[Paper Review] Construction of the generalized Cech complex

Ngoc-Khuyen Le, Philippe Martins+2 more|arXiv (Cornell University)|Sep 29, 2014

Topological and Geometric Data Analysis7 references4 citations

TL;DR

This paper presents a centralized algorithm to construct the minimal generalized Čech complex for wireless networks with cells of varying sizes, using geometric intersection verification to ensure topological accuracy. The method efficiently builds the complex up to dimension 2 in polynomial time, enabling precise detection of coverage holes and network topology analysis.

ABSTRACT

In this paper, we introduce an algorithm which constructs the generalized Cech complex. The generalized Cech complex represents the topology of a wireless network whose cells are different in size. This complex is often used in many application to locate the boundary holes or to save energy consumption in wireless networks. The complexity of a construction of the Cech complex to analyze the coverage structure is found to be a polynomial time.

Motivation & Objective

• To address the lack of efficient methods for constructing the generalized Čech complex in wireless networks with cells of different sizes.
• To provide a minimal simplicial complex that accurately represents network topology, including coverage holes.
• To ensure the algorithm runs in polynomial time for practical deployment in large-scale networks.
• To support applications such as coverage hole detection, disaster recovery, and energy-efficient cell management.

Proposed method

• The algorithm constructs the generalized Čech complex by verifying non-empty intersection among groups of cells using geometric intersection checks.
• For each candidate k-simplex, it checks whether the smallest cell in the group is contained within all other cells in the group.
• It further validates higher-dimensional simplices by checking if intersection points of pairs of cells lie within all other cells in the group.
• The construction proceeds iteratively from 0-simplices (cells) through 1-simplices (neighbors) to higher-dimensional simplices, stopping when no new simplices are found.
• The method uses a recursive verification procedure (Algorithm 2) to test whether a set of cells has a common intersection point.
• The algorithm is designed to build the minimal complex satisfying topological requirements, avoiding redundant simplices.

Experimental results

Research questions

• RQ1 How can the generalized Čech complex be constructed efficiently for wireless networks with cells of varying radii?
• RQ2 What is the computational complexity of constructing the minimal generalized Čech complex in such networks?
• RQ3 Can the algorithm detect coverage holes that are missed by the Rips complex due to its reliance on pairwise neighbor relationships?
• RQ4 To what extent does the algorithm scale with network density and cell size variation?

Key findings

• The proposed algorithm constructs the minimal generalized Čech complex in polynomial time, with a complexity of O(N² + Nn²) up to dimension 2, where N is the number of cells and n is the average number of neighbors.
• For networks with high cell density, the algorithm's execution time increases significantly, reaching 10.36 ms at density 1.5 and 30.97 ms at density 2 when building up to dimension 2.
• When constructing the complex up to dimension 10, execution time grows sharply, reaching 10,208.10 ms at density 2, indicating scalability challenges for high-dimensional complexes.
• The algorithm successfully identifies coverage holes, as demonstrated by the visualization in Figure 8, where a white space surrounded by colored simplices indicates a topological hole.
• The method outperforms the Rips complex in detecting coverage holes, as the Rips complex fails to represent holes due to its pairwise neighbor-based construction.
• The algorithm is extendable to 3D space, though it is currently designed for 2D deployments.