Solving All-k-Nearest Neighbor Problem without an Index

Lecture Notes in Computer Science, 2005

Proximity searching consists in retrieving from a database, objects that are close to a query. For this type of searching problem, the most general model is the metric space, where proximity is defined in terms of a distance function. A solution for this problem consists in building an offline index to quickly satisfy online queries. The ultimate goal is to use as few distance computations as possible to satisfy queries, since the distance is considered expensive to compute. Proximity searching is central to several applications, ranging from multimedia indexing and querying to data compression and clustering. In this paper we present a new approach to solve the proximity searching problem. Our solution is based on indexing the database with the knearest neighbor graph (knng), which is a directed graph connecting each element to its k closest neighbors. We present two search algorithms for both range and nearest neighbor queries which use navigational and metrical features of the knng graph. We show that our approach is competitive against current ones. For instance, in the document metric space our nearest neighbor search algorithms perform 30% more distance evaluations than AESA using only a 0.25% of its space requirement. In the same space, the pivot-based technique is completely useless.

Nearest neighbor queries can be satisfied, in principle, with a greedy algorithm under a proximity graph. Each object in the database is represented by a node, and proximal nodes in this graph will share an edge. To find the nearest neighbor the idea is quite simple, we start in a random node and get iteratively closer to the nearest neighbor following only adjacent edges in the proximity graph. Every reachable node from current vertex is reviewed, and only the closer-to-the-query node is expanded in the next round. The algorithm stops when none of the neighbors of the current node is closer to the query. The number of revised objects will be proportional to the diameter of the graph times the average degree of the nodes. Unfortunately the degree of a proximity graph is unbounded for a general metric space [1], and hence the number of inspected objects can be linear on the size of the database, which is the same as no indexing at all. In this paper we introduce a quasi-proximity graph induced by the all-knearest neighbor graph. The degree of the above graph is bounded but we will face local minima when running the above greedy algorithm, which boils down to have false positives in the queries. We show experimental results for high dimensional spaces. We report a recall greater than 90% for most configurations, which is very good for many proximity searching applications, reviewing just a tiny portion of the database. The space requirement for the index is linear on the database size, and the construction time is quadratic in worst case. Relaxations of our method are sketched to obtain practical subquadratic implementations.

2006

The reverse k-nearest neighbor (RkNN) problem, i.e. finding all objects in a data set the k-nearest neighbors of which include a specified query object, is a generalization of the reverse 1-nearest neighbor problem which has received increasing attention recently. Many industrial and scientific applications call for solutions of the RkNN problem in arbitrary metric spaces where the data objects are not Euclidean and only a metric distance function is given for specifying object similarity. Usually, these applications need a solution for the generalized problem where the value of k is not known in advance and may change from query to query. However, existing approaches, except one, are designed for the specific R1NN problem. In addition -to the best of our knowledge -all previously proposed methods, especially the one for generalized RkNN search, are only applicable to Euclidean vector data but not for general metric objects. In this paper, we propose the first approach for efficient RkNN search in arbitrary metric spaces where the value of k is specified at query time. Our approach uses the advantages of existing metric index structures but proposes to use conservative and progressive distance approximations in order to filter out true drops and true hits. In particular, we approximate the k-nearest neighbor distance for each data object by upper and lower bounds using two functions of only two parameters each. Thus, our method does not generate any considerable storage overhead. We show in a broad experimental evaluation on real-world data the scalability and the usability of our novel approach.

XXII Congreso Argentino de Ciencias de la Computación (CACIC 2016)., 2016

Given a collection of objects in a metric space, the Nearest Neighbor Graph (NNG) associate each node with its closest neighbor under the given metric. It can be obtained trivially by computing the nearest neighbor of every object. To avoid computing every distance pair an index could be constructed. Unfortunately, due to the curse of dimensionality the indexed and the brute force methods are almost equally inefficient. This bring the attention to algorithms computing approximate versions of NNG. The DiSAT is a proximity searching tree. It is hierarchical. The root computes the distances to all objects, and each child node of the root computes the distance to all its subtree recursively. Top levels will have accurate computation of the nearest neighbor, and as we descend the tree this information would be less accurate. If we perform a few rebuilds of the index, taking deep nodes in each iteration, keeping score of the closest known neighbor, it is possible to compute an Approximate NNG (ANNG). Accordingly, in this work we propose to obtain de ANNG by this approach, without performing any search, and we tested this proposal in both synthetic and real world databases with good results both in costs and response quality.

Proceedings of the 12th International Conference on Extending Database Technology Advances in Database Technology - EDBT '09, 2009

In this paper, we propose an original solution for the general reverse k-nearest neighbor (RkNN) search problem. Compared to the limitations of existing methods for the RkNN search, our approach works on top of any hierarchically organized tree-like index structure and, thus, is applicable to any type of data as long as a metric distance function is defined on the data objects. We will exemplarily show how our approach works on top of the most prevalent index structures for Euclidean and metric data, the R-Tree and the M-Tree, respectively. Our solution is applicable for arbitrary values of k and can also be applied in dynamic environments where updates of the database frequently occur. Although being the most general solution for the RkNN problem, our solution outperforms existing methods in terms of query execution times because it exploits different strategies for pruning false drops and identifying true hits as soon as possible.

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.