Solving All-k-Nearest Neighbor Problem without an Index

2009 Second International Workshop on Similarity Search and Applications, 2009

Retrieving the k-nearest neighbors of a query object is a basic primitive in similarity searching. A related, far less explored primitive is to obtain the dataset elements which would have the query object within their own k-nearest neighbors, known as the reverse k-nearest neighbor query. We already have indices and algorithms to solve k-nearest neighbors queries in general metric spaces; yet, in many cases of practical interest they degenerate to sequential scanning. The naive algorithm for reverse k-nearest neighbor queries has quadratic complexity, because the k-nearest neighbors of all the dataset objects must be found; this is too expensive. Hence, when solving these primitives we can tolerate trading correctness in the solution for searching time. In this paper we propose an efficient approximate approach to solve these similarity queries with high retrieval rate. Then, we show how to use our approximate k-nearest neighbor queries to construct (an approximation of) the k-nearest neighbor graph when we have a fixed dataset. Finally, combining both primitives we show how to dynamically maintain the approximate k-nearest neighbor graph of the objects currently stored within the metric dataset, that is, considering both object insertions and deletions.

British National Conference on Databases, 1998

Building an index tree is a common approach to speed upthe k nearest neighbour search in large databases of many-dimensionalrecords. Many applications require varying distance metrics by putting aweight on different dimensions. The main problem with k nearest neighboursearches using weighted euclidean metrics in a high dimensionalspace is whether the searches can be done efficiently We present a solutionto this

Proceedings of the fourth annual ACM-SIAM …, 1993

We consider the computational problem of finding nearest neighbors in general metric spaces. Of particular interest are spaces that may not be conveniently embedded or approxi-mated in Euclidian space, or where the dimensionality of a Euclidian representation 1s ...

Lecture Notes in Computer Science, 1998

Building an index tree is a common approach to speed up the k nearest neighbour search in large databases of many-dimensional records. Many applications require varying distance metrics by putting a weight on di erent dimensions. The main problem with k nearest neighbour searches using weighted euclidean metrics in a high dimensional space is whether the searches can be done e ciently We present a solution to this problem which uses the bounding rectangle of the nearestneighbour disk instead of using the disk directly. The algorithm is able to perform nearest-neighbour searches using distance metrics di erent from the metric used to build the search tree without having to rebuild the tree. It is e cient for weighted euclidean distance and extensible to higher dimensions.

2010

Numerous techniques have been proposed in the past for supporting efficient k-nearest neighbor (k-NN) queries in continuous data spaces. Limited work has been reported in the literature for k-NN queries in a non-ordered discrete data space (NDDS). Performing k-NN queries in an NDDS raises new challenges. The Hamming distance is usually used to measure the distance between two vectors (objects) in an NDDS. Due to the coarse granularity of the Hamming distance, a k-NN query in an NDDS may lead to a high degree of non-determinism for the query result. We propose a new distance measure, called Granularity-Enhanced Hamming (GEH) distance, that effectively reduces the number of candidate solutions for a query. We have also implemented k-NN queries using multidimensional database indexing in NDDSs. Further, we use the properties of our multidimensional NDDS index to derive the probability of encountering new neighbors within specific regions of the index. This probability is used to develop a new search ordering heuristic. Our experiments on synthetic and genomic data sets demonstrate that our index-based k-NN algorithm is efficient in finding k-NNs in both uniform and non-uniform data sets in NDDSs and that our heuristics are effective in improving the performance of such queries.

2007 IEEE 23rd International Conference on Data Engineering, 2007

A k-nearest neighbor (k-NN) query retrieves k objects from a database that are considered to be the closest to a given query point. Numerous techniques have been proposed in the past for supporting efficient k-NN searches in continuous data spaces. No such work has been reported in the literature for k-NN searches in a non-ordered discrete data space (NDDS). Performing k-NN searches in an NDDS raises new challenges. The Hamming distance is usually used to measure the distance between two vectors (objects) in an NDDS. Due to the coarse granularity of the Hamming distance, a k-NN query in an NDDS may lead to a large set of candidate solutions, creating a high degree of nondeterminism for the query result. We propose a new distance measure, called Granularity-Enhanced Hamming (GEH) distance, that effectively reduces the number of candidate solutions for a query. We have also considered using multidimensional database indexing for implementing k-NN searches in NDDSs. Our experiments on synthetic and genomic data sets demonstrate that our index-based k-NN algorithm is effective and efficient in finding k-NNs in NDDSs.

Lecture Notes in Computer Science, 2006

Let U be a set of elements and d a distance function defined among them. Let NN k (u) be the k elements in U − {u} having the smallest distance to u. The k-nearest neighbor graph (knng) is a weighted

Lecture Notes in Computer Science, 2007

Similarity search algorithms that directly rely on index structures and require a lot of distance computations are usually not applicable to databases containing complex objects and defining costly distance functions on spatial, temporal and multimedia data. Rather, the use of an adequate multi-step query processing strategy is crucial for the performance of a similarity search routine that deals with complex distance functions. Reducing the number of candidates returned from the filter step which then have to be exactly evaluated in the refinement step is fundamental for the efficiency of the query process. The state-of-the-art multi-step k-nearest neighbor (kNN) search algorithms are designed to use only a lower bounding distance estimation for candidate pruning. However, in many applications, also an upper bounding distance approximation is available that can additionally be used for reducing the number of candidates. In this paper, we generalize the traditional concept of R-optimality and introduce the notion of RI-optimality depending on the distance information I available in the filter step. We propose a new multi-step kNN search algorithm that utilizes lower-and upper bounding distance information (I lu) in the filter step. Furthermore, we show that, in contrast to existing approaches, our proposed solution is RI luoptimal. In an experimental evaluation, we demonstrate the significant performance gain over existing methods.

SIAM Journal on Computing, 2013

We study the Approximate Nearest Neighbor problem for metric spaces where the query points are constrained to lie on a subspace of low doubling dimension, while the data is high-dimensional. We show that this problem can be solved efficiently despite the high dimensionality of the data.

Lecture Notes in Computer Science, 2006

Proximity searching consists in retrieving from a database those elements that are similar to a query. As the distance is usually expensive to compute, the goal is to use as few distance computations as possible to satisfy queries. Indexes use precomputed distances among database elements to speed up queries. As such, a baseline is AESA, which stores all the distances among database objects, but has been unbeaten in query performance for 20 years. In this paper we show that it is possible to improve upon AESA by using a radically different method to select promising database elements to compare against the query. Our experiments show improvements of up to 75% in document databases. We also explore the usage of our method as a probabilistic algorithm that may lose relevant answers. On a database of faces where any exact algorithm must examine virtually all elements, our probabilistic version obtains 85% of the correct answers by scanning only 10% of the database.

Multimedia Tools and Applications, 2003

In order to speedup retrieval in large collections of data, index structures partition the data into subsets so that query requests can be evaluated without examining the entire collection. As the complexity of modern data types grows, metric spaces have become a popular paradigm for similarity retrieval. We propose a new index structure, called D-Index, that combines a novel clustering technique and the pivot-based distance searching strategy to speed up execution of similarity range and nearest neighbor queries for large files with objects stored in disk memories. We have qualitatively analyzed D-Index and verified its properties on actual implementation. We have also compared D-Index with other index structures and demonstrated its superiority on several real-life data sets. Contrary to tree organizations, the D-Index structure is suitable for dynamic environments with a high rate of delete/insert operations.

1999

We investigate the problem of approximate similarity (nearest neighbor) search in high-dimensional metric spaces, and describe how the distance distribution of the query object can be exploited so as to provide probabilistic guarantees on the quality of the result. This leads to a new paradigm for similarity search, called PAC-NN (probably approximately correct nearest neighbor) queries, aiming to break the "dimensionality curse". PAC-NN queries return, with probability at least 1 ; , a (1 + )-approximate NN -an object whose distance from the query q is less than (1 + ) times the distance between q and its NN. Analytical and experimental results obtained for sequential and index-based algorithms show that PAC-NN queries can be efficiently processed even on very high-dimensional spaces and that control can be exerted in order to tradeoff the accuracy of the result and the cost.

Lecture Notes in Computer Science, 2012

We propose a novel approach for solving the approximate nearest neighbor search problem in arbitrary metric spaces. The distinctive feature of our approach is that we can incrementally build a non-hierarchical distributed structure for given metric space data with a logarithmic complexity scaling on the size of the structure and adjustable accuracy probabilistic nearest neighbor queries. The structure is based on a small world graph with vertices corresponding to the stored elements, edges for links between them and the greedy algorithm as base algorithm for searching. Both search and addition algorithms require only local information from the structure. The performed simulation for data in the Euclidian space shows that the structure built using the proposed algorithm has navigable small world properties with logarithmic search complexity at fixed accuracy and has weak (power law) scalability with the dimensionality of the stored data.

Lecture Notes in Computer Science, 2004

A number of problems in computer science can be solved efficiently with the so called memory based or kernel methods. Among this problems (relevant to the AI community) are multimedia indexing, clustering, non supervised learning and recommendation systems. The common ground to this problems is satisfying proximity queries with an abstract metric database. In this paper we introduce a new technique for making practical indexes for metric range queries. This technique improves existing algorithms based on pivots and signatures, and introduces a new data structure, the Fixed Queries Trie to speedup metric range queries. The result is an O(n) construction time index, with query complexity O(n α), α ≤ 1. The indexing algorithm uses only a few bits of storage for each database element.

Given a set P of N points in a ddimensional space, along with a query point q, it is often desirable to find k points of P that are with high probability close to q. This is the Approximate k-Nearest-Neighbors problem. We present two algorithms for AkNN. Both require O(N 2 d) preprocessing time. The first algorithm has a query time cost that is O(d+log N ), while the second has a query time cost that is O(d). Both algorithms create an undirected graph on the points of P by adding edges to a linked list storing P in Hilbert order. To find approximate nearest neighbors of a query point, both algorithms perform bestfirst search on this graph. The first algorithm uses standard one dimensional indexing structures to find starting points on the graph for this search, whereas the second algorithm using random starting points. Despite the quadratic preprocessing time, our algorithms have the potential to be useful in machine learning applications where the number of query points that need to be processed is large compared to the number of points in P . The linear dependence in d of the preprocessing and query time costs of our algorithms allows them to remain effective even when dealing with highdimensional data.