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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
Yufei Tao, Dimitris Papadias, Jimeng
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
    for known updates.

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
  • Tightening more entries for free.
  • 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
  • Tightening more entries for free.
  • 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
    updates its information.

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)
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