi at its top-left or bottom-left corner, and it does not obscure any other point in P. The objective is to label the maximum number of points on the map so that the placed labels are mutually non-overlapping. 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 a priori. We show that the time complexity of this problem is Ω(n log n), and then propose an algorithm for this problem which runs in O(n log n) time. If the corner specifications of the points in P are not known, our algorithm is a 2-approximation algorithm. Here we propose an efficient heuristic algorithm that is easy to implement. Experimental evidences show that it produces optimal solutions for most of the randomly generated instances and for all the standard benchmarks available in http://www.math-inf.uni-greifswald.de/map-labeling/.

Recognition of minimum width color-spanning corridor and minimum area color-spanning rectangle

Given a set of n colored points with a total of m (≥ 3) colors in 2D, the problem of identifying the smallest color-spanning object is studied. We have considered two different shapes: (i) corridor, and (ii) rectangle of arbitrary orientation. Our proposed algorithms for the problems (i) and (ii) run in time O(n2logn) and O(n3logm) respectively.