Arcs with increasing chords in $mathbf{R}^d$

2025-09-01

AI Summary

This paper investigates upper bounds on the length of curves satisfying the Increasing Chord Property (ICP) in high-dimensional Euclidean space $mathbb{R}^d$ ($d geq 3$), and the efficient recognition of ICP polygonal chains. Addressing the longstanding absence of theoretical bounds and efficient algorithms in dimensions beyond two, we derive the first explicit length bound for ICP curves: $2left(frac{e}{2}(d+4) ight)^{d-1}|st|$. Furthermore, we design the first subquadratic-time algorithm for ICP verification—achieving expected time $O(n^{2-1/(k+1)} operatorname{polylog} n)$ for $d geq 4$. Our approach integrates high-dimensional geometric analysis, permutation-order detection, and multi-scale distance comparison techniques, thereby overcoming prior limitations confined to the planar case. The results establish foundational tools for modeling and verifying geometrically constrained curves in high dimensions.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

A curve $γ$ that connects $s$ and $t$ has the increasing chord property if $|bc| leq |ad|$ whenever $a,b,c,d$ lie in that order on $γ$. For planar curves, the length of such a curve is known to be at most $2π/3 cdot |st|$. Here we examine the question in higher dimensions and from the algorithmic standpoint and show the following: (I) The length of any $s-t$ curve with increasing chords in $mathbf{R}^d$ is at most $2 cdot left( e/2 cdot (d+4) ight)^{d-1} cdot |st|$ for every $d geq 3$. This is the first bound in higher dimensions. (II) Given a polygonal chain $P=(p_1, p_2, dots, p_n)$ in $mathbf{R}^d$, where $d geq 4$, $k =lfloor d/2 floor$, it can be tested whether it satisfies the increasing chord property in $Oleft(n^{2-1/(k+1)} { m polylog} (n) ight)$ expected time. This is the first subquadratic algorithm in higher dimensions.

Problem

• Bounding curve length with increasing chords in higher dimensions
• Developing subquadratic algorithm for testing increasing chord property
• Extending planar curve results to multidimensional Euclidean space

Innovation

• Establishes first length bound for high-dimensional curves
• Provides subquadratic algorithm for testing property
• Uses polygonal chain analysis in R^d