TY - GEN
T1 - Volume in general metric spaces
AU - Abraham, Ittai
AU - Bartal, Yair
AU - Neiman, Ofer
AU - Schulman, Leonard J.
PY - 2010
Y1 - 2010
N2 - A central question in the geometry of finite metric spaces is how well can an arbitrary metric space be "faithfully preserved" by a mapping into Euclidean space. In this paper we present an algorithmic embedding which obtains a new strong measure of faithful preservation: not only does it (approximately) preserve distances between pairs of points, but also the volume of any set of k points. Such embeddings are known as volume preserving embeddings. We provide the first volume preserving embedding that obtains constant average volume distortion for sets of any fixed size. Moreover, our embedding provides constant bounds on all bounded moments of the volume distortion while maintaining the best possible worst-case volume distortion. Feige, in his seminal work on volume preserving embeddings defined the volume of a set S = {v 1, ..., v k } of points in a general metric space: the product of the distances from v i to { v 1, ..., v i - 1 }, normalized by 1/(k-1)!, where the ordering of the points is that given by Prim's minimum spanning tree algorithm. Feige also related this notion to the maximal Euclidean volume that a Lipschitz embedding of S into Euclidean space can achieve. Syntactically this definition is similar to the computation of volume in Euclidean spaces, which however is invariant to the order in which the points are taken. We show that a similar robustness property holds for Feige's definition: the use of any other order in the product affects volume 1/(k - 1) by only a constant factor. Our robustness result is of independent interest as it presents a new competitive analysis for the greedy algorithm on a variant of the online Steiner tree problem where the cost of buying an edge is logarithmic in its length. This robustness property allows us to obtain our results on volume preserving embeddings.
AB - A central question in the geometry of finite metric spaces is how well can an arbitrary metric space be "faithfully preserved" by a mapping into Euclidean space. In this paper we present an algorithmic embedding which obtains a new strong measure of faithful preservation: not only does it (approximately) preserve distances between pairs of points, but also the volume of any set of k points. Such embeddings are known as volume preserving embeddings. We provide the first volume preserving embedding that obtains constant average volume distortion for sets of any fixed size. Moreover, our embedding provides constant bounds on all bounded moments of the volume distortion while maintaining the best possible worst-case volume distortion. Feige, in his seminal work on volume preserving embeddings defined the volume of a set S = {v 1, ..., v k } of points in a general metric space: the product of the distances from v i to { v 1, ..., v i - 1 }, normalized by 1/(k-1)!, where the ordering of the points is that given by Prim's minimum spanning tree algorithm. Feige also related this notion to the maximal Euclidean volume that a Lipschitz embedding of S into Euclidean space can achieve. Syntactically this definition is similar to the computation of volume in Euclidean spaces, which however is invariant to the order in which the points are taken. We show that a similar robustness property holds for Feige's definition: the use of any other order in the product affects volume 1/(k - 1) by only a constant factor. Our robustness result is of independent interest as it presents a new competitive analysis for the greedy algorithm on a variant of the online Steiner tree problem where the cost of buying an edge is logarithmic in its length. This robustness property allows us to obtain our results on volume preserving embeddings.
UR - http://www.scopus.com/inward/record.url?scp=78349235865&partnerID=8YFLogxK
U2 - 10.1007/978-3-642-15781-3_8
DO - 10.1007/978-3-642-15781-3_8
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AN - SCOPUS:78349235865
SN - 3642157807
SN - 9783642157806
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 87
EP - 99
BT - Algorithms, ESA 2010 - 18th Annual European Symposium, Proceedings
T2 - 18th Annual European Symposium on Algorithms, ESA 2010
Y2 - 6 September 2010 through 8 September 2010
ER -