The small-world phenomenon: an algorithmic perspective
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Finding nearest neighbors in growth-restricted metrics
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Fault-tolerant routing in peer-to-peer systems
Proceedings of the twenty-first annual symposium on Principles of distributed computing
Efficient Routing in Networks with Long Range Contacts
DISC '01 Proceedings of the 15th International Conference on Distributed Computing
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Navigating nets: simple algorithms for proximity search
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Know thy neighbor's neighbor: the power of lookahead in randomized P2P networks
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Bypassing the embedding: algorithms for low dimensional metrics
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Eclecticism shrinks even small worlds
Proceedings of the twenty-third annual ACM symposium on Principles of distributed computing
Analyzing Kleinberg's (and other) small-world Models
Proceedings of the twenty-third annual ACM symposium on Principles of distributed computing
Measured Descent: A New Embedding Method for Finite Metrics
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Triangulation and Embedding Using Small Sets of Beacons
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Fast construction of nets in low dimensional metrics, and their applications
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
On hierarchical routing in doubling metrics
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Distance estimation and object location via rings of neighbors
Proceedings of the twenty-fourth annual ACM symposium on Principles of distributed computing
Name independent routing for growth bounded networks
Proceedings of the seventeenth annual ACM symposium on Parallelism in algorithms and architectures
Metric Embeddings with Relaxed Guarantees
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Close to optimal decentralized routing in long-range contact networks
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
Could any graph be turned into a small-world?
Theoretical Computer Science - Complex networks
Object location using path separators
Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing
Routing in Networks with Low Doubling Dimension
ICDCS '06 Proceedings of the 26th IEEE International Conference on Distributed Computing Systems
Universal augmentation schemes for network navigability: overcoming the √n-barrier
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Greedy routing in tree-decomposed graphs
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Asymptotically optimal solutions for small world graphs
DISC'05 Proceedings of the 19th international conference on Distributed Computing
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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 whether such an augmentation is possible for all graphs. In this paper, we answer this question negatively by exhibiting an infinite family of graphs that 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 $\gg\log\log n$, an infinite family of graphs cannot be augmented to become navigable. Finally, we present a positive navigability result by studying the special case of square meshes of arbitrary dimension that we prove to always be augmentable to become navigable. This latter result complements Kleinberg's original result and shows that adding extra links can sometimes break the navigability.