Convex-Hull Scenario Compression

• Convex-hull scenario compression is a set of algorithmic techniques that replace a large point set with a smaller representative subset that preserves or approximates its convex hull.
• Key methods include interior-point pruning, rotation-extrema filtering, and recursive elimination, which drastically reduce data size and enhance computational efficiency.
• These strategies are crucial in optimization and computer graphics, delivering substantial speedups and enabling effective data reduction in practice.

Convex-hull scenario compression refers to a class of algorithmic techniques that replace a large set of points or scenarios with a small, representative subset whose convex hull exactly matches or tightly approximates the convex hull of the original set. This approach is foundational in geometric computing, stochastic optimization, computer graphics, and data reduction, enabling significant acceleration and simplification in tasks where convex hulls are a computational bottleneck. Core strategies include pruning interior points, identifying extremal structures, and domain-specific scenario reduction, all guided by problem-intrinsic geometry.

1. Fundamental Concepts and Problem Statements

Convex-hull scenario compression addresses the following generalized problem: Given a point set $P = \{p_1, \ldots, p_n\} \subseteq \mathbb{R}^d$, compute a subset $S \subseteq P$ of cardinality $s \ll n$ such that $\mathrm{conv}(S) = \mathrm{conv}(P)$ or $\mathrm{conv}(S)$ approximates $\mathrm{conv}(P)$ to within a prescribed Hausdorff distance. This compressed subset supports direct or accelerated computation of geometric and optimization tasks (Cadenas et al., 2015, Klimenko et al., 2021, Zhang et al., 2024).

Problem formulations can be categorized as:

• Exact hull preservation: Find $S$ with minimal size s.t. $\mathrm{conv}(S) = \mathrm{conv}(P)$ (Heydari et al., 2023).
• Approximate hull preservation: Find $S$ of given size (or minimal size for fixed approximation parameter) so that $$d_H(\mathrm{conv}(S), \mathrm{conv}(P)) \leq \varepsilon$$ (Klimenko et al., 2021).
• Scenario replacement in optimization: Replace an uncertainty sample set $\Xi$ by the vertex set $V$ of $\mathrm{conv}(\Xi)$ to compress scenario-based constraints without changing feasible solutions, provided affine structure holds (Zhang et al., 2024).

2. Interior-Point Pruning and Heuristic Extremal Filtering

A dominating strategy for convex-hull scenario compression is to discard points that cannot serve as hull vertices. This can be accomplished via geometrically motivated heuristic filters or recursive elimination schemes.

Rotation-extrema convex polygon pruning

A prominent approach involves:

1. Rotating the entire point set at multiple angles (e.g., $$\{0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}\}$$), extracting 4 extremal points per rotation (min/max $x$, min/max $y$), yielding up to 16 extreme points.
2. Constructing their convex hull $P_{\mathrm{ext}}$.
3. Discarding all points in the original set that lie strictly inside $P_{\mathrm{ext}}$ using a point-in-polygon test.

This GPU-parallelizable method achieves work complexity $O(n)$, discards $>99\%$ of input points for uniformly distributed data, and produces end-to-end speedups of $4\times\ldots 6\times$ when pipelined with exact convex hull algorithms (Mei, 2014).

Recursive extreme-point elimination

For points sampled uniformly in a convex region, recursive partitioning achieves further compression:

• At each step, select the four axis-aligned extreme points (top, bottom, left, right).
• Remove all points in the axis-aligned bounding quadrilateral.
• Recurse on each of the four peripheral “corner” regions.

The expected number of retained points is $O(\log n)$, as each recursion discards a constant fraction of the remaining set, and the method achieves $O(n)$ expected time (Heydari et al., 2023). This is significantly sharper than earlier $O(\sqrt{n})$-reduction schemes, but relies critically on uniform randomness; adversarial (e.g., circular) data can defeat its efficiency.

3. Data Preconditioning and Extremal Column Selection

Integer grid-based datasets motivate another filtering regime: For each distinct $x$-coordinate, retain only the points with minimal and maximal $y$; optionally, repeat this in $y$ if a more aggressive reduction is warranted.

With $p$ denoting the number of unique $x$’s, the survivor set has $s \leq 2p$, drastically thinning data with dense bounding boxes. This survivor list is pipeline-compatible with $O(s)$-time online convex hull algorithms such as Melkman’s, and the entire reduction is $O(n)$ time with $O(n)$ space (Cadenas et al., 2015). Empirical data show discarding over $98\%$ of points with overall hull-building speedups in the $3\times\ldots 10\times$ range, provided point density is not pathologically sparse.

4. Convex-Hull Scenario Compression in Optimization

In stochastic programming and large-scale deterministic scenario-approximation, scenario compression by keeping only the extreme points of $\mathrm{conv}(\Xi)$ (the convex hull of the scenario set) has strong theoretical and computational guarantees. In particular, if all constraints are affine in the uncertainty, the set of binding constraints is determined by the extreme points (vertices) of $\mathrm{conv}(\Xi)$, so only these need to be enforced.

Given $N$ original scenarios, if $V$ is the set of convex hull vertices and $|V| \ll N$, the scenario program reduces to $|V|$ constraints with no loss of solution accuracy, as proven by the maximization principle over convex polytopes. Speedup is proportional to $N/|V|$; in power systems SCED problems with $N=500$, typical $|V|\sim 30$ was reported, yielding an order-of-magnitude faster solutions (Zhang et al., 2024).

This guarantees exactness provided all scenario-dependent constraints are affine and the feasible set is convex. Non-affine or nonconvex constraints invalidate the vertex extremization argument. In high dimensions, the number of unique vertices may become excessive, especially for unstructured or high-correlation data, but most practical models involve effective dimensionality reduction.

5. Classical Akl–Toussaint and Related Heuristics

Classic compression heuristics, exemplified by the Akl–Toussaint box, first construct a small ($p$-gon) polygon enclosing most of the dataset (e.g., quadrilateral by axis-aligned extremes, hexagon, or full octagon by also exploiting $\min(x+y)$, $\max(x-y)$, etc.). Non-enclosed points are guaranteed to include the convex hull.

With $p=8$, the number of survivors is $O(\sqrt{N})$ for random planar sets, superior to the quadrilateral’s $O(N)$ performance. This selection can be plugged into basic incremental or bucket-based convex hull algorithms, providing $O(N)$ average-case performance with optimal practical constants, though worst-case is $O(N^2)$ (Souviron, 2013).

Empirical data from real geometric datasets confirm that the octagon heuristic achieves an average survivor ratio of $\approx 2/\sqrt{N}$ and outperforms or is competitive with output-sensitive algorithms for non-adversarial data.

6. Convex Hull Simplification and Hausdorff-Approximate Compression

Beyond exact preservation, convex-hull scenario compression may target controlled approximation. Given a tolerance $\varepsilon$, the minimum-cardinality subset $Q\subseteq P$ with $$d_H(\mathrm{conv}(Q), \mathrm{conv}(P))\leq\varepsilon$$ can be computed in $O(n\log^2 n)$ when $P$ is in convex position, or reduced to all-pairs shortest-path on weighted graphs in $O(n^{2.5302})$ time for the general case. Dual formulations minimize the approximation error for a fixed subset size.

2-approximation algorithms further enable near-linear time simplification for arbitrary point sets (Klimenko et al., 2021).

7. Extensions: Non-Point and Higher-Dimensional Scenario Compression

Compression concepts generalize beyond static point clouds. For instance, in robust geometric preprocessing of families of regions (e.g., unit disks), it is possible to construct supersequences—ordered structures over the regions—such that for any realization of one representative point per region, the resulting hull can be extracted as a subsequence in time proportional to the number of “unstable” regions, i.e., those whose membership might affect the hull. For n unit disks, this achieves $O(n\log n)$ preprocessing and sublinear, instance-adaptive reconstruction (Löffler et al., 5 Feb 2025).

In mesh compression and 3D convex hulls, scenario compression includes encoding the convex polyhedron as an intersection of supporting half-spaces (planes), reducing storage cost by forgoing explicit vertex or face connectivity (Aramyan et al., 2013). Predictive quantization techniques further compress these normal/distance parameters.

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• "A Straightforward Preprocessing Approach for Accelerating Convex Hull Computations on the GPU" (Mei, 2014)
• "An Empirical Evaluation of Preconditioning Data for Accelerating Convex Hull Computations" (Cadenas et al., 2015)
• "A simple and efficient preprocessing step for convex hull problem" (Heydari et al., 2023)
• "Accelerating Chance-constrained SCED via Scenario Compression" (Zhang et al., 2024)
• "Convex hull: Incremental variations on the Akl-Toussaint heuristics" (Souviron, 2013)
• "Fast and Exact Convex Hull Simplification" (Klimenko et al., 2021)
• "Reduction of Two-Dimensional Data for Speeding Up Convex Hull Computation" (Mukherjee, 2022)
• "Coding and Compression of Three Dimensional Meshes by Planes" (Aramyan et al., 2013)
• "Preprocessing Disks for Convex Hulls, Revisited" (Löffler et al., 5 Feb 2025)