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

2025-10-10

AI Summary

This work addresses the absence of tight approximation ratios for tree embeddings and L1-oblivious routing—particularly the lack of theoretical guarantees in parallel and distributed settings. We introduce a novel analytical framework based on multi-scale random shift decomposition. By employing exponentially growing scale partitions, refined separation probability analysis, and a constructive tree embedding technique, we achieve, for the first time, an $O(log n)$ expected stretch and competitive ratio—matching the known theoretical lower bound and closing the gap left by single-scale analyses. The resulting algorithm is both simple and scalable, attaining near-optimal work, depth, and communication rounds across sequential, parallel, and distributed models. Beyond providing a new proof pathway for classical results, this work establishes the first tight bounds for oblivious routing in parallel and distributed environments.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Develops tight approximations for tree embeddings in graphs
• Analyzes random-shift decompositions with constant trade-off bounds
• Provides efficient algorithms for oblivious routing problems

Innovation

• Random-shift decompositions with tight constant trade-off
• Tree embeddings achieving O(log n) stretch
• Fast algorithms for sequential parallel distributed settings