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THE TPR*-TREE: AN OPTIMIZEDSPATIO-TEMPORAL ACCESSMETHOD FOR PREDICTIVE QUERIES

Dimitris Papadias Yufei TaoJimeng Sun

VLDB Conference 2003

Outline

Introduction The TPR-tree and The TPR*-tree Experiments Conclusions

Introduction

Spatio-temporal databases record moving objects’ geographical locations (sometimes also shapes)

at various timestamps. support queries that explore their historical and future (predictive)

behaviors. Applications. applications: flight control systems, weather forecast and mobile

computing

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. our goal: index moving objects so that a predictive window query can

be answered with as few disk I/Os as possible.

Examples Find all airplanes that will be over Florida in the next 10 minutes. Report all vessels that will enter the United States in the next hour.

Motion Function

We consider linear motion.

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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)={8,9,3,4}, VBR(c)={-2,0,0,2}; An update is necessary only when an object’s VBR changes.

The Time Parameterized R-Tree (TPR-Tree)

Extends the R-tree by introducing the velocity bounding rectangle (VBR) in all entries.

Queries are compared with conservative MBRs of non-leaf entries. N1v={-2,1,-2,1} and N2v={-2,0,-1,2}

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TPR*-Tree

Goal: index moving objects so that a predictive window query can be

answered with as few disk I/O as possible. 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.

Exam the algorithms of the TPR-tree, identify their deficiencies, and propose new ones. The TPR*-tree.

TPR*-Tree Insertion

Choose Path Node Insert

Pick Worst (if overflow) Node split

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).

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inserting p at time 2 May result in inserting an entry into a bad sub-tree; this

problem is increasingly serious as time evolves.

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).

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inserting p at time 2

Maintain a priority queue:

[(g),0], [(h),0], [(i),20]

the path expanded so far

the accumulated penalty so far


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Visit node g:

[(h),0], [(a,g),3], [(i),20], [(b,g),32]

complete paths already although nodes a and b are not visited


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Visit node h:

[(a,g),3], [(d,h),9], [(c,h),17], [(i),20], [(b,g),32]The algorithm stops now.

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.

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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) The second step performs sorting on this dimension and

decides the entries to be removed.

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– 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.

TPR* solution: Node Split

Computes the overall perimeter for each dimension

Select the split axis as the smallest overall perimeter

Perimeter defined as the perimeter of the sweeping region of the corresponding transformed rectangle

Perimeter computation is very efficient The number of vertices of a sweeping region is small

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).

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deleting e at time 1


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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.

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deleting e at time 1


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after deleting e at time 1


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.

n1 n2

n5 n6

n3 n 4

root

...

...e

to be writtenback to disk

N1 N2 N3 N4

N5 N6

Challenge of Migration

3 Operating Systems: Microsoft Windows Sun Solaris Redhat Fedora Core 1

2 Compilers: CL, GCC (2.9.5, 3.3.2) Difference of Code Conversion

How close the compilers to the standard? Compatibility of Library

Experiments: Settings (query and tree)

Dataset 50,000 sampled objects’ MBRs are taken from a real spatial dataset NJ [Tiger] each object is associated with a VBR such that on each dimension

The velocity extent is zero (i.e., the object does not changespatial extents during its movement)

the velocity value distribution is randomed in range [0,8] the velocity can be positive or negative with equal probability.

We compare TPR*- with TPR-trees. Disk page size=1k bytes (node capacity=27 for both trees). For each object update, perform a deletion followed by an insertion on each

tree. Each predictive query is a moving rectangle, and has these parameters:

qRlen: The length of the query’s MBR qVlen: The length of the query’s VBR qTlen: The number of timestamps covered.

Conclusions

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. Cost model for performance analysis The optimal performance of a “hypothetically best

structure” Reduce disk I/Os for predictive queries

Science

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