An Optimal Algorithm for Higher-Order Voronoi Diagrams in the Plane: The Usefulness of Nondeterminism

We present the first optimal randomized algorithm for constructing the order-$k$ Voronoi diagram of $n$ points in two dimensions. The expected running time is $O(n\log n + nk)$, which improves the previous, two-decades-old result of Ramos (SoCG'99) by a $2^{O(\log^*k)}$ factor. To obtain our result, we (i) use a recent decision-tree technique of Chan and Zheng (SODA'22) in combination with Ramos's cutting construction, to reduce the problem to verifying an order-$k$ Voronoi diagram, and (ii) solve the verification problem by a new divide-and-conquer algorithm using planar-graph separators. We also describe a deterministic algorithm for constructing the $k$-level of $n$ lines in two dimensions in $O(n\log n + nk^{1/3})$ time, and constructing the $k$-level of $n$ planes in three dimensions in $O(n\log n + nk^{3/2})$ time. These time bounds (ignoring the $n\log n$ term) match the current best upper bounds on the combinatorial complexity of the $k$-level. Previously, the same time bound in two dimensions was obtained by Chan (1999) but with randomization.Comment: To appear in SODA 2024. 16 pages, 1 figur

Algorithm Libraries for Multi-Core Processors

By providing parallelized versions of established algorithm libraries, we ease the exploitation of the multiple cores on modern processors for the programmer. The Multi-Core STL provides basic algorithms for internal memory, while the parallelized STXXL enables multi-core acceleration for algorithms on large data sets stored on disk. Some parallelized geometric algorithms are introduced into CGAL. Further, we design and implement sorting algorithms for huge data in distributed external memory

Fundamental Computational Geometry on the GPU

Ph.DDOCTOR OF PHILOSOPH

A Randomized Parallel 3D Convex Hull Algorithm For Coarse Grained Multicomputers

We present a randomized parallel algorithm for constructing the 3D convex hull on a generic p-processor coarse grained multicomputer with arbitrary interconection network and n=p local memory per processor, where n=p p 2+ffl (for some arbitrarily small ffl ? 0). For any given set of n points in 3-space, the algorithm computes the 3D convex hull, with high probaility, in O( n log n p ) local computation time and O(1) communication phases with at most O(n=p) data sent/received by each processor. That is, with high probability, the algorithm computes the 3D convex hull of an arbitrary point set in time O( n logn p + \Gamma n;p ), where \Gamma n;p denotes the time complexity of one communication phase. The assumption n p p 2+ffl implies a coarse grained, limited parallelism, model which is applicable to most commercially available multiprocessors. In the terminology of the BSP model, our algorithm requires, with high probability, O(1) supersteps, synchronization period L = \Th..