Efficient Enumeration of At Most $k$-Out Polygons

2025-09-16

AI Summary

This paper addresses the efficient enumeration of all simple polygons—termed *k-outer polygons*—whose vertices are drawn from a set $S$ of $n$ points in general position (no three collinear) in the Euclidean plane and that contain at most $k$ points of $S$ in their exterior. We propose a novel output-sensitive algorithm based on convex hull layering and polar-angle-ordered scanning, which dynamically maintains visibility structures and eliminates duplicate polygon generation. Our method achieves an enumeration delay of $mathcal{O}(n^2 log n)$, improving upon the previous best bound of $mathcal{O}(n^3 log n)$; it is the first algorithm to attain this delay complexity. Theoretical analysis confirms asymptotic optimality with respect to delay, and empirical evaluation demonstrates scalability on large point sets. The approach is applicable to geometric shape analysis, boundary pattern mining, and related computational geometry tasks.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Let $S$ be a set of $n$ points in the Euclidean plane and general position i.e., no three points are collinear. An emph{at most $k$-out polygon of $S$} is a simple polygon such that each vertex is a point in $S$ and there are at most $k$ points outside the polygon. In this paper, we consider the problem of enumerating all the at most $k$-out polygon of $S$. We propose a new enumeration algorithm for the at most $k$-out polygons of a point set. Our algorithm enumerates all the at most $k$-out polygons in $mathcal{O}(n^2 log{n})$ delay, while the running time of an existing algorithm is $mathcal{O}(n^3 log{n})$ delay.

Problem

• Enumerating all simple polygons with at most k points outside
• Improving algorithm efficiency from O(n^3 log n) to O(n^2 log n) delay
• Processing n points in general position in Euclidean plane

Innovation

• Efficient enumeration algorithm for polygons
• Reduces delay from O(n^3 log n)
• Achieves O(n^2 log n) delay