3,384 research outputs found
Metric structures in L_1: Dimension, snowflakes, and average distortion
We study the metric properties of finite subsets of L_1. The analysis of such
metrics is central to a number of important algorithmic problems involving the
cut structure of weighted graphs, including the Sparsest Cut Problem, one of
the most compelling open problems in the field of approximation algorithms.
Additionally, many open questions in geometric non-linear functional analysis
involve the properties of finite subsets of L_1.Comment: 9 pages, 1 figure. To appear in European Journal of Combinatorics.
Preliminary version appeared in LATIN '0
Approximate Near Neighbors for General Symmetric Norms
We show that every symmetric normed space admits an efficient nearest
neighbor search data structure with doubly-logarithmic approximation.
Specifically, for every , , and every -dimensional
symmetric norm , there exists a data structure for
-approximate nearest neighbor search over
for -point datasets achieving query time and
space. The main technical ingredient of the algorithm is a
low-distortion embedding of a symmetric norm into a low-dimensional iterated
product of top- norms.
We also show that our techniques cannot be extended to general norms.Comment: 27 pages, 1 figur
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