TY - GEN

T1 - Approximate nearest neighbor search in metrics of planar graphs

AU - Abraham, Ittai

AU - Chechik, Shiri

AU - Krauthgamer, Robert

AU - Wieder, Udi

N1 - Publisher Copyright:
© Ittai Abraham, Shiri Checik, Robert Krauthgamer, and Uli Wieder.

PY - 2015/8/1

Y1 - 2015/8/1

N2 - Investigate the problem of approximate Nearest-Neighbor Search (NNS) in graphical metrics: The task is to preprocess an edge-weighted graph G = (V,E) on m vertices and a small "dataset" D ∪ V of size n ≤ m, so that given a query point q ∈ V, one can quickly approximate dG(q,D) (the distance from q to its closest vertex in D) and find a vertex a ∈ D within this approximated distance. We assume the query algorithm has access to a distance oracle, that quickly evaluates the exact distance between any pair of vertices. For planar graphs G with maximum degree Δ, we show how to efficiently construct a compact data structure - of size Õ(n(Δ + 1/∈)) - that answers (1 + ∈)-NNS queries in time Õ(Δ + 1/∈). Thus, as far as NNS applications are concerned, metrics derived from bounded-degree planar graphs behave as low-dimensional metrics, even though planar metrics do not necessarily have a low doubling dimension, nor can they be embedded with low distortion into ℓ2. We complement our algorithmic result by lower bounds showing that the access to an exact distance oracle (rather than an approximate one) and the dependency on Δ (in query time) are both essential.

AB - Investigate the problem of approximate Nearest-Neighbor Search (NNS) in graphical metrics: The task is to preprocess an edge-weighted graph G = (V,E) on m vertices and a small "dataset" D ∪ V of size n ≤ m, so that given a query point q ∈ V, one can quickly approximate dG(q,D) (the distance from q to its closest vertex in D) and find a vertex a ∈ D within this approximated distance. We assume the query algorithm has access to a distance oracle, that quickly evaluates the exact distance between any pair of vertices. For planar graphs G with maximum degree Δ, we show how to efficiently construct a compact data structure - of size Õ(n(Δ + 1/∈)) - that answers (1 + ∈)-NNS queries in time Õ(Δ + 1/∈). Thus, as far as NNS applications are concerned, metrics derived from bounded-degree planar graphs behave as low-dimensional metrics, even though planar metrics do not necessarily have a low doubling dimension, nor can they be embedded with low distortion into ℓ2. We complement our algorithmic result by lower bounds showing that the access to an exact distance oracle (rather than an approximate one) and the dependency on Δ (in query time) are both essential.

KW - Data structures

KW - Nearest neighbor search

KW - Planar graphs

KW - Planar metrics

KW - Planar separator

UR - http://www.scopus.com/inward/record.url?scp=84956614759&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.APPROX-RANDOM.2015.20

DO - 10.4230/LIPIcs.APPROX-RANDOM.2015.20

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AN - SCOPUS:84956614759

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 20

EP - 42

BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015

A2 - Garg, Naveen

A2 - Jansen, Klaus

A2 - Rao, Anup

A2 - Rolim, Jose D. P.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015

Y2 - 24 August 2015 through 26 August 2015

ER -