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Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2020.33
URN: urn:nbn:de:0030-drops-124405
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12440/
Chuzhoy, Julia ;
Parter, Merav ;
Tan, Zihan
On Packing Low-Diameter Spanning Trees
Abstract
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 ? of ⌊k/2⌋ edge-disjoint spanning trees, that we refer to as a tree packing; the diameter of the tree packing ? is the largest diameter of any tree in ?. 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 ? 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.
BibTeX - Entry
@InProceedings{chuzhoy_et_al:LIPIcs:2020:12440,
author = {Julia Chuzhoy and Merav Parter and Zihan Tan},
title = {{On Packing Low-Diameter Spanning Trees}},
booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
pages = {33:1--33:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-138-2},
ISSN = {1868-8969},
year = {2020},
volume = {168},
editor = {Artur Czumaj and Anuj Dawar and Emanuela Merelli},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12440},
URN = {urn:nbn:de:0030-drops-124405},
doi = {10.4230/LIPIcs.ICALP.2020.33},
annote = {Keywords: Spanning tree, packing, edge-connectivity}
}
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Keywords: |
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Spanning tree, packing, edge-connectivity |
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Collection: |
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47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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29.06.2020 |
