Franco P. Preparata

On the Connection Assignment Problem of Diagnosable Systems

This paper treats the problem of automatic fault diagnosis for systems with multiple faults. The system is decomposed into n units u 1 , u 2 , . . . , u n , where a unit is a well-identifiable portion of the system which cannot be further decomposed for the purpose of diagnosis. By means of a given arrangement of testing links (connection assignment) each unit of the system tests a subset of units, and a proper diagnosis can be arrived at for any diagnosable fault pattern. Methods for optimal assignments are given for instantaneous and sequential diagnosis procedures.

The cube-connected cycles: a versatile network for parallel computation

An interconnection pattern of processing elements, the cube-connected cycles (CCC), is introduced which can be used as a general purpose parallel processor. Because its design complies with present technological constraints, the CCC can also be used in the layout of many specialized large scale integrated circuits (VLSI). By combining the principles of parallelism and pipelining, the CCC can emulate the cube-connected machine and the shuffle-exchange network with no significant degradation of performance but with a more compact structure. We describe in detail how to program the CCC for efficiently solving a large class of problems that include Fast Fourier transform, sorting, permutations, and derived algorithms.

Convex hulls of finite sets of points in two and three dimensions

The convex hulls of sets of n points in two and three dimensions can be determined with O(n log n) operations. The presented algorithms use the “divide and conquer” technique and recursively apply a merge procedure for two nonintersecting convex hulls. Since any convex hull algorithm requires at least O(n log n) operations, the time complexity of the proposed algorithms is optimal within a multiplicative constant.

Euclidean Shortest Paths in the Presence of Rectilinear Barriers

In this paper we address the problem of constructing a Euclidean shortest path between two specified points (source, destination) in the plane, which avoids a given set of barriers. This problem had been solved earlier for polygonal obstacles with the aid of the visibility graph. This approach however, has an Ω(n2) time lower bound, if n is the total number of vertices of the obstacles. Our goal is to find interesting cases for which the solution can be obtained without the explicit construction of the entire visibility graph. The two cases are (i) the path must lie within an n-vertex simple polygon; (ii) the obstacles are n disjoint and parallel line segments. In both instances greedy O(n log n) time algorithms can be developed which solve the problems by constructing the shortest-path tree from the source to all the vertices of the obstacles and to the destination.

Plane-sweep algorithms for intersecting geometric figures

Algorithms in computational geometry are of increasing importance in computer-aided design, for example, in the layout of integrated circuits. The efficient computation of the intersection of several superimposed figures is a basic problem. Plane figures defined by points connected by straight line segments are considered, for example, polygons (not necessarily simple) and maps (embedded planar graphs). The regions into which the plane is partitioned by these intersecting figures are to be processed in various ways such as listing the boundary of each region in cyclic order or sweeping the interior of each region. Let m be the total number of points of all the figures involved and s be the total number of intersections of all line segments. A two plane-sweep algorithm that solves the problems above is presented; in the general case (non convexity) in time O((n+s)log-n) and space O(n+s); when the regions of each given figure are convex, the same can be achieved in time O(n log n +s) and space O(n)

Finding the intersection of two convex polyhedra

Abstract Given two convex polyhedra in three-dimensional space, we develop an algorithm to (i) test whether their intersection is empty, and (ii) if so to find a separating plane, while (iii) if not to find a point in the intersection and explicitly construct their intersection polyhedron. The algorithm runs in timeO (n log n), where n is the sum of the numbers of vertices of the two polyhedra. The part of the algorithm concerned with (iii) (constructing the intersection) is based upon the fact that if a point in the intersection is known, then the entire intersection is obtained from the convex hull of suitable geometric duals of the two polyhedra taken with respect to this point.

Location of a Point in a Planar Subdivision and Its Applications

Given a subdivision of the plane induced by a planar graph with n vertices, in this paper we consider the problem of identifying which region of the subdivision contains a given test point. We present a search algorithm, called point-location algorithm, which operates on a suitably preprocessed data structure. The search runs in time at most

Optimal off-line detection of repetitions in a string

O((\log n)^{2})

Bounds to Complexities of Networks for Sorting and for Switching

, while the preprocessing task runs in time at most

An Optimal Algorithm for Finding the Kernel of a Polygon

O(n \log n)