[Paper Review] Connectivity of Graphs Induced by Directional Antennas

Mirela Damian, Robin Flatland|arXiv (Cornell University)|Aug 23, 2010

Structural Analysis and Optimization1 references3 citations

TL;DR

This paper studies the problem of orienting directional antennas with 90° or 180° beamwidths at planar points to achieve strong connectivity in the resulting communication graph, under the assumption that the unit disk graph is connected. It establishes tight bounds: for 90° antennas, a radius of at most 7 suffices and at least 2 is necessary; for 180° antennas, a radius of at most 1+√3 suffices and at least √3 is necessary.

ABSTRACT

This paper addresses the problem of finding an orientation and a minimum radius for directional antennas of a fixed angle placed at the points of a planar set S, that induce a strongly connected communication graph. We consider problem instances in which antenna angles are fixed at 90 and 180 degrees, and establish upper and lower bounds for the minimum radius necessary to guarantee strong connectivity. In the case of 90-degree angles, we establish a lower bound of 2 and an upper bound of 7. In the case of 180-degree angles, we establish a lower bound of sqrt(3) and an upper bound of 1+sqrt(3). Underlying our results is the assumption that the unit disk graph for S is connected.

Motivation & Objective

• To determine the minimum radius required for directional antennas with fixed 90° or 180° beamwidths to induce a strongly connected communication graph.
• To analyze the trade-off between angular aperture and transmission radius under the constraint that the underlying unit disk graph is connected.
• To provide provable upper and lower bounds on the minimum radius for strong connectivity in both 90° and 180° antenna configurations.
• To develop a constructive method based on minimum spanning trees to orient antennas and ensure bidirectional and strong connectivity.
• To establish tight theoretical bounds that are independent of specific point distributions, relying only on the connectivity of the unit disk graph.

Proposed method

• Construct a minimum spanning tree (MST) of the point set S, leveraging its structural properties to guide antenna placement and orientation.
• Partition the MST into node groups of size at most five, with representative nodes selected to ensure coverage and connectivity within each group.
• Use geometric analysis of wedge coverage to orient antennas such that each node can reach its neighbors and representative nodes in its group.
• Apply inductive reasoning on the number of groups in the MST, proving that a radius of r = 7 suffices for 90° antennas and r = 1 + √3 for 180° antennas.
• Leverage the fact that antennas with radius r = d_max + 1 can cover the entire plane when placed at representative nodes with maximum pairwise distance d_max.
• Ensure that the root of the MST can reach any node within unit distance by using intermediate nodes in the communication path, especially when direct coverage is not possible.

Experimental results

Research questions

• RQ1 What is the minimum radius required to achieve strong connectivity in a wireless network using directional antennas with a fixed 90° beamwidth?
• RQ2 What is the minimum radius required when using 180° directional antennas, assuming the unit disk graph of the point set is connected?
• RQ3 Can tight theoretical bounds be established for the minimum radius in both 90° and 180° antenna configurations?
• RQ4 How can antenna orientations be assigned to ensure strong connectivity while minimizing the required transmission radius?
• RQ5 Does the structure of the minimum spanning tree enable a constructive and bounded-radius solution for strong connectivity?

Key findings

• For 90° directional antennas, a radius of at most 7 is sufficient to achieve strong connectivity in any planar point set whose unit disk graph is connected.
• For 90° antennas, a radius of at least 2 is necessary, as demonstrated by a construction where smaller radii fail to connect all components.
• For 180° directional antennas, a radius of at most 1 + √3 ≈ 2.732 is sufficient to achieve strong connectivity under the same connectivity assumption.
• For 180° antennas, a radius of at least √3 ≈ 1.732 is necessary, as shown by a geometric construction where smaller radii fail to enable inter-component communication.
• The proposed method ensures that the root of the MST can reach any node within unit distance, even when direct antenna coverage is not available.
• The inductive construction using representative nodes and group partitioning ensures that communication paths exist between all nodes with bounded radius, achieving strong connectivity.