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A Trajectory Splitting
Model for Efficient
Spatio-Temporal
Indexing
Presented by YuQing Zhang
 Slobodan Rasetic Jorg Sander James Elding Mario A. Nascimento
AAU
Contents
1
Introduction
2
Background and Motivation
3
Trajectory Splitting Methods
4
Experimental Results
5
Related Work
6
Conclusion
7
Strengths and Weaknesses
8
Relate to my Project
2
AAU
Contents
1
Introduction
2
Background and Motivation
3
Trajectory Splitting Methods
4
Experimental Results
5
Related Work
6
Conclusion
7
Strengths and Weaknesses
8
Relate to my Project
3
AAU
Introduction
 Problem
 Splitting trajectories optimally with the goal of minimizing the expected number of
I/Os
 Focus on
 Spatio-temporal queries over historical trajectory data Using index structures that
use MBRs
 Past Solutions
 A single MBR
 Each line segment an MBR
 Split trajectories and the resulting sub-trajectories independently by MBRs
 Main contributions
 An analytical cost model and a dynamic programming solution for splitting a given
set of trajectories optimally (in terms of expected I/Os).
 Another cost model and algorithm for segments updated incrementally
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Contents
1
Introduction
2
Background and Motivation
3
Trajectory Splitting Methods
4
Experimental Results
5
Related Work
6
Conclusion
7
Strengths and Weaknesses
8
Relate to my Project
5
AAU
Background and Motivation
 R-tree Review
 Leaf nodes: MBRs of data objects and pointers to the object
 Internal nodes: sequence of pairs of an MBR and a pointer to a child node
 Why spilt trajectories?
 Offer a great potential for improving the performance of sptatio-temporal
range queries.
 Splitting a trajectory → total volume of MBRs↓ → intersect range queries ↓
→ data pages to be retrieved ↓
 The actual amount of volume reduction depends on: the number of splits
and split points
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Background and Motivation
 Trajectory splits and query sizes
Not only minimize the volume of trajectory approximations
But also take into account query sizes
Advantages:
Include past solutions
Offers a potential for tuning the index
An average query size is not restricted to static datasets
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Contents
1
Introduction
2
Background and Motivation
3
Trajectory Splitting Methods
4
Experimental Results
5
Related Work
6
Conclusion
7
Strengths and Weaknesses
8
Relate to my Project
8
AAU
Trajectory splitting Methods - Optimal Trajectory Splitting
 Objectives
 Derive an analytical cost model
Estimate the expected number of I/Os
Yielded by a given split of a trajectory and a given query size
 Introduce an algorithm based on this cost model
 A cost model for splitting trajectories
 Definition1: a trajectory T=<p1,p2,…,pt> with pi=(xi , yi ,ti )
T[u,v]=<pu ,…pv] T=T[1,t]
T is split into m segments(1≦m ≦t-1) :
T=(T[I,i1],…,T[im-1,t]) i1,…im-1 :split positions
BT =(MBR(T[1,i1]),…, MBR(T[im-1,t]))
 The set of MBR approximations of all possible decompositions of T into m segments
Decomp(T,m)={(B1,…,Bm)}, B1=MBR(T[1,i1]),…, Bm=MBR(T[im-1.t]}

q intersects BT→ q intersects k segments →yield k I/Os →P(q∩BT;k)
m
 Definition2: E BT(q)=  k  P ( q  BT ; k )
k 1
The overall expected number of I/Os.
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Trajectory splitting Methods - Optimal Trajectory Splitting
 A cost model for splitting trajectories
 Lemma 1:
P(q∩Bi): the probability that a q intersects the ith segment in BT
 S: the area where a query q can fall
 Extended MBR Extq(Bi)
 Lemma 1:
 Minimizing the performance means finding
 Definition3:
 The minimal number of I/Os over all possible splits:
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Trajectory splitting Methods - Optimal Trajectory Splitting
 Dynamic Programming Algorithm

Object: Solve
Finds the best possible split of T for each value of m


Theorem 1
Proof
↓




Last segment (starting at u) is fixed by assumption, T[1,u] be split into m-1 segments
The sum of volumes of the extended MBRs for the first m-1segments is minimal
The whole sum to be minimal.
Consider all possible values of start positions u in the range 1<u<t for the last segment of T
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Trajectory splitting Methods – Heuristic Trajectory Splitting
 Objective
Find a more efficient and incremental method can produce near optimal results for
trajectories are updated continuously
large datasets containing long trajectories
 A Cost Model for Optimal Segment Size


“Constant-slope trajectories”: segments of equal size
Definition:

Lemma 2:

Definition: C: the number of elementary segments in Bi
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Trajectory splitting Methods – Heuristic Trajectory Splitting
 A Cost Model for Optimal Segment Size

Theorem 2. Given a query q, and increments (△x, △y, △t), the function g has a global real
minimum Copt with respect to c.

Theorem3. Copt is a solution to


Proof: Copt gives smaller f(s) values → ∑f(si) is minimal → By Lemma 2 ,Therorem3
Apply this model to an arbitrary trajectory: compute the average of increments(△x, △y, △t)
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Trajectory splitting Methods – Heuristic Trajectory Splitting
 Linear Time Trajectory Splitting



Object: Apply the Copt method.
Determine a suitable number of points that should be buffered before applying the split policy.
Iteratively collecting points until
becomes true.

Means at least one possible split will result in better I/O expection.

Linear time trajectory splitting Algorithm --- LinearSplit
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Contents
1
Introduction
2
Background and Motivation
3
Trajectory Splitting Methods
4
Experimental Results
5
Related Work
6
Conclusion
7
Strengths and Weaknesses
8
Relate to my Project
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AAU
Experimental Results
 Comparison Between





OptimalSplit: dynamic programming-based algorithm
LinearSplit: linear time algorithm
NoSplit: approximate each trajectory by a single MBR
FullSplit: approximate each line segment of a trajectory individually by an MBR
HKTG: DPSplit, volume oriented split policy
3
 Results
4

Number of Disk I/Os
No matter varying query size and varying database size, the I/O performance of our algorithms
is always significantly better than the others,.
Except FullSplit in varying database size, but its performance degrades much faster with
increasing database size.

Index Building Time
1. Varying query size:
Our algorithms exhibit a good balance between trajectory splitting time and insertion time. As
query size increases, our index building times decreases.
2. Varying Database Size
Our algorithms scale linearly at much slower rate than all other ones.
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Contents
1
Introduction
2
Background and Motivation
3
Trajectory Splitting Methods
4
Experimental Results
5
Related Work
6
Conclusion
7
Strengths and Weaknesses
8
Relate to my Project
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AAU
Related Work
 Most spatio-temporal index structures proposed are based on R-tree.
---Mokbel.. Saptio-temporal Access Methods
 Three main approaches

Time is simply treated as an additional spatial dimension.
---Theodoridis,.. Spatio-Temporal Indexing for Large Multimedia Applications
TB-tree: Insertion split strategy is oriented towards trajectory preservation.
---Pfoser,… Novel Approaches to the Indexing of Moving Object Trajectories Leads to

Inefficient indices and leading to a high degree of overlap among the MBRS.
Time and space are treated differently within a combined indexing scheme.
---Chakke,.. Indexing Large Trajectory Sets with SETI

SEB Tree
---Songs,… An Approach to Index Continuously Moving Objects
They are not compatible with our cost models since they don’t use MBRs.
Time is also treated as differently from space and it is to have virtual and incrementally
maintained 2-dimensional R-trees for each point in time.
---Nascimento,.. Towards Historical R-tree
Suffers from a prohibitively large overhead when indexing very dynamic scenarios, not suite for
trajectory data.
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Related Work
 Recent Work--- improve the first approach

Replace MBRs by different approximation
Trim the corners of trajectories’ MBRs to obtain a bounding octagon prism.
---Zhu,…Discovering Similar Multidimensional Trajectories

Splitting trajectories
Give a total number of allowed splits for a whole set of trajectories to reduce the amount of
approximations’ empty space.
---Hadjieleftheriou,.. Efficient indexing of Spatiotemporal Objects
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Contents
1
Introduction
2
Background and Motivation
3
Trajectory Splitting Methods
4
Experimental Results
5
Related Work
6
Conclusion
7
Strengths and Weaknesses
8
Relate to my Project
20
AAU
Conclusion
 Split trajectories and take into account average query sizes
 A cost model for predicating the number of data page accesses and a
trajectory splitting algorithm
 A linear time splitting algorithm
 The algorithms scale well respect to database size for both query
performance and index building time.
 Future Works


Extending the cost model to better understand the effect of directory level page accesses.
Designing optimized split policies for directory pages of spatio-temporal indices.
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Contents
1
Introduction
2
Background and Motivation
3
Trajectory Splitting Methods
4
Experimental Results
5
Related Work
6
Conclusion
7
Strengths and Weaknesses
8
Relate to my Project
22
AAU
Strengthens and Weaknesses
 Strengthens
 The proof of each lemma and theorem is quite clearly and detailed.
 Good related work
 Weaknesses
 Some definition is not clear.
S and Extq(Bi)
 Less pseudecode.
No pseudecode for dynamic Programming algorithm
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Contents
1
Introduction
2
Background and Motivation
3
Trajectory Splitting Methods
4
Experimental Results
5
Related Work
6
Conclusion
7
Strengths and Weaknesses
8
Relate to my Project
24
AAU
Relate to my Project
 My project
 Compare some methods to index the historical trajectories data by Oracle
 Find a new method to improve the indexing in some aspects.
 Relate…
 Give me a method about the realm I’m researching.
 Give me an optional orientation about my new method.
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Presented by YuQing Zhang
AAU
Questions?
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