Time- and VLSI-optimal convex hull computation on meshes with multiple broadcasting

Podality-based time-optimal computations on enhanced meshes

“…This somewhat counter-intuitive result motivated us to look at the following problem: Suppose that we are given a mesh, or enhanced mesh, of size n n ¥ and the goal is to solve instances of size m n m n £ £ Our next contribution is to show that this lower bound is tight by providing podality-based, time-optimal algorithms for the problems mentioned. In this direction, we show that once we fix a constant e > 0, all instances of size m, n m n Recently, the authors have demonstrated that the task of computing the convex hull of a set of m points in the plane can be solved time-optimally on a mesh with multiple broadcasting of size n n ¥ [11]. However, the algorithm in [11] does not use copodality and, as a result, is rather involved.…”

“…In this direction, we show that once we fix a constant e > 0, all instances of size m, n m n Recently, the authors have demonstrated that the task of computing the convex hull of a set of m points in the plane can be solved time-optimally on a mesh with multiple broadcasting of size n n ¥ [11]. However, the algorithm in [11] does not use copodality and, as a result, is rather involved. By contrast, the time-optimal convex hull algorithm developed in this paper is copodality-based and offers much more insight than the algorithm in [11].…”

“…The convex hull of a set of points is the smallest convex set that contains the original set [29], [32]. Being central to many application areas, the convex hull problem has been extensively studied in the literature, both sequentially and in parallel [1], [11], [14], [18], [29]. The diameter of a set of points is useful in agglomerative clustering [1], [14], computer graphics, path planning [22], and in a number of facility location problems [29].…”

“…However, the algorithm in [11] does not use copodality and, as a result, is rather involved. By contrast, the time-optimal convex hull algorithm developed in this paper is copodality-based and offers much more insight than the algorithm in [11]. Section 2 presents preliminaries and establishes terminology and notation; Section 3 discusses our copodalitybased convex hull algorithm; Section 4 presents novel results concerning the antipodality of a convex polygon; Section 5 presents our time-optimal diameter-finding algorithm; Section 6 discusses our time-optimal algorithm for the width and the smallest-area enclosing rectangle; Section 7 addresses the problem of computing the maximum Euclidian distance between two sets of points; Section 8 discusses the problem of computing the Minkowski sum of two convex polygons.…”

The mesh with hybrid buses: an efficient parallel architecture for digital geometry

“…Being of theoretical interest, as well as commercially available, the MMB has attracted a great deal of attention. Applications ranging from image processing [33], [44], [48], to computer graphics and robotics [43], to computational geometry and pattern recognition [6], [7], [9], [10], [11], [13], [32], [42], to optimization [20], to query processing and mobile computing [15], and to other fundamental problems [4], [8], [19] have found efficient solutions on this architecture.…”