Reconstructing a three-dimensional model with arbitrary errors
Journal of the ACM (JACM)
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
The Cricket location-support system
MobiCom '00 Proceedings of the 6th annual international conference on Mobile computing and networking
Graph Drawing: Algorithms for the Visualization of Graphs
Graph Drawing: Algorithms for the Visualization of Graphs
Lectures on Discrete Geometry
Approximation algorithm for embedding metrics into a two-dimensional space
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
GPS-Free Positioning in Mobile ad-hoc Networks
HICSS '01 Proceedings of the 34th Annual Hawaii International Conference on System Sciences ( HICSS-34)-Volume 9 - Volume 9
On the Impossibility of Dimension Reduction in \ell _1
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Low-dimensional embedding with extra information
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Cuts, Trees and ℓ1-Embeddings of Graphs*
Combinatorica
Low-distortion embeddings of general metrics into the line
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Approximation algorithms for low-distortion embeddings into low-dimensional spaces
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
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Embedding metrics into constant-dimensional geometric spaces, such as the Euclidean plane, is relatively poorly understood. Motivated by applications in visualization, ad-hoc networks, and molecular reconstruction, we consider the natural problem of embedding shortest-path metrics of unweighted planar graphs (planar graph metrics) into the Euclidean plane. It is known that, in the special case of shortest-path metrics of trees, embedding into the plane requires Θ(√n) distortion in the worst case [19, 1], and surprisingly, this worst-case upper bound provides the best known approximation algorithm for minimizing distortion. We answer an open question posed in this work and highlighted by Matoušek [21] by proving that some planar graph metrics require Ω(n2/3) distortion in any embedding into the plane, proving the first separation between these two types of graph metrics. We also prove that some planar graph metrics require Ω(n) distortion in any crossing-free straight-line embedding into the plane, suggesting a separation between low-distortion plane embedding and the well-studied notion of crossing-free straight-line planar drawings. Finally, on the upper-bound side, we prove that all outerplanar graph metrics can be embedded into the plane with O(√n) distortion, generalizing the previous results on trees (both the worst-case bound and the approximation algorithm) and building techniques for handling cycles in plane embeddings of graph metrics.