[Paper Review] Almost All Even Yao-Yao Graphs Are Spanners

Jian Li, Wei Zhan|arXiv (Cornell University)|Apr 20, 2016

Advanced Graph Theory Research24 references3 citations

TL;DR

This paper proves that for any integer $k \geq 42$, the Yao-Yao graph $\mathsf{YY}_{2k}$ is a geometric spanner with stretch factor $t_k = 6.03 + O(k^{-1})$, resolving a long-standing conjecture for even $k$ by introducing intermediate graphs—overlapping Yao graphs and trapezoidal Yao graphs—and showing $\mathsf{YY}_{2k}$ spans the trapezoidal graph, which inherits its spanner property.

ABSTRACT

It is an open problem whether Yao-Yao graphs $\mathsf{YY}_k$ (also known as sparse-Yao graphs) are all spanners when the integer parameter $k$ is large enough. In this paper we show that, for any integer $k\geq 42$, the Yao-Yao graph $\mathsf{YY}_{2k}$ is a $t_k$-spanner, with stretch factor $t_k=6.03+O(k^{-1})$ when $k$ tends to infinity. Our result generalizes the best known result which asserts that all $\mathsf{YY}_{6k}$ are spanners for $k$ large enough [Bauer and Damian, SODA'13]. Our proof is also somewhat simpler.

Motivation & Objective

• To resolve the long-standing conjecture that Yao-Yao graphs $\mathsf{YY}_k$ are geometric spanners for sufficiently large $k$.
• To establish that even-parameter Yao-Yao graphs $\mathsf{YY}_{2k}$ are spanners when $k \geq 42$, extending prior results limited to $\mathsf{YY}_{6k}$.
• To simplify and generalize the proof techniques used in earlier works on Yao-Yao spanner properties.
• To provide a tighter asymptotic stretch factor bound for $\mathsf{YY}_{2k}$, improving upon previous results.

Proposed method

• Introduce overlapping Yao graphs ($\mathsf{OY}_k$) as intermediate graphs with overlapping cones, proving they are geometric spanners.
• Define trapezoidal Yao graphs ($\mathsf{TY}_k$) using curved trapezoids to model path segments, showing $\mathsf{OY}_k \subseteq \mathsf{TY}_k$ implies $\mathsf{TY}_k$ is a spanner.
• Establish that $\mathsf{YY}_{2k}$ spans $\mathsf{TY}_{2k}$ by demonstrating that every edge in $\mathsf{TY}_{2k}$ is either in $\mathsf{YY}_{2k}$ or can be replaced by a bounded-length path in $\mathsf{YY}_{2k}$.
• Use induction on path length and geometric inequalities involving angles and distances to bound the stretch factor.
• Derive the stretch factor $t_k = \tau_k' \cdot \tau_{2k}$, where $\tau_k = (1 - 2\sin(\pi/8))^{-1} + O(k^{-1})$ and $\tau_k' = \sqrt{2} + O(k^{-1})$, yielding $t_k = 6.03 + O(k^{-1})$.
• Leverage centrosymmetric cone division and bounded cone angles to ensure path convergence and stretch control.

Experimental results

Research questions

• RQ1 Are Yao-Yao graphs $\mathsf{YY}_k$ geometric spanners for sufficiently large even $k$?
• RQ2 Can the stretch factor of $\mathsf{YY}_{2k}$ be bounded by a constant that approaches 6.03 as $k \to \infty$?
• RQ3 Is it possible to prove that $\mathsf{YY}_{2k}$ is a spanner using a simpler proof than prior works on $\mathsf{YY}_{6k}$?
• RQ4 Can intermediate graphs like $\mathsf{OY}_k$ and $\mathsf{TY}_k$ be used to bridge the gap between Yao graphs and Yao-Yao graphs in spanner analysis?
• RQ5 What is the tightest possible asymptotic stretch factor for $\mathsf{YY}_{2k}$ as $k$ increases?

Key findings

• For all integers $k \geq 42$, the Yao-Yao graph $\mathsf{YY}_{2k}$ is a geometric spanner with stretch factor $t_k = 6.03 + O(k^{-1})$.
• The proof introduces overlapping Yao graphs ($\mathsf{OY}_k$) and trapezoidal Yao graphs ($\mathsf{TY}_k$), showing $\mathsf{OY}_k \subseteq \mathsf{TY}_k$ and that $\mathsf{TY}_k$ is a spanner.
• It is proven that $\mathsf{YY}_{2k}$ spans $\mathsf{TY}_{2k}$, which implies $\mathsf{YY}_{2k}$ inherits the spanner property.
• The stretch factor $t_k$ is derived as the product $\tau_k' \cdot \tau_{2k}$, where $\tau_k' = \sqrt{2} + O(k^{-1})$ and $\tau_{2k} = (1 - 2\sin(\pi/8))^{-1} + O(k^{-1})$.
• The result improves upon prior work showing $\mathsf{YY}_{6k}$ is a spanner with stretch factor 11.76 for $k \geq 6$, and provides a tighter asymptotic bound.
• The proof technique is simpler than previous approaches and generalizes the spanner result to almost all even $k$.