Spherical coverage verification

TL;DR

This paper addresses the complex problem of verifying spherical coverage by hypercaps, demonstrating its NP-hardness, and proposing a recursive algorithm that performs well in practice despite theoretical complexity.

Contribution

The paper introduces a recursive algorithm for spherical coverage verification that reduces the problem to lower dimensions, providing practical solutions despite NP-hardness.

Findings

The problem is NP-hard due to its reduction to concave quadratic programming.

The proposed recursive algorithm is effective in practical scenarios.

QP solver heuristics may lead to false positives due to numerical instability.

Abstract

We consider the problem of covering hypersphere by a set of spherical hypercaps. This sort of problem has numerous practical applications such as error correcting codes and reverse k-nearest neighbor problem. Using the reduction of non degenerated concave quadratic programming (QP) problem, we demonstrate that spherical coverage verification is NP hard. We propose a recursive algorithm based on reducing the problem to several lower dimension subproblems. We test the performance of the proposed algorithm on a number of generated constellations. We demonstrate that the proposed algorithm, in spite of its exponential worst-case complexity, is applicable in practice. In contrast, our results indicate that spherical coverage verification using QP solvers that utilize heuristics, due to numerical instability, may produce false positives.