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
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
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
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
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
Routing in Networks with Low Doubling Dimension
ICDCS '06 Proceedings of the 26th IEEE International Conference on Distributed Computing Systems
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
Universal augmentation schemes for network navigability: overcoming the √n-barrier
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Polylogarithmic network navigability using compact metrics with small stretch
Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures
Recovering the Long-Range Links in Augmented Graphs
SIROCCO '08 Proceedings of the 15th international colloquium on Structural Information and Communication Complexity
Graph Augmentation via Metric Embedding
OPODIS '08 Proceedings of the 12th International Conference on Principles of Distributed Systems
Combinatorial algorithms for nearest neighbors, near-duplicates and small-world design
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Universal augmentation schemes for network navigability
Theoretical Computer Science
Navigable Small-World networks with few random bits
Theoretical Computer Science
Combinatorial Framework for Similarity Search
SISAP '09 Proceedings of the 2009 Second International Workshop on Similarity Search and Applications
Recovering the long-range links in augmented graphs
Theoretical Computer Science
Small worlds as navigable augmented networks: model, analysis, and validation
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Depth of field and cautious-greedy routing in social networks
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Content search through comparisons
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
Brief announcement: on augmented graph navigability
DISC'06 Proceedings of the 20th 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 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 (loglogn) are navigable. We show that for doubling dimension ≫loglogn, 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.