Norm Dadoun

A simple parallel tree contraction algorithm

Abstract A simple reduction from the tree contraction problem to the list ranking problem is presented. The reduction takes O(log n) time for a tree with n nodes, using O( n log n ) EREW processors. Thus tree contraction can be done as efficiently as list ranking. A broad class of parallel tree computations to which the tree contraction techniques apply is described. This subsumes earlier characterizations. Applications to the computation of certain properties of cographs are presented in some detail.

Parallel processing for efficient subdivision search

The study of parallelism in computational geometry has been largely confined to individual case studies and isolated results with the exception of the recent comprchcnsivc paper of Aggarwal CI al [ACGOY]. Aggarwal et al presented a number of techniques and tools which have laid the foundation for the study of parallelism in computational geometry. Among their results were parallel solutions to such familiar geometric problems as convex hull construction (in 2 and 3-d). Voronoi diagram construction (in 2-d) and closest point search, and segment intersecti&. More recently, Atallah and Goodrich [AG] elaborated on one technique which they proposed (parallel plane sweep).

The geometry of beam tracing

A solution to the hidden surface elimination problem called Beam Tracing is described. Beam tracing is related to ray tracing but uses spatial coherence within the scene, and area coherence within the image to batch computations. Beam tracing is an object space solution to the hidden surface problem. Beam tracing is formulated in terms of its principal subprocesses: intersection, sorting, and clipping. A Hierarchical Scene Representation is proposed. This incorporates the space decomposition idea of the BSP tree [Fuchs, Kedem and Naylor, 80] along with the convex polytope intersection detection technique of [Dobkin and Kirkpatrick, 83] to interleave and efficiently solve the intersection and sorting subproblems of beam tracing.

Parallel construction of subdivision hierarchies

A direct, simple and general parallel algorithm is described for the preprocessing of a planar subdivision for fast (sequential) search. In essence, the hierarchical subdivision search structure described by Kirkpatrick [K] is constructed in parallel. The method relies on an efficient parallel algorithm for constructing large independent sets in planar graphs. This is accomplished by a simple reduction to the same problem for lists. Applications to the manipulation of convex polyhedra are described including an \( O(\log^{2}n \log^{*}n) \) parallel time algorithm for constructing the convex hull of

Parallel algorithms for fractional and maximal independent sets in planar graphs

n

Hierarchical approaches to hidden surface intersection testing

points in

Optimal Parallel Algorithms for Convex Polygon Separation

R^{3}

A Parallel Tree Contraction Algorithm

and an \( O( \log n \log^{*}n) \) parallel time algorithm for detecting the separation of convex polyhedra.

Coopera-tive SubdivIsion Search Algorithms with Applica, tions

Abstract The problem of computing maximal independent sets in graphs on parallel models of computation has received considerable attention. We present simple efficient parallel algorithms for the maximal independent set problem—and a relaxation that we call the fractional independent set problem—restricted to planar graphs. Our algorithms rely on an efficient parallel algorithm for constructing large independent sets in graphs of bounded degree. The latter is accomplished by a simple reduction to the same problem for lists. Using a linear number of EREW processors, the algorithm identifies a maximal independent set in an arbitrary planar graph in O( log n log ∗n) parallel time. A randomized version of the algorithm runs in O(logn) expected parallel time.