Towards Digital Earth
— Proceedings of the International Symposium on Digital Earth
Science Press ,1999
1
Dynamic Spatial Indexing Model Based on Voronoi
Xuesheng Zhao1,2 Jun Chen2
Renliang Zhao2,3
1
China University of Minging and Technology (Beijing) D11 Xueyuan Road, Beijing, China, 100083
E-mail: [email protected]
2
National Geomatics Center of ChinaNo1. Zizhuyuan, Baishengcun, Beijing, China, 100044
E-mail: [email protected]
3
Central South of University of Technology, 410083
E-mail: [email protected]
ABSTRACT This paper presents a dynamic spatial-data index model that enhances the spatial-data indexing efficiency
of hierarchical tree with Voronoi data structure. In “global GIS”, the Earth data is very large and changed continuously in
local areas. Current GIS researches represent data model based on either raster or vector models. In these models
hierarchy of space and object are separately handled and can be hierarchically grouped using different kinds of tree
structure (such as B-tree, kd-tree, Qua-tree, R-tree and R+-tree, etc). Different users may be careful for different areas on
different hierarchy. How to pick-up data of the interested area in run time is a key problem in “the global GIS”? The major
limitation of current spatial-data hierarchical index models lacks of the spatial relationships (such as adjacency, etc.)
among the same level. When spatial objects are dynamic changed in a spatial process, maintenance of spatial-data
index among spatial objects at different of hierarchy are difficult. To solve this problem, Voronoi data model is introduced
in this paper. Voronoi spatial model combines the features of both vector (object modeling) and raster (field modeling)
model. Hierarchical index that organizes both the object and space can be integrated in the Voronoi spatial model.
Additionally, Voronoi structure has a feature of stability on the local dynamic change. This paper demonstrates how to set
up dynamic spatial-data index mechanism by combining Voronoi spatial structure and hierarchical tree.
KEY WORDS VR-tree, spatial indexing, Voronoi structure, hierarchical partition
1. Introduction
As the development of data-acquisition technique,
more and more data about the Earth can be
picked-up by various methods, such as, Space
shuttle, satellite Remote Sense (RS), the Global
Positioning System (GPS), Interfere Radar, Aerial
photograph, Ocean Drilling, etc. Most of them are
storage in electric-warehouse and not utilized. One
of reasons is related to the mode of data deal and
expression [Gore A，1998]. Faced with so large
quantity of earth-data, how to set up efficient
spatial-data indexing structure will be one of the key
problems in “global GIS”.
The conception “Hierarchy” is often used when
the efficient data-index is to be set up. Hierarchy,
which is used in managing and understanding
complexity, is an important concept in data-index.
Part-whole relations among objects at different
levels maintain a well-known hierarchical structure.
Part-whole structure progressively decomposes a
complex object into simpler components of different
levels. Hence, part-whole structure implies
descending orders of complexity [William, 1981].
For the dynamic Indexing structures, there is a
great different between aspatial data and spatial
data in a GIS. Spatial relationships exist among the
objects and it is possible to represent some of the
frequently referenced relationships using specially
constructed relations, parts of existing relations or
by maintaining conventional links. However,
although it is possible to prematerialize some
spatial relationships in this manner, it is not
pragmatic to store all such relationships explicitly.
Consequently, the dynamic evaluation of spatial
relationships is necessary.
Conventional database management systems are
able to process aspatial selection criteria efficiently,
however, they are not well suited to the task of
efficient evaluation of spatial relationships. Some
sort of spatial indexing mechanism must be
supported. Without a spatial index, a query such as
“find all objects that within a radius of 5km of
Tian-an-men Square” may require a search of the
whole database. This will be grossly inefficient
compared to retrieving only objects in the vicinity of
Tian-an-men Square; and a spatial indexing
mechanism based on proximity can be used to
prune the search space in this manner. Many
structures have been proposed for spatial indexing,
and a detailed survey is unnoticed in this paper. As
a result of analyzing the strengths and weaknesses
of existing structures, we propose a new data
structure model for spatial index, known as the
spatial V-tree. This new structure model is based on
the hierarchical tree and Voronoi data model.
The Voronoi diagram has many excellent
properties. It seems three main reasons are
responsible. First, Voronoi diagrams arise in
nutures in various situations. Indeed, several
national processes can be used to define particukar
classes of Voronoi diagrams. Second, Voronoi
diagrams
have
interesting
and
surprising
mathematical properties. Finally, Voronoi diagrams
have proved to be a powerful tool in solving
seemingly unrelated computational problems
[F.Aurenhammer, 1991].
This paper presents our work in modeling
2
Xuesheng Zhao et al./Dynamic Spatial Indexing Model Based on Voronoi
hierarchical structure of spatial-data dynamic
indexing using hierarchical tree and Voronoi data
model. Section 2 of this paper presents the current
approaches in hierarchical partition of space and
objects in vector and raster model and their
limitations in spatial data hierarchical index. Section
3 analyses the advantages and disadvantages of
Voronoi structure and hierarchical tree in dynamic
hierarchical indexing structure. The fourth section
presents the methods and applications of
combining Voronoi spatial model and hierarchical
tree in modeling hierarchical structure of spatial
dynamic processes. The final section is the
summations and conclusions.
Fig.1 Spatial object and the corresponding
spatial hierarchy described by quad-tree

a
2. The Limitations of Hierarchical Patition in
Vector and Raster Model
Volume Earth data can be hierarchically organized
into different levels of objective, for example, local,
regional and national levels. There is beneficial for
GIS
to
organize
spatial
objects
into
hierarchical-index structures in solving these
problems. Generally, raster and vector approaches
for representing hierarchical spatial structure are
commonly used in GIS.
2.1. Hierarchical Spatial Structure in Raster Model
The raster model is based on hierarchically
decomposition of space. Quad-tree representation
of region, proposed by Klinger [1971], has been the
focus of research [D.Abel, J.Smith, 1984, J.Martin,
1982,] in the fields of both image processing and
database management. The quad-tree is a variant
of
the
maximal
block
representation:
a
representation where the blocks must be disjoint
and have a standard size, which is a power of two.
These characteristics allow a systematic way for
representing homogeneous parts of an image and
are known as regular decomposition. The repetitive
pattern of partition of a quadrant into subquadrant
enables quad-trees to represent images of any
desired degree of resolution. The properties of
the quad-tree make the data structure very suitable
for image processing.
2.2. Hierarchical Spatial Structure in Vector Model
The vector model approach of hierarchical partition
is based on recursively organizing of spatial objects.
Object-oriented and deductive modeling are
commonly used in this approach. Hierarchical
structures (consist of, contained in, shape
simplification) are used to connect layers of
different representations [M.Pang & W.Shi 1998].
The object-oriented framework represents the
spatial objects obtained from different sources and
scales. Hierarchies of spatial objects are divided
into object hierarchy and geometrical object
I
e
c
III

II
d

f
Points:

Arcs:
abcdef
Polygons:
I, II, III
b
Fig.2 Hierarchical Spatial Structure in Vector
Model based on node-arc-polygon structure
hierarchy. Relations between the two hierarchies
are maintained using an object directory. Objects in
regional and classification hierarchies are organized
by spatial (e.g. sub-divide by, contained of, includes
of) and aspatial relations (e.g. a kind of, place class)
respectively.
2.3. Limits of Hierarchical Structure on Vector and
Raster Model
The major limitation of current hierarchical partition
approaches is the separation of hierarchy for space
segmentation and spatial objects [Edwards, 1993],
raster partition-based hierarchy (e.g. quad-tree,
hierarchical TIN) provides a natural representation
of hierarchical space. This approach maintains the
spatial approach describes hierarchical space as a
collection of fundamental units instead of spatial
objects (i.e. point line, and area). If a single spatial
object is changed, the entire hierarchical structure
has to be re-organized (figure 3) [M.Pang & W.Shi
1998]. Therefore, this approach is inappropriate in
modeling hierarchical spatial-data index in which
spatial data are very large and continuously
changing in local areas.
Grouping and organizing spatial objects
according to some defined relations produces
hierarchical structure of spatial objects on vector
model. In this case, changes are referred to spatial
objects themselves; hence, hierarchy of spatial
object does not suffer object is maintained using
explicitly defined relations among spatial objects
instead of recursive decomposition of space. Since
Xuesheng Zhao et al./Dynamic Spatial Indexing Model Based on Voronoi
space is not hierarchically organized, there is
nospatial connection among space at different level
3
these discrete points.
The another important feature of Voronoi
P
Fig.3 The change of spatial object and the
corresponding spatial hierarchy described by
quad-tree [M.Pang & W.Shi 1998]
of hierarchy. When a spatial object changes at
oneparticular level, these changes cannot be
propagated to its adjacent levels [M.Pang & W.Shi
1998]. Hence, hierarchy of spatial objects is still an
unsuitable solution to dynamic hierarchical spatial
index structure.
3. Analyzing Voronoi Structure and R-Tree in
Dynamic Spatial Indexing.
From above, we can see that both the hierarchy of
space and the hierarchy of spatial objects are
inadequacy in modeling dynamic spatial indexing. A
preferred spatial model is one, which can combine
both the hierarchies of space and object. The
spatial model should not only represent spatial
objects as entire objects but also propagate spatial
changes across different levels of hierarchy. Spatial
model that extended from vector and raster model
must be the preferred choice. Voronoi spatial model
is a better alternative to spatial process modeling
since the model posses an exciting feature that
combines the advantages of vector (object
modeling) and raster model (space modeling) [Gold
& Edwards 1992]. In this section, essential features
of Voronoi spatial model as well as hierarchical tree
(R-tree as an example) based on dynamic spatial
indexing are described. Consequently, applications
of Voronoi spatial model in extending R-tree based
model for hierarchical spatial-data index modeling
will be presented.
3.1. Voronoi Structure on Spatial Dynamic Indexing
and Search
The concept, “Voronoi”, is more than a century old,
discussed in 1850s by Dirichlet and in a 1908 paper
of Voronoi [Aurenhammer, 1991]. Voronoi diagrams
have proved to be a powerful tool in solving
seemingly unrelated computational problems. For
an example, the discrete points seem to have no
relations in planar, but from Voronoi diagram (such
as Fig 4) we can know the proximity relationships of
Fig.4 Voronoi diagram of discrete points
and point dynamic inserting.
diagrams is local stability of the topological
structure
under sufficiently small continuous motions of
underlying sites. The constrict functions are as
follows [T.Roos, 1991]:
Additionally, the function INCIRCLE (Pi,Pj,Pk,Pl)
is cyclically alternating. Form the functions, we can
know that local stability of the topological structure
under sufficiently small continuous motions of
underlying sites. When the topological events
happen (such as fig 4: insert a point P), only the
adjacent Voronoi structures are changed, the others
are stability. In the following section, we will use this
property to set up the dynamic spatial indexing.
Designing an additional spatial indexing system
over the data structure and implementing efficient
search algorithms could enhance searching through
the spatial database. Search structures for
Delaunay Triangulated data (the dual of the Voronoi
diagram) have been extensively discussed in the
area of computational geometry. For an incremental
construction approach, which random input order, it
is accepted that an irregular search tree can be built
while the triangulation is processing [Watson 1994].
A simple walk through the triangles can locate the
nearest neighbor in a hardly noticeable time. An
algorithm for the range search can also be
developed without additional data structure [W.Yang
& C.Gold, 1994].
The Voronoi diagram contains sufficient
topological relationships among spatial objects from
which any spatial queries can be answered at run
time. This information rich structure comes with an
extra overhead of storage space, representation
complexity and computation time because each
point and line segment needs to be fully structured
in the diagram. For a heavily populated map
composed of complex curves, a flat organization of
spatial objects could result in an occupation of large
memory and
awkward
performance.
This
4
Xuesheng Zhao et al./Dynamic Spatial Indexing Model Based on Voronoi
shortcoming has motivated research on a compact
Voronoi diagram for realistic GIS arcs and
x pi
y pi
x
y pj
INCIRCLE ( Pi, Pj, Pk , Pl ) : pj
x pk
y pk
x pl
y pl
ccw  pi , pj , pk 
polygons.In a word, the advantages of Voronoi data
x 2pi  y 2pi
x
 y
x
 y
2
pj
2
pk
2
pl
x
x pi
y pi
1
: x pj
y pj
1
x pk
y pk
1
 y
2
pj
2
pk
2
pl
1
1
1
(Eq.1)
1
(Eq.2)
These two functions possess the following properties:
1
2 
3
4 
INCIRCLE Pi, Pj, Pk, Pl   0  Pi, Pj, Pk and Pl are cocircular

 left 
INCIRCLE Pi, Pj, Pk , Pl  0  Pl lies to the
 of the oriented circumcirc le Cijk

right 
CCW Pi, Pj, Pk ,  0  Pi, Pj, and Pk are collinear

 left 
CCW Pi, Pj, Pk  0  Pk lies to the
 of the oriented line lij

right 
structure in dynamic process are as follows:
Advantages:
 posses spatial adjacent relationship
 search scope quickly
 Point-set arithmetic is easy & data structure
issimple.
 Dynamic stability of data structure.
Disadvantages:
 require large storage space
 Arithmetic of curves or areas is very difficult.
3.2. The R-Tree on Dynamic Spatial Index and
Search
The R-tree [B.Ooi, 1990] is a multi-dimensional
generalization of the B-tree, and hence the tree is
height-balanced. Like the B-tree, nodes splitting
and merging are required for inserting and deleting
objects.
In order to locate all objects, which intersect a
query rectangle, the search algorithm descends the
tree from the root. For all rectangles in a non-leaf
node that intersect with the search object, the
corresponding child-pointer becomes the root of a
subtree that will be searched subsequently.
To insert an object, the tree is traversed and all
the rectangles in the current non-leaf node are
examined. The constraint of least coverage is
employed to insert an object: the rectangle that
needs least enlargement to enclose the new object
is selected, the one with smallest area is chosen if
more than one rectangle meets the first criterion.
The nodes in the subtree indexed by the selected
entry are examined recursively. Once a leaf node is
obtained, a straightforward insertion is made if the
leaf node is not full.
In a word, the advantages of R-tree in dynamic
process are as follows:
Advantage:
 Simple structure
 Query exactly and quantification
Disadvantage:
 Lack of spatial relationship at same hierarchy.
 Maintenance the structure stability is difficult in
local dynamic process.
 Whole structure will be reconstructed when
local objects change.
4. Set Up Dynamic Indexing Model by
Combining R-Tree and Voronoi
Voronoi diagram and hierarchical tree both has
advantages and disadvantages separately. It is
necessary to combine their advantages to deal with
the volume Earth data. The main idea of VR-tree is
5
Xuesheng Zhao et al./Dynamic Spatial Indexing Model Based on Voronoi
R2
R1
that: R-tree is regarded as basic structure. Use
point-Voronoi to set up the adjacent relationship of
objects on the upper level of R-tree. This can speed
up the spatial indexing and maintenance the
stability of the spatial indexing structure when local
objects move. A simple example is as follow [Fig 6
and Fig 7]. If the province A break out flood, which
province is to be danger next? All provinces must
be search through if R-tree is used only. But using
1-order adjacent relationship of the Voronoi diagram,
only neighbor province (such as province
B,C,I,J,L,M) need to be searched first. The others
can be ignored.
Be noticed, the Voronoi edges and the province
boundary are not must be matched. The points
which Voronoi diagram based are capitals of
provinces. If the province boundaries are very
irregular (for example, long and narrow), it is
possible that boundaries adjacent between two
provinces and their Voronoi edges are dis-adjacent.
That requires search the neighbor’s neighbors. That
is the second adjacent search.
K
O
J
L
H
I
N
G
A
M
F
B
C
Q
P
E
D
Legend
Capital of province
Province boundary
Voronoi edge
Fig.6 Voronoi diagram based on some
province-capitals along the Changjiang River
in China.
R-tree
A
B
C
D
E
F
G
Adjacent
relationship
Fig. 7 The structure diagram of VB-tree
…
…
r1
r4 P3
p1
P4
p2
R3
p10
R6
R4
R5
R7
r3
r2
p7
p9
r5
p8
p6
p5
R1 R2
R3 R4 R5
R6 R7
r1 p1 p2 r31p7 p8 r2 p9 p10 r4 p3 p4 r5 p5 p6
Fig.5 The planar representation and
structure of an R-tree [B.Ooi, 1990]
5. Conclusion
In “global GIS”, large quantity of and changed data
need to be deal with. A query cannot be processed
efficiently without efficient indexing mechanisms.
Like conventional databases, global database
requires dynamic indexing structures to provide fast
access to volume Earth databased on hierarchical
tree and Voronoi spatial proximity. Current
approaches in hierarchical partition of space and
objects in vector and raster model Indexing
structures were reviewed, and their weaknesses
identified. In the following, analyzed the advantages
and disadvantages of Voronoi structure and
hierarchical tree (R-tree as an example) in dynamic
hierarchical indexing structure, and presents the
idea and method of combining Voronoi spatial
model and hierarchical tree in modeling hierarchical
structure of spatial dynamic processes. The final
gave an example to illuminate the principle of
VR-tree structure.
Acknowledgements
This research was supported by the National
Science Foundation of China under grant No.
69833010.
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