Improved Bounds on the Stretch Factor of $Y_4$

TL;DR

This paper improves the upper bounds on the stretch factors of Yao graphs in both the $L_inefty$ and Euclidean metrics, demonstrating tighter guarantees on their path lengths relative to direct distances.

Contribution

The paper provides new, significantly lower upper bounds on the stretch factors of $Y_4^inefty$ and $Y_4$ Yao graphs, advancing understanding of their efficiency.

Findings

Upper bound of 4.94 for $Y_4^inefty$

Upper bound of 54.62 for $Y_4$ in Euclidean metric

Improved previous bounds of 6.31 and 662.16 respectively

Abstract

We establish an upper bound of 4.94 on the stretch factor of the Yao graph $Y_4^\infty$ defined in the $L_\infty$-metric, improving upon the best previously known upper bound of 6.31. We also establish an upper bound of 54.62 on the stretch factor of the Yao graph $Y_4$ defined in the Euclidean metric, improving upon the best previously known upper bound of 662.16.