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The TPRTree: An Optimized SpatioTemporal Access Method for Predictive Queries

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Yufei Tao , Dimitris Papadias , Jimeng Sun. Department of Computer Science ... picks the sub-tree incurring the minimum penalty (smallest MBR/VBR enlargement) ... – PowerPoint PPT presentation

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Title: The TPRTree: An Optimized SpatioTemporal Access Method for Predictive Queries

1
The TPR-Tree An Optimized Spatio-Temporal
Access Method for Predictive Queries
Sun Department of Computer Science City
University of Hong Kong, Hong Kong University of
Science and Technology

2
Outline
• Problem definition and related work
• The TPR-tree
• Motivation
• The TPR-tree
• Experiments
• Summary

3
Problem definition
• The database stores the motion functions of
moving objects.
• For each object o, its motion function gives its
location o(t) at any future time t.
• A predictive window query
• specifies a query region qR and a future time
interval qT
• retrieves the set of all objects that will fall
in qR during qT.
• Examples
• Find all airplanes that will be over California
in the next 10 minutes.
• Report all vessels that will enter the United
States in the next hour.

4
Motion function
• We consider linear motion.
• For each object, the database stores
• Its minimum bounding rectangle (MBR) at the
reference time 0
• Its current velocity bounding rectangle (VBR)
• Examples MBR(a)2,4,3,4, VBR(a)1,1,1,1
MBR(c)8,9,8,9, VBR(c)-2,0,-2,0
• An update is necessary only when an objects VBR
changes.

5
Goal and related work
• Goal Index moving objects so that a predictive
window query can be answered with as few I/Os as
possible.
• Related work
• Theoretical Kollios et al. PODS99, Agarwal et
al. PODS00, Procopiuc et al. ALENEX, 2002
• Practical
• Tayeb et al. The Computer Journal, 1998 1D
objects
• Saltenis et al. SIGMOD00 The TPR-Tree
• Saltenis et al. ICDE02 Extension of TPR-Trees

6
The Time Parameterized R-Tree Saltenis et al.
2000
• Extends the R-tree by introducing the velocity
bounding rectangle (VBR) in non-leaf entries.
• Queries are compared with conservative MBRs of
non-leaf entries.

7
Contribution of this paper
• A mathematical model that estimates the cost of
answering a predictive window query using
TPR-like structures.
• Number of node accesses.
• Application of the model to derive the optimal
performance.
• The TPR-tree is much worse than the optimal
structure.
• We scrutinize the algorithms of the TPR-tree,
identify their deficiencies, and propose new
ones.
• The TPR-tree.

8
TPR deficiency 1 Choosing sub-tree to insert
• To insert an entry, the TPR-tree picks the
sub-tree incurring the minimum penalty (smallest
MBR/VBR enlargement).
• May result in inserting an entry into a bad
sub-tree this problem is increasingly serious as
time evolves.

9
TPR solution Choose path
• Aims at finding the best insertion path globally,
namely, among all possible paths.
• Observation We can find this path by accessing
only a few more nodes (than the TPR-tree
algorithm).

Maintain a heap (g),0, (h),0, (i),20
the path expanded so far
the accumulated penalty so far
10
TPR solution Choose path
• Aims at finding the best insertion path globally,
namely, among all possible paths.
• Observation We can find this path by accessing
only a few more nodes (than the TPR-tree
algorithm).

Visit node g (h),0, (a,g),3, (i),20,
(b,g),32
complete paths already although nodes a and b are
not visited
11
TPR solution Choose path
• Aims at finding the best insertion path globally,
namely, among all possible paths.
• Observation We can find this path by accessing
only a few more nodes (than the TPR-tree
algorithm).

Visit node h (a,g),3, (d,h),9, (c,h),17,
(i),20, (b,g),32
The algorithm stops now.
12
TPR deficiency 2 Which entries to re-insert
• When a node overflows, some of its entries are
re-inserted to defer node split (the ones that
diverge most from the node centroid).
• The entries chosen by the TPR-tree are very
likely to be re-inserted back to the same node,
so that a node split is still necessary.

13
TPR solution Pick worst
• Aims at selecting entries that can most
effectively shrink the MBR or VBR of the node
for re-insertion.
• The first step picks an appropriate dimension
(either spatial or velocity) based purely on
estimation using our cost model (see the paper
for details).
• The second step performs sorting on this
dimension and decides the entries to be removed .
• Example If the axis chosen in the first step is
the x-axis, then the sorting list is b,d,a,c.
Either b or c is removed.

14
TPR deficiency 3 Tightening MBR in deletion
• Entry deletion requires first finding the entry,
which accesses many nodes of the tree. The
TPR-tree uses this fact to tighten the MBR of
non-leaf entries.
• Assume nodes h and i are accessed before e is
found then the TPR-tree will tighten the MBR of
i only (enclosing g and f).

15
TPR deficiency 3 Tightening MBR in deletion
• Entry deletion requires first finding the entry,
which accesses many nodes of the tree. The
TPR-tree uses this fact to tighten the MBR of
non-leaf entries.
• Assume nodes h and i are accessed before e is
found then the TPR-tree will tighten the MBR of
i only (enclosing g and f).

16
TPR solution Active tightening
• Assume nodes h and i are accessed before e is
found then the TPR-tree will tighten the MBR of
both h and i.

17
TPR solution Active tightening
• Assume nodes h and i are accessed before e is
found then the TPR-tree will tighten the MBR of
both h and i.

18
TPR solution Active tightening (Cont.)
• Another example Assume the shaded nodes are
accessed to find e.
• The active tightening can tighten the MBR of n5,
n6, n3, and n4.
• But not n1 and n2.

19
Experiments Settings (data)
• We generate synthetic datasets simulating flight
traffic.
• The 2D data space has length 10000 on each axis.
• Sample 5k points from a real dataset to act as
airports.
• At timestamp 0, the positions of 100k aircrafts
(each represented as a point) are generated
randomly at these airports.
• Each airplane is associated with a destination
airport, and a velocity value uniformly obtained
in 20,50.
• After an airplane reaches the destination, a new
destination airport is assigned and a new
velocity is generated. Accordingly, the database

20
Experiments Settings (query and tree)
• We compare TPR- with TPR-trees.
• Disk size1k bytes (node capacity27 for both
trees).
• For each object update, perform a deletion
followed by an insertion on each tree.
• The heights of both trees remain 4, and the
number of leaf nodes remains around 5400.
• Each predictive query is a moving rectangle, and
has these parameters
• qRlen The length of the querys MBR
• qVlen The length of the querys VBR
• qTlen The number of timestamps covered.

21
Result 1 Query cost vs. number of updates
qRlen100, qVlen5, qTlen50
qRlen1600, qVlen5, qTlen50
22
Result 2 Query cost vs. number of updates
qRlen400, qVlen0, qTlen50
qRlen400, qVlen10, qTlen50
23
Result 3 Query cost vs. number of updates
qRlen400, qVlen5, qTlen1
qRlen400, qVlen5, qTlen100
24
Result 4 Update cost vs. number of updates
• Observation 1 The update cost is dominated by
that of deletion.
• Observation 2 The (more complex) TPR-tree
update algorithm pays off as time evolves,
because it leads to a better overall structure.

25
Summary
• The TPR-tree combines the idea of conservative
MBR directly with the tree construction
algorithms of R-trees.
• The TPR-tree improves it by designing algorithms
that take into account the special features for
moving objects.
• In the paper
• Performance analysis
• The optimal performance of a hypothetically best
structure
• Complete experiments (particularly, the
comparison of TPR- and TPR-trees with the best
possible performance)