A new range query algorithm for Universal B-trees

Abstract

In multi-dimensional databases the essential tool for accessing data is the range query (or window query). In this paper we introduce a new algorithm of processing range query in universal B-tree (UB-tree), which is an index structure for searching in multi-dimensional databases. The new range query algorithm (called the DRU algorithm) works efficiently, even for processing high-dimensional databases. In particular, using the DRU algorithm many of the UB-tree inner nodes need not to be accessed. We explain the DRU algorithm using a simple geometric model, providing a clear insight into the problem. More specifically, the model exploits an interesting relation between the Z-curve and generalized quad-trees. We also present experimental results for the DRU algorithm implementation.

Introduction

In the area of database systems, the emergence of new database forms requires a development of appropriate access methods. Unlike single-dimensional databases, which are indexed/searched according to a simple key (e.g. using B-trees), we often intend to access data according to a composite key (according to several attributes generally). We call such databases multi-dimensional, since data instance is represented by a vector of simple values. A collection of data vectors (data tuples) can be interpreted as set of points in a multi-dimensional vector space.

Let us specify several necessary notations needed for further discussion:

Definition 1 vector space

A discrete vector space $\Omega$ is defined as the cartesian product of finite domains $D_i$, i.e. $\Omega =D_1\times D_2\times \cdots \times D_n$. The vector space $\Omega$ has n dimensions, while each particular domain $D_i$ is associated with the ith dimension of the space. Each point (in the universe $\Omega$) or data tuple (in the database) is represented by a vector $o=[o_1,o_2,\dots ,o_n]$, $o_i\in D_i$.

For the sake of simplicity, we assume that the vector space $\Omega$ is a hyper-cube determined as the nth power of a single domain D, i.e. $\Omega =D^n$, where D is a linearly ordered interval of integers $D=〈0,2^p-1〉$. The cardinality of D is $\left|D\right|=2^p$ for some integer p.

One of the most popular queries required for access to multi-dimensional databases is the range query (also called window query or rectangular query), by which the user specifies an interval of values $〈a_i,b_i〉$ (for each attribute $A_i$), which the retrieved data tuples have to match. The range query can be represented by a hyper-box QB in the space $\Omega$. The ranges of query box QB are defined by two boundary points, the lower bound ${\mathrm{QB}}_{\mathrm{low}}=[a_1,a_2,\dots ,a_n]$ and the upper bound ${\mathrm{QB}}_{\mathrm{up}}=[b_1,b_2,\dots ,b_n]$, where $a_1⩽b_1,a_2⩽b_2,\dots ,a_n⩽b_n$. The purpose of range query is to select all data tuples inside the query box QB, i.e. to select all such tuples o satisfying $a_i⩽o_i⩽b_i$, for $1⩽i⩽n$ (see Fig. 1).

However, for range queries the classic indexing methods, maintaining n single-dimensional indices, are inefficient. For that reason, a class of indexing methods has been developed, called spatial access methods¹ (SAMs), allowing to efficiently index and query multi-dimensional data.

In this paper we introduce a new range query algorithm for the universal B-tree (UB-tree), which is a spatial access method based on the B${}^{+}$-tree and the Z-ordering. With the new algorithm (called the DRU algorithm), we address two issues. First, the existing algorithms are either inefficient or vaguely described. Our approach, on the other side, is deeply described in the geometric as well as in the algorithmic way. Second, the DRU algorithm works more efficiently in high-dimensional indices.

The paper is organized as follows. In Section 2 we overview the basic concepts of UB-tree. The problem of range query processing and some related work is discussed in Section 3. The description of the DRU algorithm is presented in Section 4, while the geometric principles behind the algorithm are explained in Section 5. In Sections 6, 7 the experimental results are analysed and concluded.