Random-Shift Revisited: Tight Approximations for Tree Embeddings and L1-Oblivious Routings

TL;DR

This work revisits random-shift decompositions to obtain tight, scalable guarantees for probabilistic tree embeddings and $\ell_1$-oblivious routings. By analyzing both exact and approximate SSSP implementations, the authors show that hierarchical random-shift decompositions achieve an average $O(\log n)$ trade-off between separation probability and distance-to-centers, yielding $O(\log n)$-approximate tree embeddings that match the classical $\Omega(\log n)$ lower bound. They further adapt these insights to provide a tight $O(\log n)$-competitive $\ell_1$-oblivious routing, with efficient sequential, parallel, distributed, and streaming algorithms. The results unify and simplify the construction of FRT-like trees while delivering optimal or near-optimal performance guarantees in various computational models. Together, these contributions offer practical, scalable tools for distance-preserving representations and routing in large graphs.

Abstract

We present a new and surprisingly simple analysis of random-shift decompositions -- originally proposed by Miller, Peng, and Xu [SPAA'13]: We show that decompositions for exponentially growing scales $D = 2^0, 2^1, \ldots, 2^{\log_2(\operatorname{diam}(G))}$, have a tight constant trade-off between distance-to-center and separation probability on average across the distance scales -- opposed to a necessary $Ω(\log n)$ trade-off for a single scale. This almost immediately yields a way to compute a tree $T$ for graph $G$ that preserves all graph distances with expected $O(\log n)$-stretch. This gives an alternative proof that obtains tight approximation bounds of the seminal result by Fakcharoenphol, Rao, and Talwar [STOC'03] matching the $Ω(\log n)$ lower bound by Bartal [FOCS'96]. Our insights can also be used to refine the analysis of a simple $\ell_1$-oblivious routing proposed in [FOCS'22], yielding a tight $O(\log n)$ competitive ratio. Our algorithms for constructing tree embeddings and $\ell_1$-oblivious routings can be implemented in the sequential, parallel, and distributed settings with optimal work, depth, and rounds, up to polylogarithmic factors. Previously, fast algorithms with tight guarantees were not known for tree embeddings in parallel and distributed settings, and for $\ell_1$-oblivious routings, not even a fast sequential algorithm was known.

Theorems & Definitions (10)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3: Main Result for Probabilistic Tree Embeddings
• Theorem 1.4: Main Result for $\ell_1$-Oblivious Routing
• Lemma 3.1: Distances-to-Center
• Lemma 3.2: Sum-of-Deviations
• Lemma 3.3: Lemma 4.4 of randomshift2013
• proof : Sketch of proof
• Lemma 4.1
• proof