2025/10/10 by Rasmus Kyng, Maximilian Probst Gutenberg, Kyng, Rasmus +3 · 1 citation
Computer Science · #Advanced Graph Theory Research #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2510.09124
openalex publication_date 2025/10/10 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
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 = 20, 21, …, 2log2(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 ℓ1-oblivious routing proposed in [FOCS'22], yielding a tight O(log n) competitive ratio. Our algorithms for constructing tree embeddings and ℓ1-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 ℓ1-oblivious routings, not even a fast sequential algorithm was known.