STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Theoretical Computer Science
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Cuts, Trees and ℓ1-Embeddings of Graphs*
Combinatorica
Low distortion maps between point sets
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
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
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Approximation algorithms for embedding general metrics into trees
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
Theoretical Computer Science
An exact algorithm for minimum distortion embedding
Theoretical Computer Science
Bandwidth and distortion revisited
Discrete Applied Mathematics
Slightly superexponential parameterized problems
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
What's next? future directions in parameterized complexity
The Multivariate Algorithmic Revolution and Beyond
Distortion is Fixed Parameter Tractable
ACM Transactions on Computation Theory (TOCT)
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We study low-distortion embedding of metric spaces into the line, and more generally, into the shortest path metric of trees, from the parameterized complexity perspective. Let M = M (G ) be the shortest path metric of an edge weighted graph G , with the vertex set V (G ) and the edge set E (G ), on n vertices. We give the first fixed parameter tractable algorithm that for an unweighted graph metric M and integer d either constructs an embedding of M into the line with distortion at most d , or concludes that no such embedding exists. Our algorithm requires O (nd 4(2d + 1)2d ) time which is a significant improvement over the best previous algorithm of Bădoiu et al. that runs in time O (n 4d + 2 d O (1)). We find it surprising that this problem turns out to be fixed parameter tractable, because of its apparent similarity to the notoriously hard Bandwidth Minimization problem. We extend our results on embedding unweighted graph metric into the line in two ways. First, we give an algorithm to construct small distortion embeddings of weighted graph metrics. The running time of our algorithm is O (n (dW )4(2d + 1)2dW ) where W is the largest edge weight of the input graph. To complement this result, we show that the exponential dependence on the maximum edge weight is unavoidable. In particular, we show that deciding whether a weighted graph metric M (G ) with maximum weight W V (G )| can be embedded into the line with distortion at most d is NP-Complete for every fixed rational d *** 2. This rules out any possibility of an algorithm with running time O ((nW ) h (d )) where h is a function of d alone. Secondly, we consider more general host metrics for which analogous results hold. In particular, we prove that for any tree T with maximum degree Δ , embedding M into a shortest path metric of T is fixed parameter tractable, parameterized by (Δ ,d ).