[Paper Review] A Note on Interference in Random Point Sets

Luc Devroye, Pat Morin|arXiv (Cornell University)|Feb 27, 2012

Computational Geometry and Mesh Generation8 references3 citations

TL;DR

This paper analyzes interference in geometric graphs formed by n points uniformly and independently distributed in the unit d-cube. It establishes that the minimum spanning tree has interference Θ((log n)^{1/2}) with high probability, while a better-connected graph via bucketing achieves O((log n)^{1/3}) interference, and no graph can go below Ω((log n)^{1/4}). These results show interference grows slowly in random point sets, contrasting sharply with worst-case constructions where interference can be linear in n.

ABSTRACT

The (maximum receiver-centric) interference of a geometric graph (von Rickenbach etal (2005)) is studied. It is shown that, with high probability, the following results hold for a set, V, of n points independently and uniformly distributed in the unit d-cube, for constant dimension d: (1) there exists a connected graph with vertex set V that has interference O((log n)^{1/3}); (2) no connected graph with vertex set V has interference o((log n)^{1/4}); and (3) the minimum spanning tree of $V$ has interference Theta((\log n)^{1/2}).

Motivation & Objective

• To understand the behavior of maximum receiver-centric interference in random geometric networks under uniform i.i.d. point distributions.
• To determine tight bounds on the minimum possible interference I(V) for connected graphs on random point sets in [0,1]^d.
• To compare the interference of the minimum spanning tree (MST) with that of more carefully constructed graphs.
• To close the gap between existing upper and lower bounds on interference in random point sets.

Proposed method

• Uses probabilistic analysis and the second moment method to bound the expected number of interfering nodes around each point.
• Applies the concept of Zeno configurations to model dense local neighborhoods that contribute to interference.
• Employs a bucketing strategy to partition the unit cube and construct a graph with low interference by limiting long-range connections.
• Derives upper bounds using concentration inequalities and asymptotic analysis for i.i.d. uniform point processes in d-dimensional space.
• Establishes lower bounds via contradiction and extremal constructions, showing that interference cannot be asymptotically smaller than Ω((log n)^{1/4}).
• Uses the family of graphs T(V) from prior work to extend results to locally computable graph constructions.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the minimum interference I(V) for a connected geometric graph on n i.i.d. uniform points in [0,1]^d?
• RQ2 How does the interference of the minimum spanning tree compare to that of other graph constructions in random point sets?
• RQ3 Can interference be bounded below by a function strictly smaller than (log n)^{1/4} in random point sets?
• RQ4 Is there a structural relationship between the interference of the MST and the interference of other graphs, such as I(V) = O(√I(MST(V)))?

Key findings

• The minimum spanning tree of n i.i.d. uniform points in [0,1]^d has interference Θ((log n)^{1/2}) with high probability, for any constant d ≥ 1.
• A graph constructed via a bucketing strategy achieves interference O((log n)^{1/3}) with high probability for d ∈ {1,2}, and O((log n)^{1/3}(log log n)^{1/2}) for d ≥ 3.
• No connected graph on such a point set can have interference o((log n)^{1/4}) with high probability, establishing a lower bound.
• The interference of the MST is tightly concentrated around (log n)^{1/2}, and this value is asymptotically optimal up to logarithmic factors.
• The results extend to the unit disk graph model when the transmission range r(n) is Ω(√(log n / n)), a necessary condition for connectivity.
• The LocalRadiusReduction algorithm applied to unit disk graphs yields interference O((log n)^{1/2}) with high probability when r(n) ∈ O(2^{√(log n)} / √n).