High-Dimensional Convex Hulls: Speed-Optimized Algorithm for N-D Spaces

Accelerating N-Dimensional Convex Hulls: A Faster Hull Algorithm

Overview

This covers a faster algorithmic approach for computing convex hulls in N-dimensional Euclidean space (R^N). The goal is to reduce runtime and memory overhead compared with classic methods (e.g., Quickhull, gift wrapping, incremental algorithms) especially on higher-dimensional or large datasets, while preserving numerical robustness and correctness.

Key ideas

• Dimensional reduction with careful pruning: Project or reduce the problem size by eliminating interior points early using fast approximate tests (randomized hashing, spatial partitioning, or quick antipodal checks) so only candidate extreme points remain for exact hull construction.
• Incremental face-local updates: Insert points in batches and update hull facets locally rather than rebuilding globally; maintain adjacency graphs of facets to limit updates to affected regions.
• Output-sensitive complexity: Aim for algorithms whose running time depends on input size n and hull complexity h (number of facets/vertices) — e.g., O(n log n + nh^(1-2/(N))) or similar output-sensitive bounds depending on approach and assumptions.
• Parallelization: Partition dataset spatially or by random sampling; construct partial hulls in parallel and merge using pairwise hull merges or convex hull unions via facet stitching.
• Numeric robustness: Use exact predicates (adaptive precision arithmetic) or robust floating-point techniques (Shewchuk-style predicates, epsilon handling) to avoid degeneracies and incorrect facet orientation.
• Data structures: Maintain facet adjacency graphs, conflict lists (points vs facets), and spatial indices (kd-tree, bounding-volume hierarchies) for fast point-to-facet queries.

Algorithm sketch (batch-incremental with pruning)

1. Preprocess: remove duplicates, compute bounding box, optionally normalize coordinates.
2. Quick elimination: use random sampling to compute an approximate hull; discard points strictly inside this approximate hull via fast point-in-convex-hull tests (e.g., barycentric or facet half-space checks).
3. Build initial hull from a small affinely independent subset (N+1 points).
4. Batch insertion: split remaining candidate points into batches. For each batch:
   • For each point, find visible facets using conflict lists or spatial index.
   • Create new facets by connecting horizon edges; update adjacency and conflict lists.
   • Prune facets with small solid angles or near-degenerate geometry using robustness checks.
5. Merge partial hulls if computed in parallel: compute union by inserting facets/vertices from one hull into another using facet-stitching operations.
6. Final clean-up: remove coplanar redundant facets, reorient facets consistently, and optionally simplify output.

Complexity and practical performance

• Best practical performance combines pruning and output-sensitivity: when h << n, runtime approaches O(n log n + h poly(N)). In worst case (h large, e.g., random points on sphere), complexity can approach O(n^{ceil(N/2)}) for some algorithms; improved methods and parallelism mitigate this.
• Parallel implementations scale well for large n, with communication cost dominated by boundary merging and conflict resolution.

Implementation notes

• Use robust geometry libraries where possible (CGAL, Qhull as reference) or implement exact predicates (Shewchuk).
• Represent facets by oriented simplices (indices to vertices) and store adjacency for local updates.
• Use memory pools for dynamic facet/vertex allocation; carefully manage concurrency in parallel merges.
• Test on varied distributions (uniform, Gaussian, points on sphere, clustered) to tune pruning thresholds.

When to use

• High-dimensional datasets (N > 3) where classic 2D/3D algorithms don’t scale.
• Large n with relatively small hull size h (sparse extremes).
• Applications: computational geometry, collision detection in high-dim spaces, machine learning (convex hull for support vectors, outlier detection), multi-objective optimization (Pareto front convexification).

Limitations

• As N grows, combinatorial complexity increases rapidly; practical limits often N ≤ 10–20 depending on n and h.
• Numerical issues and degenerate configurations require careful handling.
• Worst-case inputs can still be expensive; algorithmic guarantees vary based on assumptions (randomness, general position).

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