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An Algorithmic Solution for Computing Circle Intersection Areas and its Applications to Wireless Communications

Federico Librino, Marco Levorato, Michele Zorzi

TL;DR

This work tackles the problem of exactly computing the areas of intersections among an arbitrary number of circles, with direct relevance to wireless-network analysis where coverage regions are circular. It introduces a trellis-based iterative algorithm anchored by two geometric theorems that let higher-order intersections be inferred from lower-order ones, enabling efficient computation even for many circles. The method provides both non-exclusive and exclusive intersection areas, offering exact results where Monte-Carlo methods yield approximations. A practical wireless-network design application demonstrates how the computed intersection areas support outage probability analysis under cooperative access-point scenarios, with potential for broader connectivity and coverage assessments in heterogeneous and sensor networks. Overall, the algorithm offers a scalable tool for precise geometric modeling in network design and performance evaluation.

Abstract

A novel iterative algorithm for the efficient computation of the intersection areas of an arbitrary number of circles is presented. The algorithm, based on a trellis-structure, hinges on two geometric results which allow the existence-check and the computation of the area of the intersection regions generated by more than three circles by simple algebraic manipulations of the intersection areas of a smaller number of circles. The presented algorithm is a powerful tool for the performance analysis of wireless networks, and finds many applications, ranging from sensor to cellular networks. As an example of practical application, an insightful study of the uplink outage probability of in a wireless network with cooperative access points as a function of the transmission power and access point density is presented.

An Algorithmic Solution for Computing Circle Intersection Areas and its Applications to Wireless Communications

TL;DR

This work tackles the problem of exactly computing the areas of intersections among an arbitrary number of circles, with direct relevance to wireless-network analysis where coverage regions are circular. It introduces a trellis-based iterative algorithm anchored by two geometric theorems that let higher-order intersections be inferred from lower-order ones, enabling efficient computation even for many circles. The method provides both non-exclusive and exclusive intersection areas, offering exact results where Monte-Carlo methods yield approximations. A practical wireless-network design application demonstrates how the computed intersection areas support outage probability analysis under cooperative access-point scenarios, with potential for broader connectivity and coverage assessments in heterogeneous and sensor networks. Overall, the algorithm offers a scalable tool for precise geometric modeling in network design and performance evaluation.

Abstract

A novel iterative algorithm for the efficient computation of the intersection areas of an arbitrary number of circles is presented. The algorithm, based on a trellis-structure, hinges on two geometric results which allow the existence-check and the computation of the area of the intersection regions generated by more than three circles by simple algebraic manipulations of the intersection areas of a smaller number of circles. The presented algorithm is a powerful tool for the performance analysis of wireless networks, and finds many applications, ranging from sensor to cellular networks. As an example of practical application, an insightful study of the uplink outage probability of in a wireless network with cooperative access points as a function of the transmission power and access point density is presented.

Paper Structure

This paper contains 15 sections, 2 theorems, 34 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction and Problem Statement
  2. Applications to Wireless Networks
  3. Problem Statement and Contribution
  4. Geometric results
  5. The Algorithm
  6. Trellis Structure
  7. Auxiliary Variables
  8. Computation of the Intersection Areas
  9. Proof of algorithm correctness
  10. Algorithm complexity
  11. Comparison with Monte-Carlo approximation
  12. Network Design Application
  13. Conclusion
  14. Proof of Theorem \ref{['teo:intinun']}
  15. Derivation of $\textbf{A}_{N_c}$ for $N_c = 4$

Key Result

Theorem 1

Consider a subset of $n{\leq}N_c$ circles $\mathcal{S}{=}\{\gamma_{i_1},\ldots,\gamma_{i_n}\}{\subseteq}\mathcal{C}$ and the associated intersection region $\mathcal{I}^{(n)}_{\{i_1,\ldots,i_n\}}$. With a slight abuse of notation, we refer to the considered subset of circles when denoting the inters then also $\phi(\mathcal{I}^{(n)}_{\{i_1,\ldots,i_{n}\}})>0$.

Figures (13)

  • Figure 1: Example of configuration with three circles and identification of the regions of interest.
  • Figure 2: Intersection of $n$ circles. Here each circumference contains one arc belonging to the circular polygon $\Delta$, which delimits the intersection.
  • Figure 3: Example of circles configuration and areas considered in the first four steps of the algorithm.
  • Figure 4: The trellis structure for $n=5$. Some of the binary labels are reported. The $i$--th column of vertices corresponds to the subset $\mathcal{V}_i$.
  • Figure 5: Example of disposition of 4 circles.
  • ...and 8 more figures

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • proof