On performance of quadrant-recursive spatial orders

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Abstract

The performance of many computational paradigms can be considerably improved by using appropriate quadrant-recursive spatial orders. The Hilbert order has received intensive interest in literature. Its encoding and decoding processes, however, are time-consuming. It is desired to design new spatial orders that are competitive with the Hilbert order in performance yet require simpler encoding and decoding procedures. In this paper, several new quadrant-recursive spatial orders are proposed. Of them the Q4 order behaves best, and its algorithm is more efficient than the corresponding algorithm of the Hilbert order.

Introduction

In areas of applied and computational mathematics many paradigms are concerned with spatial ordering. One of the typical examples is the finite difference method in solving the ordinary or partial differential equations. The algorithmization of such procedures in a finite computing machine is in nature based on some spatial ordering. In computer applications there are also many paradigms involving spatial ordering, such as range query in database systems [5] and digital halftoning in image processing [6].

Analytically, spatial ordering is a bijection that relates a finite set DnIn to a finite set D1I1, where In and I1 stand for the n-dimensional (n-D) and the one-dimensional (1-D) integer space, respectively. Geometrically, a spatial order determines a linear traversal in In. In general, the locality of a subset of points in In is no longer preserved in I1: the neighboring points in In are scattered in I1. Most applications, however, involve the proximity issue that localizability is desired to be preserved as much as possible. Accordingly, a special class of spatial orders, which recursively exhaust a quadrant of a square before exiting it, has attained much interest. Several instances of this class were described in [9]. Two of the most popular ones are the Hilbert order and the Z order. The Hilbert order is a discrete representation of the Hilbert’s space filling curve [4], while the Z order (also known as the Morton order [11]) is associated the well-known quadtree structure [13].

As mentioned in [9], a parameter called resolution is used to describe the granularity of the domain of a spatial order. For two-dimensional (2-D) cases, a spatial order is of resolution r if it is defined on a image with 2r×2r pixels and small values of r correspond low resolutions. The Hilbert orders of resolution 1–3 and the corresponding space filling curves are depicted in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6. The Z order of resolution 3 is presented in Fig. 7.

The Hilbert order exhibits good performance in many applications [1], [5]. However, its computation is complicated [8]. Therefore, it would be significant to find new spatial orders that are competitive with the Hilbert order in performance yet require relative simple formulas.

In this paper, several new spatial orders with the quadrant-recursive structure are proposed. A metric is adopted to evaluate their performance. Of them the Q4 order behaves best and its performance is very close to the Hilbert order. It is shown that the encoding and decoding algorithms of the Q4 order only need 66.7% and 80.0% of operations of the corresponding algorithms of the Hilbert order. This impressive improvement suggests that the Q4 order has the advantage over the Hilbert order in terms of computational efficiency.

Section snippets

The Hilbert order

An analytical examination is of central importance for understanding the inherent property of a particular spatial order and its significance for applications. In this section, a comprehensive analysis for the Hilbert order, based on the recursive approach, is presented. This approach also provides a framework to evaluate the Q orders proposed in the subsequent sections. To make the problem tractable, the analysis is conducted in the 2-D space.

An appropriate metric needs to be adopted to

The Q1 order

It would be helpful to introduce a few terminology for simplifying the description. Given the coordinates of a point P in the 2-D Cartesian space, encoding is the process that determines the corresponding value of P in a particular spatial order. The inverse process is called decoding. The Hilbert order exhibits desired performance for range query retrieval because the ordering recursively exhausts a quadrant of a square before exiting it, and always proceeds from one cell to its neighbor in

The Q2 order

The Q2 order is described in this section. Its ordering patterns of resolution 2 and 3 as well as the corresponding reference paths are shown in Fig. 13, Fig. 14, Fig. 15, Fig. 16.

The results are narrated as the following theorems. The proof is similar to the Hilbert order.

Theorem 4.1

Let N(1)r be the number of templates with one cluster in the grid of resolution r, thenN(1)r=384r(r⩾2)

Theorem 4.2

Let N(2)r be the number of templates with two clusters in the grid of resolution r, thenN(2)r=184r122r(r⩾2)

Theorem 4.3

Let N(3)r be

The Q3 order

The Q3 order is introduced in this section. Its ordering pattern of resolution 2 and 3 as well as the corresponding reference paths are shown in Fig. 17, Fig. 18, Fig. 19, Fig. 20.

The properties concerned are narrated as a series of theorems. The proof is included since the proving process is slightly more complicated than those needed for the Q1 or Q2 order.

Theorem 5.1

Let N(1)r be the number of templates with one cluster in the grid of resolution r, thenN(1)r+N(1)r−1=23484r23(r⩾2)

Proof

It follows from Fig. 4,

The Q4 order

The configurations of the Q4 order of resolution 2 and 3 are shown in Fig. 21, Fig. 22, Fig. 23, Fig. 24.

The properties concerned are narrated as a series of theorems. In this case, the analysis is simpler than that needed for the Q3 order, thus the proof is not included to save the space. The interested reader may refer to [7].

Theorem 6.1

Let N(1)r be the number of templates with one cluster in the grid of resolution r, thenN(1)r=25644r(r⩾3)

Theorem 6.2

Let N(2)r be the number of templates with two clusters in the

The Q5, Q6 and Q7 orders

An eight-variable integer programming (IP) model can be constructed to determine the optimal spatial order(s) with the quadrant-exhaustive and single-type hierarchical structures. By single-type it means the iteration rule is identical for any resolution. To simplify the analysis, consider the orders developed from the rook-connected path in the system of resolution 1 only (as shown in Fig. 4, for example). Let this path be called the basic path. The sub-ordering of each quadrants in the system

Neighbor proximity

The neighbor proximity problem has been an active research topic in the field of computational geometry. Its solution process serves as a fundamental operation for core modules in many geographical information systems [2]. The problem involves searching a subset of cells with the specified attributes such that the distance between the given subset and the current cell is a minimum. In the literature, various metrics have been proposed for defining the distance [12]. Of them, the City-block

Encoding and decoding the Q4 order

Of the seven Q orders proposed above, the Q4 order behaves best on the average for the 2×2 templates. For proximity problems, the Q4 order also exhibits satisfactory performance. On the other hand, of the seven Q orders, only the Q1 and Q4 orders do not involve iterating the indexing sequence of the basic path. This property implies that their algorithms are simpler than others’ ones. However, the behavior of the Q1 order is not so good as the Q4 order. Thus the Q4 order is chosen as a

Conclusion

Analysis on spatial ordering can be conducted in three areas: algebraic structures, algebraic operations, and performance metrics. Studies in the first and second areas were reported in [9] and [10], respectively. The present work concentrates on performance metrics of a special class of spatial orders that recursively exhaust a quadrant of a square before exiting it. This type of spatial orders has proven to be very useful in many computational paradigms. Of several new quadrant-recursive

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