A Data Structure for Representing and Efficient Querying Large Scenes of Geometric Objects: MB* Trees

Abstract

Abstract. We are concerned with the problem of partitioning complex scenes of geometric objects in order to support the solutions of proximity problems in general metric spaces with an efficiently computable distance function. We present a data structure called Monotone Bisector* Tree (MB* Tree), which can be regarded as a divisive hierarchical approach of centralized clustering methods (compare [3] and [10]). We analyze some structural properties showing that MB* Trees are a proper tool for a general representation of proximity information in complex scenes of geometric objects.Given a scene ofn objects in d-dimensional space and some Minkowski-metric. We additionally demand a general position of the objects and that the distance between a point and an object of the scene can be computed in constant time. We show that a MB* Tree with logarithmic height can be constructed in optimal O(n log n) time using O(n) space. This statement still holds if we demand that the cluster radii, which appear on a path from the root down to a leaf, should generate a geometrically decreasing sequence.We report on extensive experimental results which show that M B* Trees support a large variety of proximity queries by a single data structure efficiently.

Index-driven similarity search in metric spaces (Survey Article)

“…The next object o i for which to compute d (q, o i ) is chosen as the object in S u (as defined in Section 9.1) having (1) According to their experiments, the best choice is the object with the least lowerbound distance estimate (i.e., item 1), which is the same as used in AESA. Micó et al [1996] proposed a hybrid distance-based indexing method termed TLAESA that makes use of both a distance matrix and hierarchical clustering, thereby combining aspects of LAESA [Micó et al 1994] (see Section 9.2) and the mb-tree [Noltemeier et al 1993] (see Section 6.3). The hierarchical search structure used by TLAESA applies the same variation on the gh-tree as is used in the mb-tree: two pivots are used in each node for splitting the subset associated with the node (based on which pivot is closer), where one of the pivots in each nonroot node is inherited from its parent.…”

“…Eccentricity of children is disadvantageous for pruning because the radii of the covering balls increase as the search hierarchy is descended. The potential for having eccentric children is viewed by some (e.g., Dehne and Noltemeier [1987] and Noltemeier et al [ , 1993) as a drawback of the bisector tree. Hence, it has been proposed to modify the definition of the bisector tree so that one of the two pivots in each nonleaf node n, except for the root, is inherited from its parent node-that is, of the two pivots in the parent of n, the one that is inherited is the one that is closer to each object in the subtree rooted at n. In other words, each pivot will also be a pivot in the child node corresponding to that pivot.…”

“…This approach of storing m 2 ranges should be contrasted with the approach of Kamgar-Parsi and Kanal[1985] andLarsen and Kanal [1986] who only store the m ranges formed by r i,i lo and r i,i hi while in the augmented gh-tree[Kalantari and McDonald 1983;Merkwirth et al 2000;Noltemeier et al 1993], described in Section 6.1, only r i,i hi is stored.…”

The Basic Principles of Metric Indexing

“…This guarantees that no node can have a greater covering radius than its parent. Another relative is the Monotonous Bisector * Tree (MBS * -tree) [74,75]. A more recent relative is the BU-tree [46], which is simply a static BS-tree, built bottom-up (hence the name), using clustering techniques.…”

Similarity Search The Metric Space Approach

“…A variant of the BST, called the Monotonous Bisector Tree (MBT), has been proposed in [Noltemeier et al, 1992b, Noltemeier et al, 1992a. The idea behind this structure is that one of the pivots of each internal node other than the root node is inherited from its parent node.…”