AAU A Trajectory Splitting Model for Efficient Spatio-Temporal Indexing Presented by YuQing Zhang Slobodan Rasetic Jorg Sander James Elding Mario A

AAU

A Trajectory Splitting Model for Efficient

Spatio-Temporal Indexing

Presented by YuQing Zhang

Slobodan Rasetic Jorg Sander James Elding Mario A. Nascimento

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Contents

Introduction1

Background and Motivation2

Trajectory Splitting Methods3

Experimental Results4

Related Work5

Conclusion6

Strengths and Weaknesses7

Relate to my Project8

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Contents

Introduction1




Related Work5

Conclusion6



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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

Introduction1




Related Work5

Conclusion6



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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 approximationsBut also take into account query sizesAdvantages:

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

Introduction1




Related Work5

Conclusion6



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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)

Definition2: E BT(q)=

The overall expected number of I/Os.

);(1

kBqPk T

m

k

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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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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 B i

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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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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

Introduction1




Related Work5

Conclusion6



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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

Results 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.

3 4

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Contents

Introduction1




Related Work5

Conclusion6



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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

Introduction1




Related Work5

Conclusion6



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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

Introduction1




Related Work5

Conclusion6



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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

Introduction1




Related Work5

Conclusion6



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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

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Questions?

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AAU A Trajectory Splitting Model for Efficient Spatio-Temporal Indexing Presented by YuQing Zhang Slobodan Rasetic Jorg Sander James Elding Mario A