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VQhull: a Fast Planar Quickhull

Thomas Koopman, Jordy Aaldering, Bernard van Gastel, Sven-Bodo Scholz

TL;DR

VQhull presents a vectorized, parallel implementation of Quickhull for planar convex hulls that minimizes data movement and exploits CPU bandwidth. By introducing a vectorized in-place subset extraction and a two-phase parallelization (parallel step plus cleanup), it achieves up to 1.6–16× sequential and 1.5–11× parallel speedups over the state of the art, while approaching hardware bandwidth limits on non-NUMA systems and showing meaningful energy efficiency trends. The work includes extensive benchmarking across three platforms and three PBBS datasets, with a nuanced discussion of branch behavior, vectorization, memory subsystem effects, and energy consumption. The findings highlight that performance and energy can decouple in bandwidth-bound geometric algorithms and point to future directions like memory-bandwidth–reduction heuristics and NUMA-aware parallel strategies to push further gains.

Abstract

Finding the convex hull is a fundamental problem in computational geometry. Quickhull is a fast algorithm for finding convex hulls. In this paper, we present VQhull, a fast parallel implementation of Quickhull that exploits vector instructions, and coordinates CPU cores in a way that minimizes data movement. This implementation obtains a sequential runtime improvement of 1.6--16x, and a parallel runtime improvement of 1.5-11x compared to the state of the art on the Problem Based Benchmark Suite. VQhull achieves 85--100% of non-NUMA architectures' peak bandwidth, and 66--78% on our two-CPU NUMA system. This leaves little room for further improvements. A 4x speedup on 8 cores has a parallel efficiency of 50%. This suggests a waste of energy, but our measurements show a more complicated picture: energy usage may even be lower in parallel. Quickhull serves as a case study that runtime and energy consumption do not go hand in hand.

VQhull: a Fast Planar Quickhull

TL;DR

VQhull presents a vectorized, parallel implementation of Quickhull for planar convex hulls that minimizes data movement and exploits CPU bandwidth. By introducing a vectorized in-place subset extraction and a two-phase parallelization (parallel step plus cleanup), it achieves up to 1.6–16× sequential and 1.5–11× parallel speedups over the state of the art, while approaching hardware bandwidth limits on non-NUMA systems and showing meaningful energy efficiency trends. The work includes extensive benchmarking across three platforms and three PBBS datasets, with a nuanced discussion of branch behavior, vectorization, memory subsystem effects, and energy consumption. The findings highlight that performance and energy can decouple in bandwidth-bound geometric algorithms and point to future directions like memory-bandwidth–reduction heuristics and NUMA-aware parallel strategies to push further gains.

Abstract

Finding the convex hull is a fundamental problem in computational geometry. Quickhull is a fast algorithm for finding convex hulls. In this paper, we present VQhull, a fast parallel implementation of Quickhull that exploits vector instructions, and coordinates CPU cores in a way that minimizes data movement. This implementation obtains a sequential runtime improvement of 1.6--16x, and a parallel runtime improvement of 1.5-11x compared to the state of the art on the Problem Based Benchmark Suite. VQhull achieves 85--100% of non-NUMA architectures' peak bandwidth, and 66--78% on our two-CPU NUMA system. This leaves little room for further improvements. A 4x speedup on 8 cores has a parallel efficiency of 50%. This suggests a waste of energy, but our measurements show a more complicated picture: energy usage may even be lower in parallel. Quickhull serves as a case study that runtime and energy consumption do not go hand in hand.

Paper Structure

This paper contains 28 sections, 9 equations, 9 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Quickhull Algorithm
  4. Performance on Central Processing Units
  5. Branch Prediction
  6. Vectorization
  7. Data Movement and Multithreading
  8. Subset Extraction
  9. Sequential Extraction
  10. Parallel Extraction
  11. Parallel step
  12. Cleanup
  13. Implementation
  14. Parallelization
  15. Bandwidth considerations
  16. ...and 13 more sections

Figures (9)

  • Figure 1: An example of the first partitioning step of the Quickhull algorithm.
  • Figure 2: The vcompresspd instruction
  • Figure 3: Invariant for extracting the subsets $S_1$ and $S_2$ from a set of points $P$.
  • Figure 4: Parallel partition step for $3$ threads $t_0$, $t_1$, $t_2$
  • Figure 5: The Kuzmin dataset of $10^8$ points. The first partition only moves $r_1$. The second partition eliminates all remaining points.
  • ...and 4 more figures