Excluded minors, network decomposition, and multicommodity flow
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Minimum 0-extensions of graph metrics
European Journal of Combinatorics
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Approximation algorithms for the 0-extension problem
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
An improved approximation algorithm for the 0-extension problem
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
The intrinsic dimensionality of graphs
Proceedings of the thirty-fifth annual ACM symposium on Theory of 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
Cuts, Trees and ℓ1-Embeddings of Graphs*
Combinatorica
Bypassing the embedding: algorithms for low dimensional metrics
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of 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
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Searching dynamic point sets in spaces with bounded doubling dimension
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Cover trees for nearest neighbor
ICML '06 Proceedings of the 23rd international conference on Machine learning
Optimal-stretch name-independent compact routing in doubling metrics
Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing
Nearest-neighbor-preserving embeddings
ACM Transactions on Algorithms (TALG)
Optimal scale-free compact routing schemes in networks of low doubling dimension
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
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In recent years, considerable advances have been made in the study of properties of metric spaces in terms of their doubling dimension. This line of research has not only enhanced our understanding of finite metrics, but has also resulted in many algorithmic applications. However, we still do not understand the interaction between various graphtheoretic (topological) properties of graphs, and the doubling (geometric) properties of the shortest-path metrics induced by them. For instance, the following natural question suggests itself: given a finite doubling metric (V, d), is there always an unweighted graph (V′,E′) with V ⊆ V′ such that the shortest path metric d′ on V′ is still doubling, and which agrees with d on V. This is often useful, given that unweighted graphs are often easier to reason about. A first hurdle to answering this question is that subdividing edges can increase the doubling dimension unboundedly, and it is not difficult to show that the answer to the above question is negative. However, surprisingly, allowing a (1 + Ɛ) distortion between d and d′ enables us bypass this impossibility: we show that for any metric space (V, d), there is an unweighted graph (V′, E′) with shortest-path metric d′ : V′ × V′ → R≥0 such that - for all x, y ∈ V , the distances d(x, y) ≤ d′(x, y) ≤ (1 + Ɛ) ċ d(x, y), and - the doubling dimension for d′ is not much more than that of d, where this change depends only on Ɛ and not on the size of the graph. We show a similar result when both (V, d) and (V′, E′) are restricted to be trees: this gives a simple proof that doubling trees embed into constant dimensional Euclidean space with constant distortion. We also show that our results are tight in terms of the tradeoff between distortion and dimension blowup.