@inproceedings{6318becfe5e74b9db09801ee89de938b,

title = "A doubling dimension threshold ⊖ (log log n) for augmented graph navigability",

abstract = "In his seminal work, Kleinberg showed how to augment meshes using random edges, so that they become navigable; that is, greedy routing computes paths of polylogarithmic expected length between any pairs of nodes. This yields the crucial question of determining wether such an augmentation is possible for all graphs. In this paper, we answer negatively to this question by exhibiting a threshold on the doubling dimension, above which an infinite family of graphs cannot be augmented to become navigable whatever the distribution of random edges is. Precisely, it was known that graphs of doubling dimension at most O(log log n) are navigable. We show that for doubling dimension ≫ log log n, an infinite family of graphs cannot be augmented to become navigable. Finally, we complete our result by studying the special case of square meshes, that we prove to always be augmentable to become navigable.",

keywords = "Doubling dimension, Greedy routing, Small world",

author = "Pierre Fraigniaud and Emmanuelle Lebhar and Zvi Lotker",

year = "2006",

month = jan,

day = "1",

doi = "10.1007/11841036_35",

language = "English",

isbn = "3540388753",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "376--386",

booktitle = "Algorithms, ESA 2006 - 14th Annual European Symposium, Proceedings",

address = "Germany",

note = "14th Annual European Symposium on Algorithms, ESA 2006 ; Conference date: 11-09-2006 Through 13-09-2006",

}