Excluded minors, network decomposition, and multicommodity flow
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Approximating the bandwidth via volume respecting embeddings
Journal of Computer and System Sciences - 30th annual ACM symposium on theory of computing
Robust algorithms for restricted domains
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
The intrinsic dimensionality of graphs
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Random Projection: A New Approach to VLSI Layout
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Virtual coordinates for ad hoc and sensor networks
Proceedings of the 2004 joint workshop on Foundations of mobile computing
Localization and routing in sensor networks by local angle information
Proceedings of the 6th ACM international symposium on Mobile ad hoc networking and computing
Distributed localization using noisy distance and angle information
Proceedings of the 7th ACM international symposium on Mobile ad hoc networking and computing
ICDCN '09 Proceedings of the 10th International Conference on Distributed Computing and Networking
Large-Scale Networked Systems: From Anarchy to Geometric Self-structuring
ICDCN '09 Proceedings of the 10th International Conference on Distributed Computing and Networking
Sensor networks continue to puzzle: selected open problems
ICDCN'08 Proceedings of the 9th international conference on Distributed computing and networking
A weakly robust PTAS for minimum clique partition in unit disk graphs
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Shifting strategy for geometric graphs without geometry
Journal of Combinatorial Optimization
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The quality of an embedding Φ : V → R2 of a graph G = (V,E) into the Euclidean plane is the ratio of max{u,v}∈E ∥Φ(u)-Φ(v)∥2 to min{u,v}∉E ∥Φ(u)-Φ(v)∥2. Given a unit disk graph G = (V,E), we seek algorithms to compute an embedding Φ : V → R2 of best (smallest) quality. This paper presents a simple, combinatorial algorithm for computing a O(log2.5 n)-quality 2-dimensional embedding of a given unit disk graph. Note that G comes with no associated geometric information. If the embedding is allowed to reside in higher dimensional space, we obtain improved results: a quality-2 embedding in RO(1). Our results extend to unit ball graphs (UBGs) in fixed dimensional Euclidean space. Constructing a "growth-restricted approximation" of the given unit disk graph lies at the core of our algorithm. This approach allows us to bypass the standard and costly technique of solving a linear program with exponentially many "spreading constraints". As a side effect of our construction, we get a constant-factor approximation to the minimum clique cover problem on UBGs, described without geometry. Our problem is a version of the well known localization problem in wireless networks.