Separator for $c$-Packed Segments and Curves

Separator for $c$-Packed Segments and Curves: A Simpler Approach

Introduction

This paper, "Separator for $c$-Packed Segments and Curves" (2604.04011), presents a novel and simpler algorithmic proof for computing balanced geometric separators for $c$-packed sets of segments in $\mathbb{R}^d$. The main contribution is to demonstrate, with increased proof simplicity, that such separators can intersect only $O(c)$ segments while partitioning the set into reasonably balanced subsets. While similar separator results were previously established, this work streamlines both the analytic foundation and the algorithmic implementation, making it more accessible and directly applicable for geometric algorithms operating on $c$-packed objects.

Definitions and Problem Statement

A set $S$ of line segments in $\mathbb{R}^d$ is termed $c$-packed if, for any center $p$ and radius $c$0, the sum of lengths of the portions of segments inside the ball $c$1 is at most $c$2. This property restricts the local complexity of the segment set and is widely applicable whenever the underlying geometric input is not too densely clustered. The separator problem aims to find a geometric primitive (here, a sphere) that both (1) intersects a small number of segments (ideally $c$3), and (2) partitions the remaining segments so that a constant fraction appears on each side.

Main Results

The central result is the following: For a $c$4-packed set $c$5 of $c$6 segments in $c$7, there exists a sphere $c$8 that can be computed in expected linear time with the following properties:

• The sphere $c$9 intersects at most $c$0 segments of $c$1.
• On each side of the sphere, at least $c$2 segments of $c$3 are disjoint from $c$4.

The proof leverages a volumetric argument, exploiting the covering and packing properties intrinsic to $c$5-packed sets. The algorithm proceeds by considering the endpoints of all segments and seeking a ball containing a controlled fraction (specifically $c$6) of these points. By drawing a sphere at a random radius between that of the ball and twice its value, the expected number of segment intersections can be strictly bounded using a key lemma relating packing to intersection counts. The application of Markov's inequality then guarantees, with high probability, the existence of a separator matching the claimed bounds.

For practicality, the algorithm uses a $c$7-approximation to the smallest enclosing ball and appeals to linear-time selection to maintain computational efficiency. The separator construction avoids dense iterative subdivision or annotation of the input, streamlining use in algorithms where separator hierarchies are required.

Technical Implications

The simplified proof method highlights several notable implications:

• Tighter Analytical Bound: The number of intersected segments is proportional to $c$8, exposing an explicit dependence on the packing parameter and not on the input size $c$9 or the dimension $\mathbb{R}^d$0 (except via constants in $\mathbb{R}^d$1 related to the covering number).
• Improved Algorithmic Simplicity: The construction's reliance on straightforward sampling and classical covering arguments reduces practical implementation complexity. This enables more direct integration with divide-and-conquer or sublinear algorithms for geometric optimization on curves, polyline simplification, and related domains.
• Robustness to Dimensionality: The $\mathbb{R}^d$2-packed assumption and the corresponding separator size bound are stable under changes in $\mathbb{R}^d$3, facilitating extension to high-dimensional subproblems or embeddings, as encountered in TDA or high-dimensional clustering contexts.

Connections and Extensions

This result is closely related to earlier works on separators for well-behaved geometric objects, but provides a more transparent path to realizing the separator construction and elucidates its dependence on standard packing and covering constants (as e.g., in doubling metrics and VC-dimension arguments). The approach also paves the way for analogous separator constructions for more general families of geometric objects, such as $\mathbb{R}^d$4-packed polygonal curves, as often studied in the analysis of Fréchet distance and motion planning.

Potential future research directions include investigating separators for dynamically evolving $\mathbb{R}^d$5-packed sets, adapting the method for non-Euclidean settings, or refining the bounds for special classes of geometric graphs derived from $\mathbb{R}^d$6-packed inputs.

Conclusion

The paper establishes that for $\mathbb{R}^d$7-packed sets of segments in Euclidean space, one can efficiently compute a balanced separator (a sphere) that meets only $\mathbb{R}^d$8 segments, with a simpler and more accessible proof and algorithm than previously known. This strengthens the toolkit available for geometric divide-and-conquer design, particularly in algorithmic settings where $\mathbb{R}^d$9-packing assumptions are warranted. Extensions of this method to broader classes of geometric objects are anticipated to further benefit the theoretical and practical development of geometric algorithms.