Journal of Parallel and Distributed Computing
Embedding tree metrics into low dimensional Euclidean spaces
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Efficient algorithms for inverting evolution
Journal of the ACM (JACM)
Low-Distortion Embeddings of Trees
GD '01 Revised Papers from the 9th International Symposium on Graph Drawing
Probabilistic approximation of metric spaces and its algorithmic applications
FOCS '96 Proceedings of the 37th 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
Dimension reduction for ultrametrics
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
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
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
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
Ordinal embeddings of minimum relaxation: general properties, trees, and ultrametrics
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
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
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
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We study the problem of minimum-distortion embedding of ultrametrics into the plane and higher dimensional spaces. Ultrametrics are a natural class of metrics that frequently occur in applications involving hierarchical clustering. Low-distortion embeddings of ultrametrics into the plane help visualizing complex structures they often represent.Given an ultrametric, a natural question is whether we can efficiently find an optimal-distortion embedding of this ultrametric into the plane, and if not, whether we can design an efficient algorithm that produces embeddings with near-optimal distortion. We show that the problem of finding minimum-distortion embedding of ultrametrics into the plane is NP-hard, and thus approximation algorithms are called for. Given an input ultrametric M, let c denote the minimum distortion achievable by any embedding of M into the plane. Our main result is a linear-time algorithm that produces an O(c3)-distortion embedding. This result can be generalized to embedding ultrametrics into Rd, for any d≥2, with distortion cO(d), where c is the minimum distortion achievable for embedding the input ultrametric into Rd.Additionally, we show that any ultrametric can be embedded into the plane with distortion O(√n), and in general, into Rd with distortion dO(1) n1/d. Combining the two results together, we obtain an O(n1/3)-approximation algorithm for the problem of minimum-distortion embedding of ultrametrics into the plane.