TY - GEN
T1 - On low dimensional local embeddings
AU - Abraham, Ittai
AU - Bartal, Yair
AU - Neiman, Ofer
PY - 2009
Y1 - 2009
N2 - We study the problem of embedding metric spaces into low dimensional ℓp spaces while faithfully preserving distances from each point to its k nearest neighbors. We show that any metric space can be embedded into ℓpO(ep log2 k) with k-local distortion of O((log k)/p). We also show that any ultrametric can be embedded into ℓpO(log k)/c3 with k-local distortion 1 + ∈. Our embedding results have immediate applications to local Distance Oracles. We show how to preprocess a graph in polynomial time to obtain a data structure of O(nk1/t log2 k) bits, such that distance queries from any node to its k nearest neighbors can be answered with stretch O(t).
AB - We study the problem of embedding metric spaces into low dimensional ℓp spaces while faithfully preserving distances from each point to its k nearest neighbors. We show that any metric space can be embedded into ℓpO(ep log2 k) with k-local distortion of O((log k)/p). We also show that any ultrametric can be embedded into ℓpO(log k)/c3 with k-local distortion 1 + ∈. Our embedding results have immediate applications to local Distance Oracles. We show how to preprocess a graph in polynomial time to obtain a data structure of O(nk1/t log2 k) bits, such that distance queries from any node to its k nearest neighbors can be answered with stretch O(t).
UR - http://www.scopus.com/inward/record.url?scp=70349110053&partnerID=8YFLogxK
U2 - 10.1137/1.9781611973068.95
DO - 10.1137/1.9781611973068.95
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AN - SCOPUS:70349110053
SN - 9780898716801
T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
SP - 875
EP - 884
BT - Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms
PB - Association for Computing Machinery (ACM)
T2 - 20th Annual ACM-SIAM Symposium on Discrete Algorithms
Y2 - 4 January 2009 through 6 January 2009
ER -