The mqr-tree for Very Large Object Sets

Wendy Osborn and Marc Moreau

Department of Mathematics and Computer Science, University of Lethbridge,

4401 University Drive West, Lethbridge, Alberta, T1K 3M4, Canada

Keywords:

Spatial Access Methods, Very Large Object Sets, Performance.

Abstract:

This paper presents an evaluation of the mqr-tree for indexing a database containing a very large number

of objects. Many spatial access methods have been proposed for handling either point and/or region data,

with the vast majority able to handle a limited number of instances of these data types efﬁciently. However,

many established and emerging application areas, such as recommender systems, require the management

and indexing of very large object sets, such as a million places of interest that are each represented with a

point. Using between one and ﬁve million points and objects, a comparison of both index construction and

spatial query evaluation is performed versus a benchmark spatial indexing strategy. We show that the mqr-tree

achieves signiﬁcantly lower overlap and overcoverage when used to index a very large collection of objects.

Also, the mqr-tree achieves signiﬁcantly improved query processing performance in many cases. Therefore,

the mqr-tree is a signiﬁcant candidate for handling very large object sets for emerging applications.

1 INTRODUCTION

Many applications exist that store and manipulate

spatial data such as objects, points and lines. A

spatial database (Samet, 1990; Shekhar and Chawla,

2003; Rigaux et al., 2001) contains a large collec-

tion of objects that are located in multidimensional

space. For example, the Geological Survey of Canada

maintains a repository of spatial data for many geo-

science applications (Geological Survey of Canada,

2006), while the Protein Data Bank (Research Collab-

oratory For Structural Bioinformatics, 2004) contains

many three-dimensional protein structures. More re-

cent applications that utilize spatial data include rec-

ommender systems such as a mobile tourist informa-

tion provider (Hinze et al., 2009; Osborn and Hinze,

2014). Recommender systems must fetch information

for a user on their mobile divide based on their chang-

ing current location. All of the above applications

must manage a signiﬁcantly large amount of data.

Regardless of the application that utilizes spatial

data, an important issue in spatial data management

is the ability to efﬁciently retrieve a subset of objects

from these very large data sets based on their location

by using a spatial access method (i.e., spatial index).

Many spatial access methods have been proposed in

the literature ((Nievergeltet al., 1984; Guttman, 1984;

Sellis et al., 1987; Beckmann et al., 1990; Kamel

and Faloutsos, 1994; Berchtold et al., 1996; Koudas,

2000), see (Gaede and G¨unther, 1998; Rigaux et al.,

2001; Shekhar and Chawla, 2003) for comprehen-

sive surveys). Most proposed spatial access methods

are approximation techniques. They manage objects

of arbitrary shape by utilizing for each a minimum

bounding rectangle (MBR), which estimates the ex-

tent of the object over all of its dimensions. The min-

imum bounding rectangle also works well for point

data (a minimum bounding rectangle of zero area). In

addition, the minimum bounding rectangles are also

used to approximate regions of space that contain ob-

jects. This is done in a hierarchical fashion. The

use of minimum bounding rectangles allows for faster

testing and ﬁltering of objects that will not qualify for

a spatial query, such as a region query or point query.

However, the use of minimum bounding rectan-

gles in approximation-based methods also leads to

some issues. The ﬁrst is the coverage of whitespace

by the index (known as overcoverage). The second is

the multiple overlap of the same region of space by

many minimum bounding rectangles. Both issues can

lead to unnecessary searching in the index structure

for objects that qualify for region and point queries.

These issues may be exacerbated when dealing with

a signiﬁcantly large number of spatial objects - say,

in the millions. Therefore, the evaluation of spatial

access methods with very large object sets is required

to ensure that the efﬁcient retrieval and querying of

spatial data is still being accomplished.

367

Osborn W. and Moreau M..

The mqr-tree for Very Large Object Sets.

DOI: 10.5220/0005463203670373

In Proceedings of the 17th International Conference on Enterprise Information Systems (ICEIS-2015), pages 367-373

ISBN: 978-989-758-096-3

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

= B

> B

= B

> B

Placement

0 0 0 0

0 0 1 0

0 0 0 1

1 0 0 1

0 1 0 0

1 0 0 0

0 1 0 1

0 1 1 0

1 0 1 0

Figure 1: Relative orientation of A with respect to B.

A recently proposed spatial access method, the

mqr-tree (Moreau et al., 2009; Moreau and Osborn,

2012), focuses on minimizing overcoverageand over-

lap when indexing arbitrary objects, while keeping

space utilization at 50% or more. In addition to ef-

ﬁciently handling objects of non-zero area, the mqr-

tree was shown to achieve zero overlap when solely

indexing point data. This leads to point queries that

can be executed without the need to search the same

regions of space several times.

Therefore, we investigate further the ability of the

mqr-tree to efﬁciently index a signiﬁcantly larger set

of objects and to provide a means to efﬁciently per-

form region and point queries on a very large ob-

ject set. A comparison versus a benchmark strat-

egy shows that: 1) the mqr-tree achieves signiﬁcantly

lower overlap and overcoverage, and 2) the mqr-tree

also achieves performance improvements when used

to execute region and point queries. This makes the

mqr-tree a signiﬁcant candidate for indexing the very

large object and point sets required for spatial appli-

cation, and in particular emerging applications such

as recommender systems.

This paper proceeds as follows. Section 2 presents

some background information on the mqr-tree. Sec-

tion 3 overviews the factors that affect the perfor-

mance of spatial access methods. Section 4 presents

the framework and results of our performance eval-

uation. Finally, Section 5 concludes this paper and

provides some directions of future research.

2 BACKGROUND

In this section, we present some background on the

mqr-tree (Moreau et al., 2009; Moreau and Osborn,

2012) approximation-based spatial access method.

Due to space limitations, more details the insertion

and region search strategies can be found in (Moreau

and Osborn, 2012).

The mqr-tree is a hierarchical spatial access

method. Unlike other proposed hierarchical spatial

access methods (Nievergelt et al., 1984; Guttman,

1984; Sellis et al., 1987; Beckmann et al., 1990;

Kamel and Faloutsos, 1994; Berchtold et al., 1996;

Koudas, 2000), the mqr-tree utilizes two-dimensional

nodes. These nodes allow spatial objects and points

to be inserted in a way that maintain the spatial rela-

tionships between them and exploit them when per-

forming spatial queries. Both the minimum bounding

rectangles that represent objects and those covering

regions that contain objects are organized in the same

manner. Another difference between other proposed

structures and the mqr-tree is that the mqr-tree is not a

balanced data structure. However, the cost in balance

is made up by signiﬁcant savings in other cost factors,

such as overcoverage and overlap (see Section 3).

The relative placement of a minimum bounding

rectangle with respect to other minimum bounding

rectangles in the mqr-tree is determined by using the

centroids of their corresponding minimum bounding

rectangles. Figure 1 depicts the deﬁned relationships,

where A refers to the centroid of a new minimum

bounding rectangle, and B refers to the centre of a

node region that contains other objects. The orienta-

tions (NE, SE, SW, NW) are deﬁned to include cen-

troids that fall on the axes (E, S, W, N, respectively).

Also, an equals (EQ) orientation is included, to han-

dle two centroids that overlap.

Figure 2 depicts the two-dimensional layout of a

node. A node contains 5 locations, each of which

correspond to one of the orientations (NE, SE, SW,

NW, EQ) deﬁned above. Each location can contain a

pointer to either an object or another node. A node

must have at least two locations that reference either

an object or a subtree. The origin of the node is its

centre. The centre is deﬁned by the centroid of the

minimum bounding rectangle for the node (called the

node MBR). A node MBR contains the minimum

bounding rectangles for all objects and subtrees that

can be reached from the node. As objects are added

to and removed from the node, the node MBR is ad-

justed to accommodate the new and removed objects.

Also depicted in Figure 2 are three objects that are

indexed by the node. The node MBR is depicted by

the dashed box. Object 1 is located in the NW loca-

tion of the node because its centroid is located north-

west of the centroid for the node MBR. Similarly, the

centroid of Object 2 is located northeast of the cen-

troid of the node MBR, so it is place in the NE lo-

cation of the node. Finally, the centroid of Object 3

is located directly south of the centroid of the node

MBR, so it is placed in the SE location in the node.

Figure 3 (from (Osborn and Hinze, 2014)) de-

picts an mqr-tree that contains points representing

a selection of locations in New Zealand. The root

node, which is highlighted in bold lines on the in-

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368

Figure 2: Node with Objects.

Christchurch

Bay of Plenty

Dunedin

Taupo

Rotorua

Auckland

Hamilton

Wellington

Figure 3: mqr-tree example.

dex, represents the node MBR that encompasses all

other minimum bounding rectangles on the map (and

also highlighted in bold lines). The root node refer-

ences two minimum bounding rectangles with corre-

sponding subtrees,which contain points for commu-

nities on the North Island and South Island respec-

tively. Since the South Island subtree is southwest of

the centroid of the root node MBR, it is placed in loca-

tion SW. Similarly, the North Island subtree is north-

east of the centroid of the root node MBR, and there-

fore is placed in location NE. Dunedin is located SW,

and Christchurch is located NE (respectively) of the

node MBR containing them, and therefore are placed

SW and NE (respectively) in the node. Rotorua over-

laps the centre of the node MBR containing it, and

is placed in location CTR. Finally, the node contain-

ing Wellington also contains references to two other

subtrees.

3 PERFORMANCE FACTORS

In an approximation-based spatial access method, the

performance of any spatial query is affected by differ-

ent factors, including the tree height, space utilization

in the nodes, overcoverage and overlap. We discuss

these performance factors below:

• Average space utilization is the average number of

minimum bounding rectangles per node. Ideally,

the higher the number of minimum bounding rect-

angles per node, the lower the number of nodes

and height of the tree.

• Overcoverage is the amount of whitespace (i.e.,

areas not containing any objects) containin in the

minimum bounding rectangles of a spatial index.

Ideally, this value should be very low. A higher

overcoverage will result in searches along paths

in the index that will lead to no objects.

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369

Table 1: Indexing Object Data

#objs index #nodes height coverage overcoverage overlap sp. util

mqr-tree 576012 11(14) 1119513278 76605868 51278457 55

1,000,000

R-tree 392198 10 4082116307 808197010 793017305 70

mqr-tree 1155177 11(14) 2341881055 153502467 102799437 55

2,000,000

R-tree 783755 10 9551639075 1896621703 1868199636 71

mqr-tree 1730886 12(16) 3600613232 230207810 154241842 55

3,000,000

R-tree 1176238 11 13618377689 2641489767 2596676029 71

mqr-tree 2304733 12(16) 4877583581 306173488 205199188 55

4,000,000

R-tree 1568264 11 19244154175 3817950567 3759439628 71

mqr-tree 2883025 12(16) 6183271262 383337570 256843983 55

5,000,000

R-tree 1959231 11 30882512659 6256387504 6188027877 71

• Overlap is the amount of space covered by two

or more minimum bounding rectangles. Ideally,

overlap should minimal or even zero. Signiﬁcant

amounts of overlap will cause searches to traverse

multiple paths that cover the same region of space,

with many of those times being unnecessary and

leading to no objects.

• Height is the number of nodes from the root to the

leaf node on the longest path of the index. Ide-

ally, the index should be shallow so that all paths

to the leaf nodes require few disk accesses. How-

ever, shorter paths may result in trees with higher

overlap and overcoverage.

4 EVALUATION

In this section, we present the methodology and re-

sults of our empirical evaluation of the mqr-tree for

very large data sets. We compare the mqr-tree against

the R-tree (Guttman, 1984), which is a benchmark for

spatial indexing. We ﬁrst present the overall frame-

work for evaluation, the makeup of the data sets, and

the evaluation criteria. This is followed by the results

of our evaluation on these data sets.

4.1 Data Sets

All of our evaluations were performed using gener-

ated data sets. Altogether, 25 data sets were created -

15 for constructing the indices, and an additional 10

for evaluating the search performance of the mqr-tree.

We used generated data, instead of real-world data, so

that we could control the distribution of the data and

size of the objects in our preliminary evaluations.

We ﬁrst describe the ﬁles for index creation, then

we describe the ﬁles for searching.

• Five data ﬁles contain between 1,000,000 and

5,000,000 points that were uniformly distributed

across a region that was 10 ∗

√

#points

in area.

• The next ﬁve data ﬁles contained between

1,000,000 and 5,000,000 squares that were 10

units each, and were uniformly distributed across

a region of the same size as that used for points.

• The remaining ﬁve data sets for creation contain

between 1,000,000 and 5,000,000 objects which

is a mixture of points and squares.

• In addition, 5 ﬁles were randomly generated to

evaluate the performance of region searching.

Each ﬁle contains 20 search regions that were 10

units each. Each search ﬁle corresponds to a data

set size mentioned above.

• Finally, 5 ﬁles were randomly generated to evalu-

ate the performance of point searching. Similarly,

each ﬁle contains 20 search points and correspond

to a data set size mentioned above.

4.2 Evaluation Framework

We perform evaluations on both the construction and

the search performance of the mqr-tree. The insertion

evaluation framework will be presented ﬁrst, followed

by the framework for evaluating searching.

For both the mqr-tree and R-tree, we used a fanout

of 5 entries per node. This choice is due to the mqr-

tree having a 5-node fanout limit. Reducing the R-tree

to a 5-node fanout would guarantee a fair comparison.

For each data set, we created one mqr-tree and 10

R-trees by repeated insertion of every object in the

data set. Each tree was built using a random ordering

of the object set. The number of nodes, height (both

worst-case and average-case), average space utiliza-

tion in each node, total coverage of all minimum

bounding rectangles, total overcoverage of all mini-

mum bounding rectangles, and the total overlap be-

tween all minimum bounding rectangles was calcu-

lated for each tree. With respect to tree height, the

R-tree is balanced, so therefore all paths are the same

length and therefore the worst-case and average case

height will be the same. The mqr-tree, however, is

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Table 2: Indexing Point Data.

#objs index #nodes height coverage overcoverage overlap sp. util

mqr-tree 575993 11(14) 899829674 99999647 0 55

1,000,000

R-tree 385331 10 3471592309 775006290 675006640 71

mqr-tree 1155314 11(14) 1902010284 200221974 0 55

2,000,000

R-tree 770889 10 8034028888 1761219524 1560997542 71

mqr-tree 1729328 12(14) 2940363153 300328747 0 55

3,000,000

R-tree 1156297 11 12881506927 2814777938 2514449189 71

mqr-tree 2304621 12(16) 3999071988 399999680 0 55

4,000,000

R-tree 1541392 11 21209175398 4670080750 4270081079 71

mqr-tree 2883030 12(16) 5083956406 500416712 0 55

5,000,000

R-tree 1927280 11 26404730270 5829899230 5329482524 71

Table 3: Indexing Mixed Object and Point Data.

#objs index #nodes height coverage overcoverage overlap sp. util

mqr-tree 587209 11(14) 566623167 40418311 29935198 54

1,000,000

R-tree 393018 10 2072809600 418732883 411397435 70

mqr-tree 1178897 11(16) 1181198140 80525151 60011758 54

2,000,000

R-tree 785851 10 4213706283 833688678 819272092 70

mqr-tree 1767303 12(16) 1816095721 120919999 89841540 54

3,000,000

R-tree 1178270 11 7281617131 1469500615 1448862536 70

mqr-tree 2348069 12(16) 2464062205 161339416 119849135 54

4,000,000

R-tree 1570870 11 10140414556 2036878811 2009435617 70

mqr-tree 2932339 12(16) 3119947706 201588891 149992043 54

5,000,000

R-tree 1962555 11 13755635004 2788853267 2755657374 70

not balanced. Therefore, both the longest path and the

average path length will need to be recorded. On all

result tables, this will be indicated by the average path

length, followed by the longest length in parentheses.

For the R-trees, an average for each performance

factor was calculated because each R-tree will vary

based on insertion order. This is not the case for the

mqr-tree - the same index will be created regardless

of the insertion order (Moreau and Osborn, 2012).

In addition to index construction, each index will

be evaluated for the performance of both region

searching and point searching. After each tree is con-

structed, both the corresponding region search and

point search query sets will be applied. The perfor-

mance criteria for both types of search is the number

of nodes that must be accessed (i.e. page hits) in or-

der to ﬁnd all overlapping points or objects (or ﬁnd

the lack thereof). We assume the worst-case scenario

of one disk access per node in our work. After all re-

gion and point queries have been executed, an average

number of page hits is calculated for each query type.

4.3 Index Construction Results

Table 1 depicts the results of the performance eval-

uation for index construction using the object data.

With respect to coverage, overcoverage and overlap,

that the mqr-tree achieves signiﬁcant improvements

over the R-tree for all object set sizes, especially for

overcoverage and overlap. We see an improvement

of between 90-94% for overcoverage, a 93-96% im-

provement for overlap, and a 72-80% improvement

in coverage. In all cases, the improvement in cover-

age, overcoverage and overlap improves as the num-

ber of points in the index increases. The R-tree does

achieves higher space utilization, a lower number of

notes and a lower maximum tree height over the mqr-

tree. However, the following must be noted. First,

the improvement in height of the R-tree over the mqr-

tree is between 21-23%. In addition, the improve-

ment in the number of nodes of the R-tree is approx-

imately 33%. Both are signiﬁcantly lower increases

than those achieved in coverage, overlap and over-

coverage by the mqr-tree. Finally, the mqr-tree still

achieves over 50% space utilization for all cases. Fi-

nally, the average height of the mqrtree, at approxi-

mately 75% of the maximum height, is almost identi-

cal to that for the R-tree. These are more minor sacri-

ﬁces to make for the other signiﬁcant improvements.

Table 2 depicts the performance evaluation results

when point data is used to construct the indices. Be-

cause the mqr-tree is only indexing point data here,

we achieve the expected zero overlap.We also observe

an improvement of between 87-92% in overcoverage

Themqr-treeforVeryLargeObjectSets

371

Figure 4: Search - Indexed Object Data.

Figure 5: Search - Indexed Point Data.

Figure 6: Search - Indexed Mixed Object and Point Data.

and between 74-81% in coverage. It should be noted

here that the values for coverage, overlap and over-

coverage are lower for both the mqr-tree and R-tree

over those found when both structures were index-

ing object data. Also, the space utilization and tree

heights are almost identical to those found for the

mqr-tree and R-tree when object data was indexed.

In Table 3 we present the results of the ﬁnal

construction performance evaluation when the mixed

data set was used to construct the indexes. Here for all

performance factors, we ﬁnd improvement trends that

are similar to those found for the object-indexed data.

The improvement in coverage is between 72-78%,

while the improvements for overcoverage and overlap

are between 90-93% and 92-95% respectively.

A very surprising ﬁnd, however, is with the val-

ues overall when compared to the object-indexed and

point-indexed data above. When indexing data that

is a mix of points and objects, the lowest coverage

and overcoverage are achieved for both the mqr-tree

and R-tree, and the overlap for the R-tree is also the

lowest. It would be expected that a mixed data set

would achieve lower values over the object data set,

but not over the data set consisting entirely of points!

We feel the reason for this lies in how the data is or-

ganized. For a mixed data set, many points may be

enclosed by another object, and therefore will not re-

sult in extra overcoverage or overlap. For a point data

set, because the points have no objects to cover them,

the point set will generate additional whitespace (and

therefore, overcoverage) and possibly overlap as well.

4.4 Search Results

As mentioned, after each index was constructed, each

was evaluated for search performance using 20 region

queries and 20 point queries. Figures 4 to 6 depict

the results of the search evaluation. On all charts,

mbrObj and rObj are the average number of node ac-

cesses (i.e. page hits) required for the mqr-tree and R-

tree, respectively, to process the region queries, while

mqrPoint and rPoint are the average number of node

accesses required for the mqr-tree and R-tree, respec-

tively, to process point queries. Across all ﬁgures, we

see that the mqr-tree has a lower number of node ac-

cesses required for processing all spatial queries. In

particular, for region queries, the mqr-tree achieves

between 30-40% less node accesses than the R-tree.

For point queries, the mqr-tree still outperforms the

R-tree, but the decrease in node accesses are not as

signiﬁcant in many cases. The exception to this is

when a point search is performed in point and mixed

object data when the number of objects in the index

is around 4 to 5 million. Then some more signiﬁcant

results are achieved

We also make the following observations. First,

although the mqr-tree requires a number of node ac-

cess that is larger than the average tree height, it is

always lower than the maximum height. Therefore,

even if extra paths are being searched, it involves very

few nodes. Second, in every case (both for region and

point queries), the R-tree has a number of node ac-

cesses that is larger than its tree height. In particular,

for object queries, the number of node accesses is also

larger than the maximum height for the mqr-tree. This

shows that the R-tree needs to search many paths to

locate required objects that satisfy the queries. This

is the drawback in the R-tree of having a signiﬁcant

amount of overcoverage and overlap. Conversely, this

is the advantage of the mqr-tree of having a signiﬁ-

cantly lower overlap and overcoverage.

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5 CONCLUSIONS AND FUTURE

WORK

In this paper, we present an evaluation of the ability of

the mqr-tree to handle signiﬁcantly large amounts of

object data. We evaluated index construction, region

queries and point queries. Our experiments showed

that the mqr-tree achieved signiﬁcantly lower cover-

age, overcoverage and overlap than the benchmark

R-tree strategy. In addition, the when the mqr-tree

was used for performing region and point queries, the

mqr-tree outperformed the R-tree in all cases, espe-

cially for all region queries.

Some future research include the following. The

ﬁrst is to extend the mqr-tree to support nearest neigh-

bour and continuous spatial queries. The second is

to investigate strategies to decrease the height and in-

crease the node size of the mqr-tree without increas-

ing its overlap and overcoverage. The third is to com-

pare the mqr-tree against other spatial indexes such

as the R

∗

-tree, and also with real-world data sets. Fi-

nally, it is desirable to explore the notion of a dis-

tribute mqr-tree. This would further increase its abil-

ity to index more data.

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