2026-07-26T20:56:37Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:12440 2024-03-06T10:50:08Z ddc:004 open_access

On Packing Low-Diameter Spanning Trees Chuzhoy, Julia Parter, Merav Tan, Zihan Spanning tree packing edge-connectivity Edge connectivity of a graph is one of the most fundamental graph-theoretic concepts. The celebrated tree packing theorem of Tutte and Nash-Williams from 1961 states that every k-edge connected graph G contains a collection 𝒯 of ⌊k/2⌋ edge-disjoint spanning trees, that we refer to as a tree packing; the diameter of the tree packing 𝒯 is the largest diameter of any tree in 𝒯. A desirable property of a tree packing for leveraging the high connectivity of a graph in distributed communication networks, is that its diameter is low. Yet, despite extensive research in this area, it is still unclear how to compute a tree packing of a low-diameter graph G, whose diameter is sublinear in |V(G)|, or, alternatively, how to show that such a packing does not exist. In this paper, we provide first non-trivial upper and lower bounds on the diameter of tree packing. We start by showing that, for every k-edge connected n-vertex graph G of diameter D, there is a tree packing 𝒯 containing Ω(k) trees, of diameter O((101k log n)^D), with edge-congestion at most 2. Karger’s edge sampling technique demonstrates that, if G is a k-edge connected graph, and G[p] is a subgraph of G obtained by sampling each edge of G independently with probability p = Θ(log n/k), then with high probability G[p] is connected. We extend this result to show that the diameter of G[p] is bounded by O(k^(D(D+1)/2)) with high probability. This immediately gives a tree packing of Ω(k/log n) edge-disjoint trees of diameter at most O(k^(D(D+1)/2)). We also show that these two results are nearly tight for graphs with a small diameter: we show that there are k-edge connected graphs of diameter 2D, such that any packing of k/α trees with edge-congestion η contains at least one tree of diameter Ω((k/(2α η D))^D), for any k,α and η. Additionally, we show that if, for every pair u,v of vertices of a given graph G, there is a collection of k edge-disjoint paths connecting u to v, of length at most D each, then we can efficiently compute a tree packing of size k, diameter O(D log n), and edge-congestion O(log n). Finally, we provide several applications of low-diameter tree packing in the distributed settings of network optimization and secure computation. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Julia Chuzhoy and Merav Parter and Zihan Tan 2020 Is Part Of LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2020.33 urn:nbn:de:0030-drops-124405 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.33 eng https://creativecommons.org/licenses/by/3.0/legalcode