Subhas C. Nandy

A Unified Algorithm for Finding Maximum and Minimum Object Enclosing Rectangles and Cuboids

Abstract Given a set of n points in R 2 bounded within a rectangular floor F, and a rectangular plate P of specified size, we consider the following two problems: find an isothetic position of P such that it encloses (i) maximum and (ii) minimum number of points, keeping P totally contained within F. For both of these problems, a new algorithm based on interval tree data structure is presented, which runs in O(nlogn) time and consumes O(n) space. If polygonal objects of arbitrary size and shape are distributed in R 2, the proposed algorithm can be tailored for locating the position of the plate to enclose maximum or minimum number of objects with the same time and space complexity. Finally, the algorithm is extended for identifying a cuboid, i.e., a rectangular parallelepiped that encloses maximum number of polyhedral objects in R 3. Thus, the proposed technique serves as a unified paradigm for solving a general class of enclosure problems encountered in computational geometry and pattern recognition.

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.

Largest empty rectangle among a point set

This work generalizes the classical problem of finding the largest empty rectangle among obstacles in 2D. Given a set P of n points, here a maximal empty rectangle (MER) is defined as a rectangle of arbitrary orientation such that each of its four boundaries contain at least one member of P and the interior of the rectangle is empty. We propose a very simple algorithm based on standard data structure to locate a MER of largest area in the plane. The worst-case time complexity of our algorithm is O(n3). Though the worst-case space complexity is O(n2), it reserves O(n log n) space on an average to maintain the required data structure during the execution of the algorithm.

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.

Efficient algorithms for Identifying All Maximal Isothetic Empty Rectangles in VLSI Layout Design

In this paper, we consider the following problem of computational geometry which has direct applications to VLSI layout design : given a set of n isothetic solid rectangles on a rectangular floor, identify all maximal-empty-rectangles (MERs). A tighter upper bound on the number of MERs is derived. A new algorithm based on interval trees for identifying all MERs is then presented which runs in O(nlogn+R) time in the worst case and in O(nlogn) time in the average case, where R denotes the number of MERs. The space complexity of the algorithm is O(n). Finally, we explore the problem of recognizing the maximum (area)- empty- rectangle without explicitly generating all MERs. Our analysis shows that, on an average, around 70% of MERs need not be examined in order to locate the maximum. The proposed algorithm can readily be tailored to solve the MER problem in an ensemble of points as well as within an isothetic polygon.

On finding an empty staircase polygon of largest area (width) in a planar point-set

This paper presents an algorithm for identifying a maximal empty-staircase-polygon (MESP) of largest area, among a set of n points on a rectangular floor. A staircase polygon is an isothetic polygon bounded by two monotonically rising (falling) staircases. A monotonically rising staircase is a sequence of alternatingly horizontal and vertical line segments from the bottom-left corner of the floor to its top-right corner such that for every pair of points α = (xα, yα) and β = (xβ, yβ) on the staircase, xα ≤ xβ implies yα ≤ yβ. A monotonically falling staircase can similarly be defined from the bottom-right corner of the floor to its top-left corner. An empty staircase polygon is a MESP if it is not contained in another larger empty staircase polygon. The problem of recognizing the largest MESP is formulated using permutation graph, and a simple O(n3) time algorithm is proposed. Next, based on certain novel geometric properties of the problem, an improved algorithm is developed that identifies the largest MESP in O(n2) time and space. The algorithm can be easily tailored for identifying the widest MESP in a similar environment. The general problem of locating the largest area/width MESP among a set of isothetic polygonal obstacles, can be solved easily. These geometric optimization problems have several applications to VLSI layout design, robot motion planning, to name a few.

Searching networks with unrestricted edge costs

Best-first and depth-first heuristic search algorithms often assume underlying search graphs with only nonnegative edge costs and attempt to optimize simple objective functions. Applicability of these algorithms to graphs with both positive and negative edge costs is not completely studied. In the paper, two new problems are identified: one in computational geometry and the other in the layout design of very large scale integrated (VLSI) circuits. The former problem relates to a weight-balanced bipartitioning of a given set of points in a plane. The goal of the second problem is to find an area-balanced staircase path in a VLSI floorplan. Formulations of these problems lead to an interesting directed acyclic search graph with positive, zero and negative edge costs and an objective function of general nature. These problems are NP-hard. To solve such general problems optimally, search schemes are proposed. Experimental results reveal the efficacy and versatility of the proposed schemes, the depth-first scheme being the better choice. It is shown that the classical number-partitioning problem can also be formulated in this framework.

Location of the Largest Empty Rectangle among Arbitrary Obstacles

This paper outlines the following generalization of the classical maximal-empty-rectangle (MER) problem: given n arbitrarily-oriented non-intersecting line segments of finite length on a rectangular floor, locate an empty isothetic rectangle of maximum area. Thus, the earlier restriction on isotheticity of the obstacles is relaxed. Based on the wellknown technique of matrix searching, a novel algorithm of time complexity O(nlog2n) and space complexity O(n), is proposed. Next, the technique is extended to handle the following two related open problems: locating the largest isothetic MER (i) inside an arbitrary simple polygon and (ii) amidst a set of arbitrary polygonal obstacles.

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.

Efficiency of discriminant analysis when initial samples are classified stochastically

Abstract We consider the problem of discriminant analysis of two multivariate normal populations having a common dispersion matrix, where the initial samples are classified stochastically. We assume a beta model for this classification variable and assume it to be independent of the feature vector X , given the group. We study the Efron efficiency of this procedure compared to the situation where the initial classification is done deterministically and correctly. We present tables and charts of this efficiency and conclude that stochastic supervision contains a great deal of information on the discriminant function.