On triangulation of simple networks
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
On compact routing for the internet
ACM SIGCOMM Computer Communication Review
Reconstructing approximate tree metrics
Proceedings of the twenty-sixth annual ACM symposium on Principles of distributed computing
Squarepants in a tree: sum of subtree clustering and hyperbolic pants decomposition
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Metric clustering via consistent labeling
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs
Proceedings of the twenty-fourth annual symposium on Computational geometry
Space-Time Tradeoffs for Proximity Searching in Doubling Spaces
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Packing and Covering δ-Hyperbolic Spaces by Balls
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Squarepants in a tree: Sum of subtree clustering and hyperbolic pants decomposition
ACM Transactions on Algorithms (TALG)
Greedy forwarding in dynamic scale-free networks embedded in hyperbolic metric spaces
INFOCOM'10 Proceedings of the 29th conference on Information communications
Constant approximation algorithms for embedding graph metrics into trees and outerplanar graphs
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Horoball hulls and extents in positive definite space
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
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We initiate the study of approximate algorithms on negatively curved spaces. These spaces have recently become of interest in various domains of computer science including networking and vision. The classical example of such a space is the real-hyperbolic space \mathbb{H}^d for d \geqslant 2, but our approach applies to a more general family of spaces characterized by Gromov's (combinatorial) hyperbolic condition. We give efficient algorithms and data structures for problems like approximate nearest-neighbor search and compact, low-stretch routing on subsets of negatively curved spaces of fixed dimension (including \mathbb{H}^d as a special case). In a different direction, we show that there is a PTAS for the Traveling Salesman Problem when the set of cities lie, for example, in \mathbb{H}^d. This generalizes Arora's results for \mathbb{R}^d. Most of our algorithms use the intrinsic distance geometry of the data set, and only need the existence of an embedding into some negatively curved space in order to function properly. In other words, our algorithms regard the interpoint distance function as a black box, and are independent of the representation of the input points.