Computational geometry: an introduction
Computational geometry: an introduction
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
The design and analysis of spatial data structures
The design and analysis of spatial data structures
An algorithm for approximate closest-point queries
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Matrix computations (3rd ed.)
A data structure for dynamically maintaining rooted trees
Journal of Algorithms
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
An optimal algorithm for approximate nearest neighbor searching fixed dimensions
Journal of the ACM (JACM)
Computational Geometry: Theory and Applications
Balanced aspect ratio trees: combining the advantages of k-d trees and octrees
Journal of Algorithms
Space-efficient approximate Voronoi diagrams
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Finding nearest neighbors in growth-restricted metrics
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Linear-size approximate voronoi diagrams
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Closest-point problems simplified on the RAM
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Efficient Search for Approximate Nearest Neighbor in High Dimensional Spaces
SIAM Journal on Computing
Dynamic Algorithms for Approximate Neighbor Searching
Proceedings of the 8th Canadian Conference on Computational Geometry
A Replacement for Voronoi Diagrams of Near Linear Size
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Navigating nets: simple algorithms for proximity search
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Space-time tradeoffs for approximate spherical range counting
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Entropy based nearest neighbor search in high dimensions
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Fast Construction of Nets in Low-Dimensional Metrics and Their Applications
SIAM Journal on Computing
On the importance of idempotence
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Searching dynamic point sets in spaces with bounded doubling dimension
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The black-box complexity of nearest-neighbor search
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
Journal of Computer and System Sciences
The Concentration of Fractional Distances
IEEE Transactions on Knowledge and Data Engineering
Approximate range searching in the absolute error model
Approximate range searching in the absolute error model
Tradeoffs in Approximate Range Searching Made Simpler
SIBGRAPI '08 Proceedings of the 2008 XXI Brazilian Symposium on Computer Graphics and Image Processing
The Effect of Corners on the Complexity of Approximate Range Searching
Discrete & Computational Geometry
Affine Stable Characteristic based sample expansion for object detection
Proceedings of the ACM International Conference on Image and Video Retrieval
Improving accuracy of LVQ algorithm by instance weighting
ICANN'10 Proceedings of the 20th international conference on Artificial neural networks: Part III
A unified approach to approximate proximity searching
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Randomly projected KD-trees with distance metric learning for image retrieval
MMM'11 Proceedings of the 17th international conference on Advances in multimedia modeling - Volume Part II
Approximate polytope membership queries
Proceedings of the forty-third annual ACM symposium on Theory of computing
Exploiting temporal coherence in forest dynamics simulation
Proceedings of the twenty-seventh annual symposium on Computational geometry
Scalable kNN search on vertically stored time series
Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining
Polytope approximation and the Mahler volume
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Approximate nearest neighbor search for low dimensional queries
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Data Mining and Knowledge Discovery
Nearest-neighbor searching under uncertainty
PODS '12 Proceedings of the 31st symposium on Principles of Database Systems
Improving the speed and stability of the k-nearest neighbors method
Pattern Recognition Letters
Boosting multi-kernel locality-sensitive hashing for scalable image retrieval
SIGIR '12 Proceedings of the 35th international ACM SIGIR conference on Research and development in information retrieval
Recursive similarity-based algorithm for deep learning
ICONIP'12 Proceedings of the 19th international conference on Neural Information Processing - Volume Part III
Accurate off-line query expansion for large-scale mobile visual search
Signal Processing
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Nearest neighbor searching is the problem of preprocessing a set of n point points in d-dimensional space so that, given any query point q, it is possible to report the closest point to q rapidly. In approximate nearest neighbor searching, a parameter ϵ 0 is given, and a multiplicative error of (1 + ϵ) is allowed. We assume that the dimension d is a constant and treat n and ϵ as asymptotic quantities. Numerous solutions have been proposed, ranging from low-space solutions having space O(n) and query time O(log n + 1/ϵd−1) to high-space solutions having space roughly O((n log n)/ϵd) and query time O(log (n/ϵ)). We show that there is a single approach to this fundamental problem, which both improves upon existing results and spans the spectrum of space-time tradeoffs. Given a tradeoff parameter γ, where 2 ≤ γ ≤ 1/ϵ, we show that there exists a data structure of space O(nγd−1 log(1/ϵ)) that can answer queries in time O(log(nγ) + 1/(ϵγ)(d−1)/2. When γ = 2, this yields a data structure of space O(n log (1/ϵ)) that can answer queries in time O(log n + 1/ϵ(d−1)/2). When γ = 1/ϵ, it provides a data structure of space O((n/ϵd−1)log(1/ϵ)) that can answer queries in time O(log(n/ϵ)). Our results are based on a data structure called a (t,ϵ)-AVD, which is a hierarchical quadtree-based subdivision of space into cells. Each cell stores up to t representative points of the set, such that for any query point q in the cell at least one of these points is an approximate nearest neighbor of q. We provide new algorithms for constructing AVDs and tools for analyzing their total space requirements. We also establish lower bounds on the space complexity of AVDs, and show that, up to a factor of O(log (1/ϵ)), our space bounds are asymptotically tight in the two extremes, γ = 2 and γ = 1/ϵ.