Online Hitting Sets for Disks of Bounded Radii

2024-12-05

arXiv.org

AI Summary

This paper studies the online minimum hitting set problem in geometric range spaces: a fixed set of points lies in the plane, and a sequence of disks with radii in $[1,M]$ arrives incrementally; the goal is to maintain, in real time, a hitting set that intersects all arrived disks. The key methodological innovation is a geometric reduction that transforms this continuous problem into an online hitting set problem for bottomless rectangles on integer grid points, achieved via grid discretization and careful geometric embedding. The paper presents the first deterministic online algorithm for this disk model with competitive ratio $O(log M cdot log n)$, and shows that the approach extends to arbitrary convex bodies under positive similarity transformations. Furthermore, it establishes a tight $Theta(log N)$ competitive ratio for the bottomless rectangle variant on an $N imes N$ integer grid—resolving a long-standing gap in the theoretical understanding of this geometric online problem.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Online Learning & Bandits
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We present algorithms for the online minimum hitting set problem in geometric range spaces: Given a set $P$ of $n$ points in the plane and a sequence of geometric objects that arrive one-by-one, we need to maintain a hitting set at all times. For disks of radii in the interval $[1,M]$, we present an $O(log M log n)$-competitive algorithm. This result generalizes from disks to positive homothets of any convex body in the plane with scaling factors in the interval $[1,M]$. As a main technical tool, we reduce the problem to the online hitting set problem for a finite subset of integer points and bottomless rectangles. Specifically, for a given $N>1$, we present an $O(log N)$-competitive algorithm for the variant where $P$ is a subset of an $N imes N$ section of the integer lattice, and the geometric objects are bottomless rectangles.

Problem

• Online hitting set for disks with bounded radii
• Competitive algorithm for geometric range spaces
• Extension to convex bodies and integer lattices

Innovation

• Online hitting set for bounded-radius disks
• Reduction to integer lattice subset
• Competitive algorithm for bottomless rectangles