2,639 research outputs found
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
Constant approximation algorithms for embedding graph metrics into trees and outerplanar graphs
In this paper, we present a simple factor 6 algorithm for approximating the
optimal multiplicative distortion of embedding a graph metric into a tree
metric (thus improving and simplifying the factor 100 and 27 algorithms of
B\v{a}doiu, Indyk, and Sidiropoulos (2007) and B\v{a}doiu, Demaine, Hajiaghayi,
Sidiropoulos, and Zadimoghaddam (2008)). We also present a constant factor
algorithm for approximating the optimal distortion of embedding a graph metric
into an outerplanar metric. For this, we introduce a general notion of metric
relaxed minor and show that if G contains an alpha-metric relaxed H-minor, then
the distortion of any embedding of G into any metric induced by a H-minor free
graph is at meast alpha. Then, for H=K_{2,3}, we present an algorithm which
either finds an alpha-relaxed minor, or produces an O(alpha)-embedding into an
outerplanar metric.Comment: 27 pages, 4 figires, extended abstract to appear in the proceedings
of APPROX-RANDOM 201
Metric Embedding via Shortest Path Decompositions
We study the problem of embedding shortest-path metrics of weighted graphs
into spaces. We introduce a new embedding technique based on low-depth
decompositions of a graph via shortest paths. The notion of Shortest Path
Decomposition depth is inductively defined: A (weighed) path graph has shortest
path decomposition (SPD) depth . General graph has an SPD of depth if it
contains a shortest path whose deletion leads to a graph, each of whose
components has SPD depth at most . In this paper we give an
-distortion embedding for graphs of SPD
depth at most . This result is asymptotically tight for any fixed ,
while for it is tight up to second order terms.
As a corollary of this result, we show that graphs having pathwidth embed
into with distortion . For
, this improves over the best previous bound of Lee and Sidiropoulos that
was exponential in ; moreover, for other values of it gives the first
embeddings whose distortion is independent of the graph size . Furthermore,
we use the fact that planar graphs have SPD depth to give a new
proof that any planar graph embeds into with distortion . Our approach also gives new results for graphs with bounded treewidth,
and for graphs excluding a fixed minor
- …