A robust model for finding optimal evolutionary trees
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
On the approximability of numerical taxonomy (fitting distances by tree metrics)
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
Approximating the bandwidth via volume respecting embeddings
Journal of Computer and System Sciences - 30th annual ACM symposium on theory of computing
Approximation algorithm for embedding metrics into a two-dimensional space
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
The Complexity of the Approximation of the Bandwidth Problem
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Approximating Minimum Max-Stretch spanning Trees on unweighted graphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Low-dimensional embedding with extra information
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Low distortion maps between point sets
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
The complexity of low-distortion embeddings between point sets
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for low-distortion embeddings into low-dimensional spaces
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Embedding ultrametrics into low-dimensional spaces
Proceedings of the twenty-second annual symposium on Computational geometry
Plane embeddings of planar graph metrics
Proceedings of the twenty-second annual symposium on Computational geometry
Approximation algorithms for embedding general metrics into trees
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Ultra-low-dimensional embeddings for doubling metrics
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Circular partitions with applications to visualization and embeddings
Proceedings of the twenty-fourth annual symposium on Computational geometry
Improved algorithms for optimal embeddings
ACM Transactions on Algorithms (TALG)
Distortion lower bounds for line embeddings
Information Processing Letters
Minimum Distortion Embeddings into a Path of Bipartite Permutation and Threshold Graphs
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
On the internet delay space dimensionality
Proceedings of the 8th ACM SIGCOMM conference on Internet measurement
Hardness of Embedding Metric Spaces of Equal Size
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
Information Processing Letters
Distortion Is Fixed Parameter Tractable
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Ultra-low-dimensional embeddings for doubling metrics
Journal of the ACM (JACM)
Inapproximability for planar embedding problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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
Theoretical Computer Science
Low Distortion Maps Between Point Sets
SIAM Journal on Computing
An exact algorithm for minimum distortion embedding
Theoretical Computer Science
Slightly superexponential parameterized problems
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Optimal distortion embedding of complete binary trees into lines
Information Processing Letters
Distortion is Fixed Parameter Tractable
ACM Transactions on Computation Theory (TOCT)
| Hi-index | 0.00 |
A low-distortion embedding between two metric spaces is a mapping which preserves the distances between each pair of points, up to a small factor called distortion. Low-distortion embeddings have recently found numerous applications in computer science.Most of the known embedding results are "absolute",that is, of the form: any metric Y from a given class of metrics C can be embedded into a metric X with low distortion c. This is beneficial if one can guarantee low distortion for all metrics Y in C. However, in any situations, the worst-case distortion is too large to be meaningful. For example, if X is a line metric, then even very simple metrics (an n - point star or an n -point cycle) are embeddable into X only with distortion linear in n. Nevertheless, embeddings into the line (or into low-dimensional spaces) are important for many applications.A solution to this issue is to consider "relative" (or "approximation") embedding problems, where the goal is to design an (a-approxiation) algorithm which, given any metric X from C as an input, finds an embedding of X into Y which has distortion a *cY (X), where cY (X)is the best possible distortion of an embedding of X into Y.In this paper we show algorithms and hardness results for relative embedding problems.In particular we give: •an algorith that, given a general metric M, finds an embedding with distortion O (Δ3⁄4 poly(c line (M))), where Δ is the spread of M•an algorithm that,given a weighted tree etric M, finds an embedding with distortion poly(c line (M)) •a hardness result, showing that computing minimum line distortion is hard to approximate up to a factor polynomial in n,even for weighted tree metrics with spread Δ=n O (1).