Chee-Keng Yap

A “retraction” method for planning the motion of a disc

Abstract A new approach to certain motion-planning problems in robotics is introduced. This approach is based on the use of a generalized Voronoi diagram, and reduces the search for a collision-free continuous motion to a search for a connected path along the edges of such a diagram. This approach yields an O ( n log n ) algorithm for planning an obstacle-avoiding motion of a single circular disc amid polygonal obstacles. Later papers will show that extensions of the approach can solve other motion-planning problems, including those of moving a straight line segment or several coordinated discs in the plane amid polygonal obstacles.

AnO(n logn) algorithm for the voronoi diagram of a set of simple curve segments

LetX be a given set ofn circular and straight line segments in the plane where two segments may interest only at their endpoints. We introduce a new technique that computes the Voronoi diagram ofX inO(n logn) time. This result improves on several previous algorithms for special cases of the problem. The new algorithm is relatively simple, an important factor for the numerous practical applications of the Voronoi diagram.

Some consequences of non-uniform conditions on uniform classes☆

We obtain some results of the form: If certain complexity classes satisfy a non-uniform condition, then some unlikely consequences follow. More precisely: 1. (1) If the ‘non-uniform polynomial-time hierarchy’ collapses at level i>0, i.e., Σipoly = Πipoly, then the Meyer-Stockmeyer hierarchy collapses at level i + 2, i.e., Σi+2 = Πi+2. This strengthens a generalization of a result of Karp and Lipton (1980). 2. (2) If co-NP is conjunctively reducible to a sparse set, then P = NP. This generalizes a theorem of Fortune (1979). 3. (3) If NP is conjunctively and disjunctively reducible to a sparse NP-complete set, then P = NP. This is a partial generalization of a result of Mahaney (1980). Conjuctive and disjunctive reducibility were introduced by Ladner, Lynch and Selman (1975). 4. (4) If co-NP is γ-reducible to a sparse set, then NP = co-NP. γ-reducibility was introduced 5. by Adleman and Manders (1977).

Parallel computational geometry

We present efficient parallel algorithms for several basic problems in computational geometry: convex hulls, Voronoi diagrams, detecting line segment intersections, triangulating simple polygons, minimizing a circumscribing triangle, and recursive data-structures for three-dimensional queries.

Towards exact geometric computation

Abstract Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixed-precision model, usually the machine floating-point arithmetic. Such implementations have many well-known problems, here informally called “robustness issues”. To reconcile theory and practice, authors have suggested that theoretical algorithms ought to be redesigned to become robust under fixed-precision arithmetic. We suggest that in many cases, implementors should make robustness a non-issue by computing exactly. The advantages of exact computation are too many to ignore. Many of the presumed difficulties of exact computation are partly surmountable and partly inherent with the robustness goal. This paper formulates the theoretical framework for exact computation based on algebraic numbers. We then examine the practical support needed to make the exact approach a viable alternative. It turns out that the exact computation paradigm encompasses a rich set of computational tactics. Our fundamental premise is that the traditional “BigNumber” package that forms the work-horse for exact computation must be reinvented to take advantage of many features found in geometric algorithms. Beyond this, we postulate several other packages to be built on top of the BigNumber package.

On k -hulls and related problems

For any set X of points (in any dimension) and any

New upper bounds in Klee's measure problem

k = 1,2, \cdots

Shape from probing

, we introduce the concept of the k-hull of X. The k-hull is the set of points p such that for any hyperplane containing p there...

A core library for robust numeric and geometric computation

New upper bounds for the measure problem of Klee are given which significantly improve the previous bounds for dimensions greater than two. An

Quantitative Steinitz's theorems with applications to multifingered grasping

O(n^{d / 2} \log n,n)