Approximating Euclidean Shallow-Light Trees

TL;DR

This work studies shallow-light trees (SLTs) in constant-dimensional Euclidean spaces, achieving improved bicriteria approximations for both Steiner and non-Steiner variants. The authors develop a novel tile- and net-based reduction framework that reduces global SLT construction to localized problems within trapezoidal tiles, leveraging ellipses and hitting-set ideas to bound root-stretch and total weight. They obtain two main results: a non-Steiner SLT with root-stretch 1+O(ε log ε^{-1}) and weight O(opt_ε log^2 ε^{-1}), and a Steiner SLT with the same stretch and weight O(opt_ε log ε^{-1}), both in near-linear time up to polylog factors, extendable to higher dimensions. The paper also analyzes limitations of prior existential bounds, provides lower-bound constructions, and discusses generalizations to d-space, highlighting the practical impact for Euclidean geometric networks, VLSI, and related optimization problems.

Abstract

For a weighted graph $G = (V, E, w)$ and a designated source vertex $s \in V$, a spanning tree that simultaneously approximates a shortest-path tree w.r.t. source $s$ and a minimum spanning tree is called a shallow-light tree (SLT). Specifically, an $(α, β)$-SLT of $G$ w.r.t. $s \in V$ is a spanning tree of $G$ with root-stretch $α$ (preserving all distances between $s$ and the other vertices up to a factor of $α$) and lightness $β$ (its weight is at most $β$ times the weight of a minimum spanning tree of $G$). Despite the large body of work on SLTs, the basic question of whether a better approximation algorithm exists was left untouched to date, and this holds in any graph family. This paper makes a first nontrivial step towards this question by presenting two bicriteria approximation algorithms. For any $ε>0$, a set $P$ of $n$ points in constant-dimensional Euclidean space and a source $s\in P$, our first (respectively, second) algorithm returns, in $O(n \log n \cdot {\rm polylog}(1/ε))$ time, a non-Steiner (resp., Steiner) tree with root-stretch $1+O(ε\log ε^{-1})$ and weight at most $O(\mathrm{opt}_ε\cdot \log^2 ε^{-1})$ (resp., $O(\mathrm{opt}_ε\cdot \log ε^{-1})$), where $\mathrm{opt}_ε$ denotes the minimum weight of a non-Steiner (resp., Steiner) tree with root-stretch $1+ε$.

Figures (13)

• Figure 1: Points in a trapezoid tile $\tau$, and Steiner points $s_1,\ldots ,s_4\in \mathcal{E}_{ps}$ on parallel lines.
• Figure 2: The ellipse $\mathcal{E}_{ps}$ with foci $p$ and $s$ and major axis of length $(1+\epsilon)\cdot d(p,s)$.
• Figure 3: A part of the path $\pi_{ps}$ in the halfplane $L^-$.
• Figure 4: A section of the trapezoid tiling of the plane.
• Figure 5: Left: region $R(\tau)$ is the union of ellipses $\mathcal{E}_{ps}$ for all $p\in \tau$ (including $p\in \{a,b,c\}$). Right: Region $R^-(\tau)$ and the line $L_\tau$.

Theorems & Definitions (62)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• proof
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof