Abstract
We provide the first sparse covers and probabilistic partitions for graphs excluding a fixed minor that have strong diameter bounds; i.e. each set of the cover/partition has a small diameter as an induced sub-graph. Using these results we provide improved distributed name-independent routing schemes. Specifically, given a graph excluding a minor on r vertices and a parameter ρ>0 we obtain the following results: (1) a polynomial algorithm that constructs a set of clusters such that each cluster has a strong-diameter of O(r 2 ρ) and each vertex belongs to 2O(r) r! clusters; (2) a name-independent routing scheme with a stretch of O(r 2), headers of O(log n+rlog r) bits, and tables of size 2O(r) r! log 4 n/log log n bits; (3) a randomized algorithm that partitions the graph such that each cluster has strong-diameter O(r6r ρ) and the probability an edge (u,v) is cut is O(r d(u,v)/ρ).
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C. Gavoille is supported by the projects ‘GeoComp” and “Alpage”of the ACI Masses de Données.
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Abraham, I., Gavoille, C., Malkhi, D. et al. Strong-Diameter Decompositions of Minor Free Graphs. Theory Comput Syst 47, 837–855 (2010). https://doi.org/10.1007/s00224-010-9283-6
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DOI: https://doi.org/10.1007/s00224-010-9283-6