What Is a Good Nearest Neighbors Algorithm for Finding Similar Patches in Images?
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Lecture 04 Linear Structures Sort
Algorithmics (6EAP) MTAT.03.238 Linear structures, sorting, searching, etc Jaak Vilo 2018 Fall Jaak Vilo 1 Big-Oh notation classes Class Informal Intuition Analogy f(n) ∈ ο ( g(n) ) f is dominated by g Strictly below < f(n) ∈ O( g(n) ) Bounded from above Upper bound ≤ f(n) ∈ Θ( g(n) ) Bounded from “equal to” = above and below f(n) ∈ Ω( g(n) ) Bounded from below Lower bound ≥ f(n) ∈ ω( g(n) ) f dominates g Strictly above > Conclusions • Algorithm complexity deals with the behavior in the long-term – worst case -- typical – average case -- quite hard – best case -- bogus, cheating • In practice, long-term sometimes not necessary – E.g. for sorting 20 elements, you dont need fancy algorithms… Linear, sequential, ordered, list … Memory, disk, tape etc – is an ordered sequentially addressed media. Physical ordered list ~ array • Memory /address/ – Garbage collection • Files (character/byte list/lines in text file,…) • Disk – Disk fragmentation Linear data structures: Arrays • Array • Hashed array tree • Bidirectional map • Heightmap • Bit array • Lookup table • Bit field • Matrix • Bitboard • Parallel array • Bitmap • Sorted array • Circular buffer • Sparse array • Control table • Sparse matrix • Image • Iliffe vector • Dynamic array • Variable-length array • Gap buffer Linear data structures: Lists • Doubly linked list • Array list • Xor linked list • Linked list • Zipper • Self-organizing list • Doubly connected edge • Skip list list • Unrolled linked list • Difference list • VList Lists: Array 0 1 size MAX_SIZE-1 3 6 7 5 2 L = int[MAX_SIZE] -
The Fractal Dimension Making Similarity Queries More Efficient
The Fractal Dimension Making Similarity Queries More Efficient Adriano S. Arantes, Marcos R. Vieira, Agma J. M. Traina, Caetano Traina Jr. Computer Science Department - ICMC University of Sao Paulo at Sao Carlos Avenida do Trabalhador Sao-Carlense, 400 13560-970 - Sao Carlos, SP - Brazil Phone: +55 16-273-9693 {arantes | mrvieira | agma | caetano}@icmc.usp.br ABSTRACT time series, among others, are not useful, and the simila- This paper presents a new algorithm to answer k-nearest rity between pairs of elements is the most important pro- neighbor queries called the Fractal k-Nearest Neighbor (k- perty in such domains [5]. Thus, a new class of queries, ba- NNF ()). This algorithm takes advantage of the fractal di- sed on algorithms that search for similarity among elements mension of the dataset under scan to estimate a suitable emerged as more adequate to manipulate these data. To radius to shrinks a query that retrieves the k-nearest neigh- be able to apply similarity queries over elements of a data bors of a query object. k-NN() algorithms starts searching domain, a dissimilarity function, or a “distance function”, for elements at any distance from the query center, progres- must be defined on the domain. A distance function δ() ta- sively reducing the allowed distance used to consider ele- kes pairs of elements of the domain, quantifying the dissimi- ments as worth to analyze. If a proper radius can be set larity among them. If δ() satisfies the following properties: to start the process, a significant reduction in the number for any elements x, y and z of the data domain, δ(x, x) = 0 of distance calculations can be achieved. -
Two Algorithms for Nearest-Neighbor Search in High Dimensions
Two Algorithms for Nearest-Neighbor Search in High Dimensions Jon M. Kleinberg∗ February 7, 1997 Abstract Representing data as points in a high-dimensional space, so as to use geometric methods for indexing, is an algorithmic technique with a wide array of uses. It is central to a number of areas such as information retrieval, pattern recognition, and statistical data analysis; many of the problems arising in these applications can involve several hundred or several thousand dimensions. We consider the nearest-neighbor problem for d-dimensional Euclidean space: we wish to pre-process a database of n points so that given a query point, one can ef- ficiently determine its nearest neighbors in the database. There is a large literature on algorithms for this problem, in both the exact and approximate cases. The more sophisticated algorithms typically achieve a query time that is logarithmic in n at the expense of an exponential dependence on the dimension d; indeed, even the average- case analysis of heuristics such as k-d trees reveals an exponential dependence on d in the query time. In this work, we develop a new approach to the nearest-neighbor problem, based on a method for combining randomly chosen one-dimensional projections of the underlying point set. From this, we obtain the following two results. (i) An algorithm for finding ε-approximate nearest neighbors with a query time of O((d log2 d)(d + log n)). (ii) An ε-approximate nearest-neighbor algorithm with near-linear storage and a query time that improves asymptotically on linear search in all dimensions. -
Rank-Approximate Nearest Neighbor Search: Retaining Meaning and Speed in High Dimensions
Rank-Approximate Nearest Neighbor Search: Retaining Meaning and Speed in High Dimensions Parikshit Ram, Dongryeol Lee, Hua Ouyang and Alexander G. Gray Computational Science and Engineering, Georgia Institute of Technology Atlanta, GA 30332 p.ram@,dongryel@cc.,houyang@,agray@cc. gatech.edu { } Abstract The long-standing problem of efficient nearest-neighbor (NN) search has ubiqui- tous applications ranging from astrophysics to MP3 fingerprinting to bioinformat- ics to movie recommendations. As the dimensionality of the dataset increases, ex- act NN search becomes computationally prohibitive; (1+) distance-approximate NN search can provide large speedups but risks losing the meaning of NN search present in the ranks (ordering) of the distances. This paper presents a simple, practical algorithm allowing the user to, for the first time, directly control the true accuracy of NN search (in terms of ranks) while still achieving the large speedups over exact NN. Experiments on high-dimensional datasets show that our algorithm often achieves faster and more accurate results than the best-known distance-approximate method, with much more stable behavior. 1 Introduction In this paper, we address the problem of nearest-neighbor (NN) search in large datasets of high dimensionality. It is used for classification (k-NN classifier [1]), categorizing a test point on the ba- sis of the classes in its close neighborhood. Non-parametric density estimation uses NN algorithms when the bandwidth at any point depends on the ktℎ NN distance (NN kernel density estimation [2]). NN algorithms are present in and often the main cost of most non-linear dimensionality reduction techniques (manifold learning [3, 4]) to obtain the neighborhood of every point which is then pre- served during the dimension reduction. -
Nearest Neighbor Search in Google Correlate
Nearest Neighbor Search in Google Correlate Dan Vanderkam Robert Schonberger Henry Rowley Google Inc Google Inc Google Inc 76 9th Avenue 76 9th Avenue 1600 Amphitheatre Parkway New York, New York 10011 New York, New York 10011 Mountain View, California USA USA 94043 USA [email protected] [email protected] [email protected] Sanjiv Kumar Google Inc 76 9th Avenue New York, New York 10011 USA [email protected] ABSTRACT queries are then noted, and a model that estimates influenza This paper presents the algorithms which power Google Cor- incidence based on present query data is created. relate[8], a tool which finds web search terms whose popu- This estimate is valuable because the most recent tradi- larity over time best matches a user-provided time series. tional estimate of the ILI rate is only available after a two Correlate was developed to generalize the query-based mod- week delay. Indeed, there are many fields where estimating eling techniques pioneered by Google Flu Trends and make the present[1] is important. Other health issues and indica- them available to end users. tors in economics, such as unemployment, can also benefit Correlate searches across millions of candidate query time from estimates produced using the techniques developed for series to find the best matches, returning results in less than Google Flu Trends. 200 milliseconds. Its feature set and requirements present Google Flu Trends relies on a multi-hour batch process unique challenges for Approximate Nearest Neighbor (ANN) to find queries that correlate to the ILI time series. Google search techniques. In this paper, we present Asymmetric Correlate allows users to create their own versions of Flu Hashing (AH), the technique used by Correlate, and show Trends in real time. -
R-Trees with Voronoi Diagrams for Efficient Processing of Spatial
VoR-Tree: R-trees with Voronoi Diagrams for Efficient Processing of Spatial Nearest Neighbor Queries∗ y Mehdi Sharifzadeh Cyrus Shahabi Google University of Southern California [email protected] [email protected] ABSTRACT A very important class of spatial queries consists of nearest- An important class of queries, especially in the geospatial neighbor (NN) query and its variations. Many studies in domain, is the class of nearest neighbor (NN) queries. These the past decade utilize R-trees as their underlying index queries search for data objects that minimize a distance- structures to address NN queries efficiently. The general ap- based function with reference to one or more query objects. proach is to use R-tree in two phases. First, R-tree's hierar- Examples are k Nearest Neighbor (kNN) [13, 5, 7], Reverse chical structure is used to quickly arrive to the neighborhood k Nearest Neighbor (RkNN) [8, 15, 16], k Aggregate Nearest of the result set. Second, the R-tree nodes intersecting with Neighbor (kANN) [12] and skyline queries [1, 11, 14]. The the local neighborhood (Search Region) of an initial answer applications of NN queries are numerous in geospatial deci- are investigated to find all the members of the result set. sion making, location-based services, and sensor networks. While R-trees are very efficient for the first phase, they usu- The introduction of R-trees [3] (and their extensions) for ally result in the unnecessary investigation of many nodes indexing multi-dimensional data marked a new era in de- that none or only a small subset of their including points veloping novel R-tree-based algorithms for various forms belongs to the actual result set. -
Dimensional Testing for Reverse K-Nearest Neighbor Search
University of Southern Denmark Dimensional testing for reverse k-nearest neighbor search Casanova, Guillaume; Englmeier, Elias; Houle, Michael E.; Kröger, Peer; Nett, Michael; Schubert, Erich; Zimek, Arthur Published in: Proceedings of the VLDB Endowment DOI: 10.14778/3067421.3067426 Publication date: 2017 Document version: Final published version Document license: CC BY-NC-ND Citation for pulished version (APA): Casanova, G., Englmeier, E., Houle, M. E., Kröger, P., Nett, M., Schubert, E., & Zimek, A. (2017). Dimensional testing for reverse k-nearest neighbor search. Proceedings of the VLDB Endowment, 10(7), 769-780. https://doi.org/10.14778/3067421.3067426 Go to publication entry in University of Southern Denmark's Research Portal Terms of use This work is brought to you by the University of Southern Denmark. Unless otherwise specified it has been shared according to the terms for self-archiving. If no other license is stated, these terms apply: • You may download this work for personal use only. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying this open access version If you believe that this document breaches copyright please contact us providing details and we will investigate your claim. Please direct all enquiries to [email protected] Download date: 05. Oct. 2021 Dimensional Testing for Reverse k-Nearest Neighbor Search Guillaume Casanova Elias Englmeier Michael E. Houle ONERA-DCSD, France LMU Munich, Germany NII, Tokyo, -
Efficient Nearest Neighbor Searching for Motion Planning
E±cient Nearest Neighbor Searching for Motion Planning Anna Atramentov Steven M. LaValle Dept. of Computer Science Dept. of Computer Science Iowa State University University of Illinois Ames, IA 50011 USA Urbana, IL 61801 USA [email protected] [email protected] Abstract gies of con¯guration spaces. The topologies that usu- ally arise are Cartesian products of R, S1, and projective We present and implement an e±cient algorithm for per- spaces. In these cases, metric information must be ap- forming nearest-neighbor queries in topological spaces propriately processed by any data structure designed to that usually arise in the context of motion planning. perform nearest neighbor computations. The problem is Our approach extends the Kd tree-based ANN algo- further complicated by the high dimensionality of con¯g- rithm, which was developed by Arya and Mount for Eu- uration spaces. clidean spaces. We argue the correctness of the algo- rithm and illustrate its e±ciency through computed ex- In this paper, we extend the nearest neighbor algorithm amples. We have applied the algorithm to both prob- and part of the ANN package of Arya and Mount [16] to abilistic roadmaps (PRMs) and Rapidly-exploring Ran- handle general topologies with very little computational dom Trees (RRTs). Substantial performance improve- overhead. Related literature is discussed in Section 3. ments are shown for motion planning examples. Our approach is based on hierarchically searching nodes in a Kd tree while handling the appropriate constraints induced by the metric and topology. We have proven the 1 Introduction correctness of our approach, and we have demonstrated the e±ciency of the algorithm experimentally. -
An Investigation of Practical Approximate Nearest Neighbor Algorithms
An Investigation of Practical Approximate Nearest Neighbor Algorithms Ting Liu, Andrew W. Moore, Alexander Gray and Ke Yang School of Computer Science Carnegie-Mellon University Pittsburgh, PA 15213 USA ftingliu, awm, agray, [email protected] Abstract This paper concerns approximate nearest neighbor searching algorithms, which have become increasingly important, especially in high dimen- sional perception areas such as computer vision, with dozens of publica- tions in recent years. Much of this enthusiasm is due to a successful new approximate nearest neighbor approach called Locality Sensitive Hash- ing (LSH). In this paper we ask the question: can earlier spatial data structure approaches to exact nearest neighbor, such as metric trees, be altered to provide approximate answers to proximity queries and if so, how? We introduce a new kind of metric tree that allows overlap: certain datapoints may appear in both the children of a parent. We also intro- duce new approximate k-NN search algorithms on this structure. We show why these structures should be able to exploit the same random- projection-based approximations that LSH enjoys, but with a simpler al- gorithm and perhaps with greater efficiency. We then provide a detailed empirical evaluation on five large, high dimensional datasets which show up to 31-fold accelerations over LSH. This result holds true throughout the spectrum of approximation levels. 1 Introduction The k-nearest-neighbor searching problem is to find the k nearest points in a dataset X ⊂ RD containing n points to a query point q 2 RD, usually under the Euclidean distance. It has applications in a wide range of real-world settings, in particular pattern recognition, machine learning [7] and database querying [11]. -
A Fast All Nearest Neighbor Algorithm for Applications Involving Large Point-Clouds à Jagan Sankaranarayanan , Hanan Samet, Amitabh Varshney
Best Journal Paper of 2007 Award ARTICLE IN PRESS Computers & Graphics 31 (2007) 157–174 www.elsevier.com/locate/cag A fast all nearest neighbor algorithm for applications involving large point-clouds à Jagan Sankaranarayanan , Hanan Samet, Amitabh Varshney Department of Computer Science, Center for Automation Research, Institute for Advanced Computer Studies, University of Maryland, College Park, MD - 20742, USA Abstract Algorithms that use point-cloud models make heavy use of the neighborhoods of the points. These neighborhoods are used to compute the surface normals for each point, mollification, and noise removal. All of these primitive operations require the seemingly repetitive process of finding the k nearest neighbors (kNNs) of each point. These algorithms are primarily designed to run in main memory. However, rapid advances in scanning technologies have made available point-cloud models that are too large to fit in the main memory of a computer. This calls for more efficient methods of computing the kNNs of a large collection of points many of which are already in close proximity. A fast kNN algorithm is presented that makes use of the locality of successive points whose k nearest neighbors are sought to reduce significantly the time needed to compute the neighborhood needed for the primitive operation as well as enable it to operate in an environment where the data is on disk. Results of experiments demonstrate an order of magnitude improvement in the time to perform the algorithm and several orders of magnitude improvement in work efficiency when compared with several prominent existing methods. r 2006 Elsevier Ltd. All rights reserved. -
Fast Algorithm for Nearest Neighbor Search Based on a Lower Bound Tree
in Proceedings of the 8th International Conference on Computer Vision, Vancouver, Canada, Jul. 2001, Volume 1, pages 446–453 Fast Algorithm for Nearest Neighbor Search Based on a Lower Bound Tree Yong-Sheng Chen†‡ Yi-Ping Hung†‡ Chiou-Shann Fuh‡ †Institute of Information Science, Academia Sinica, Taipei, Taiwan ‡Dept. of Computer Science and Information Engineering, National Taiwan University, Taiwan Email: [email protected] Abstract In the past, Bentley [2] proposed the k-dimensional bi- nary search tree to speed up the the nearest neighbor search. This paper presents a novel algorithm for fast nearest This method is very efficient when the dimension of the data neighbor search. At the preprocessing stage, the proposed space is small. However, as reported in [13] and [3], its algorithm constructs a lower bound tree by agglomeratively performance degrades exponentially with increasing dimen- clustering the sample points in the database. Calculation of sion. Fukunaga and Narendra [8] constructed a tree struc- the distance between the query and the sample points can be ture by repeating the process of dividing the set of sam- avoided if the lower bound of the distance is already larger ple points into subsets. Given a query point, they used than the minimum distance. The search process can thus a branch-and-bound search strategy to efficiently find the be accelerated because the computational cost of the lower closest point by using the tree structure. Djouadi and Bouk- bound, which can be calculated by using the internal node tache [5] partitioned the underlying space of the sample of the lower bound tree, is less than that of the distance. -
Search Trees
The Annals of Applied Probability 2014, Vol. 24, No. 3, 1269–1297 DOI: 10.1214/13-AAP948 c Institute of Mathematical Statistics, 2014 SEARCH TREES: METRIC ASPECTS AND STRONG LIMIT THEOREMS By Rudolf Grubel¨ Leibniz Universit¨at Hannover We consider random binary trees that appear as the output of certain standard algorithms for sorting and searching if the input is random. We introduce the subtree size metric on search trees and show that the resulting metric spaces converge with probability 1. This is then used to obtain almost sure convergence for various tree functionals, together with representations of the respective limit ran- dom variables as functions of the limit tree. 1. Introduction. A sequential algorithm transforms an input sequence t1,t2,... into an output sequence x1,x2,... where, for all n ∈ N, xn+1 de- pends on xn and tn+1 only. Typically, the output variables are elements of some combinatorial family F, each x ∈ F has a size parameter φ(x) ∈ N and xn is an element of the set Fn := {x ∈ F : φ(x)= n} of objects of size n. In the probabilistic analysis of such algorithms, one starts with a stochastic model for the input sequence and is interested in certain aspects of the out- put sequence. The standard input model assumes that the ti’s are the values of a sequence η1, η2,... of independent and identically distributed random variables. For random input of this type, the output sequence then is the path of a Markov chain X = (Xn)n∈N that is adapted to the family F in the sense that (1) P (Xn ∈ Fn) = 1 for all n ∈ N.