TY - GEN

T1 - Dynamic Data Structures for k-Nearest Neighbor Queries

AU - Berg, Sarita de

AU - Staals, Frank

PY - 2021

Y1 - 2021

N2 - Our aim is to develop dynamic data structures that support k-nearest neighbors (k-NN) queries for a set of n point sites in O(f(n) + k) time, where f(n) is some polylogarithmic function of n. The key component is a general query algorithm that allows us to find the k-NN spread over t substructures simultaneously, thus reducing a O(tk) term in the query time to O(k). Combining this technique with the logarithmic method allows us to turn any static k-NN data structure into a data structure supporting both efficient insertions and queries. For the fully dynamic case, this technique allows us to recover the deterministic, worst-case, O(log²n/log log n +k) query time for the Euclidean distance claimed before, while preserving the polylogarithmic update times. We adapt this data structure to also support fully dynamic geodesic k-NN queries among a set of sites in a simple polygon. For this purpose, we design a shallow cutting based, deletion-only k-NN data structure. More generally, we obtain a dynamic k-NN data structure for any type of distance functions for which we can build vertical shallow cuttings. We apply all of our methods in the plane for the Euclidean distance, the geodesic distance, and general, constant-complexity, algebraic distance functions.

AB - Our aim is to develop dynamic data structures that support k-nearest neighbors (k-NN) queries for a set of n point sites in O(f(n) + k) time, where f(n) is some polylogarithmic function of n. The key component is a general query algorithm that allows us to find the k-NN spread over t substructures simultaneously, thus reducing a O(tk) term in the query time to O(k). Combining this technique with the logarithmic method allows us to turn any static k-NN data structure into a data structure supporting both efficient insertions and queries. For the fully dynamic case, this technique allows us to recover the deterministic, worst-case, O(log²n/log log n +k) query time for the Euclidean distance claimed before, while preserving the polylogarithmic update times. We adapt this data structure to also support fully dynamic geodesic k-NN queries among a set of sites in a simple polygon. For this purpose, we design a shallow cutting based, deletion-only k-NN data structure. More generally, we obtain a dynamic k-NN data structure for any type of distance functions for which we can build vertical shallow cuttings. We apply all of our methods in the plane for the Euclidean distance, the geodesic distance, and general, constant-complexity, algebraic distance functions.

KW - data structure

KW - simple polygon

KW - geodesic distance

KW - nearest neighbor searching

U2 - 10.4230/LIPIcs.ISAAC.2021.14

DO - 10.4230/LIPIcs.ISAAC.2021.14

M3 - Conference contribution

VL - 212

T3 - LIPIcs

SP - 1

EP - 14

BT - 32nd International Symposium on Algorithms and Computation, ISAAC 2021, December 6-8, 2021, Fukuoka, Japan

A2 - Ahn, Hee-Kap

A2 - Sadakane, Kunihiko

PB - Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH

ER -