Optimal parallel algorithms for rectilinear link-distance problems

We provide optimal parallel solutions to several link-distance problems set in trapezoided rectilinear polygons. All our main parallel algorithms are deterministic and designed to run on the exclusive read exclusive write parallel random access machine (EREW PRAM). Let P be a trapezoided rectilinear simple polygon with n vertices. In O(log n) time using O(n/log n) processors we can optimally compute: 1. Minimum réctilinear link paths, or shortest paths in the L1 metric from any point in P to all vertices of P. 2. Minimum rectilinear link paths from any segment inside P to all vertices of P. 3. The rectilinear window (histogram) partition of P. 4. Both covering radii and vertex intervals for any diagonal of P. 5. A data structure to support rectilinear link-distance queries between any two points in P (queries can be answered optimally in O(log n) time by uniprocessor). Our solution to 5 is based on a new linear-time sequential algorithm for this problem which is also provided here. This improves on the previously best-known sequential algorithm for this problem which used O(n log n) time and space.5 We develop techniques for solving link-distance problems in parallel which are expected to find applications in the design of other parallel computational geometry algorithms. We employ these parallel techniques, for example, to compute (on a CREW PRAM) optimally the link diameter, the link center, and the central diagonal of a rectilinear polygon

Pop-stacks in parallel

A stack may be viewed as a machine with two instructions POP and PUSH that transforms an input sequence of data items into an output sequence. Pop-stacks are a stack-like container data type but the POP operation removes all the items on the stack. The permutations that can be sorted by a system of k pop-stacks in parallel are shown to be characterizable by a finite set of forbidden patterns and, for k = 2, a recurrence relation is found for their enumeration

An Optimal Algorithm for Detecting Weak Visibility of a Polygon

In 1981, Avis and Toussaint gave a linear-time algorithm for the following problem. Given a simple n-Vertex polygon P and an edge of P, determine whether each point in P can be seen by some (not necessarily the same) point on the edge. They posed the more general problem of finding a subquadratic algorithm for determining whether such an edge exists. In this paper, we present a linear-time algorithm for determining all (if any) such edges of a given simple polygon

Translation separability of sets of polygons

We consider the problem of separating a set of polygons by a sequence of translations (one such collision-free translation motion for each polygon). If all translations are performed in a common direction the separability problem so obtained has been referred to as the uni-directional separability problem; for different translation directions, the more general multi-directional separability problem arises. The class of such separability problems has been studied previously and arises e.g. in computer graphics and robotics. Existing solutions to the uni-directional problem typically assume the objects to have a certain predetermined shape (e.g., rectangular or convex objects), or to have a direction of separation already available. Here we show how to compute all directions of unidirectional separability for sets of arbitrary simple polygons. The problem of determining whether a set of polygons is multi-directionally separable had been posed by G.T. Toussaint. Here we present an algorithm for solving this problem which, in addition to detecting whether or not the given set is multidirectionally separable, also provides an ordering in which to separate the polygons. In case that the entire set is not multi-directionally separable, the algorithm will find the largest separable subset

Simple optimal algorithms for rectilinear link path and polygon separation problems

The link metric, defined on a constrained region R of the plane, sets the distance between a pair of points in R to equal the minimum number of line segments or links needed to construct a path in R between the point pair. The minimum rectilinear link path problem considered here is to compute a rectilinear path consisting of the minimum number of links between two points in R, when R is inside an n-sided rectilinear simple polygon. In this paper we present optimal sequential and parallel algorithms to compute a minimum rectilinear link path in a trapezoided region R. Our parallel algorithm requires O(log n) time using a total of O(n) operations. The complexity of our algorithm matches that of the algorithm of McDonald and Peters [19]. By exploiting the dual structure of the trapezoidation of R, we obtain a conceptually simple and easy to implement algorithm. As applications of our techniques we provide an optimal solution to the minimum nested polygon problem and the minimum polygon separation problem. The minimum nested polygon problem asks for finding a rectilinear polygon, with minimum number of sides, that is nested between two given rectilinear polygons one of which is contained in the other. The minimum polygon separation problem asks for computing a minimum number of orthogonal lines and line segments that separate two given non-intersecting simple rectilinear polygons. All parallel algorithms are deterministic, designed to run on the exclusive read exclusive write parallel random access machine (EREW PRAM), and are optimal

Disassembling two-dimensional composite parts via translations

This paper deals with the computational complexity of disassembling 2-dimensional composite parts (comprised of simple polygons) via collision-free translations. The first result of this paper is an O(Mn + M log M) algorithm for computing a sequence of translations (performed in a common direction) to disassemble composite parts. The algorithm improves on the O(Mn log Mn) bound previously established for this problem and is easily seen to be optimal. The algorithm solves the problem posed by Nurmi and by Toussaint. The second result of this paper is an Ω(Mn + M log M) lower bound proof for the problem of detecting whether a composite part can be disassembled, or contains interlocking subparts. Thus, detecting the existence of a disassembly sequence is as hard as computing one. As a consequence, the algorithm for computing a disassembly sequence is optimal also for the detecting problem

Separating a polyhedron by one translation from a set of obstacles

We present efficient algorithms for several problems of movable separability in 3-dimensional space. The first algorithm determines all directions in which a convex polyhedron can be translated through a planar convex polygonal window. The algorithm runs in linear time. This is a considerable improvement over the previous O(n2logn) time algorithm of [17], where n is the total number of vertices in the objects. The second algorithm computes, in O(n) time, all directions in which a convex polyhedron can be translated to infinity without collisions with a convex obstacle (n is the number of vertices of the polyhedra). A generalization of the planesweep technique, called “sphere-sweep”, is given and provides an efficient algorithm for the last problem which is: determine all directions in which a convex polyhedron can be separated from a set of convex obstacles. Our results are obtained by avoiding the standard technique of motion planning problems, the Ω(n2) time computation of the Minkowski differences of the polyhedra

Minimum Decompositions of Polygonal Objects

This chapter discusses techniques for decompositions of objects into the minimum number of some component type. In many applications, the objects encountered are rectilinear polygons. In image processing, the boundaries of objects are often stored on a grid that usually implies that digitized images are rectilinear polygons. Very large-scale integration (VLSI) designs are also often stored on a grid and typically contain many rectilinear polygons. The chapter discusses the decompositions of rectilinear polygons when Steiner points are allowed and when Steiner points are disallowed. It also discusses the decomposition of arbitrary simple polygons when Steiner points are allowed and when Steiner points are disallowed. The result in this study concerns the partitioning of a polygonal region into a minimum number of trapezoids with two horizontal sides. Triangles with a horizontal side are considered to be trapezoids with two horizontal sides, one of which is degenerate. This problem is closely related to VLSI artwork data processing systems of electron-beam lithography for VLSI microfabrication. The chapter also discusses the decomposition of object which contains holes