Location-aware Query Processing and Optimization: A Tutorial

1
Location-aware Query Processing and Optimization
A Tutorial

• Mohamed F. Mokbel Walid G. Aref
• Department of Computer Science and Engineering,
  University of Minnesota
• Minneapolis, MN, USA
• mokbel_at_cs.umn.edu
• Department of Computer Science, Purdue University
• West Lafayette, Indiana, USA.
• aref_at_cs.purdue.edu

2
Motivation
3
Applications Traffic Monitoring

• How many cars are in the downtown area?
• Send an alert if a non-friendly vehicle enters a
  restricted region
• Report any congestion in the road network
• Once an accident is discovered, immediately send
  alarm to the nearest police and ambulance cars
• Make sure that there are no two aircrafts with
  nearby paths

4
Applications (Cont.) Location-based Store Finder
/ Advertisement

• Where is my nearest Gas station?
• What are the fast food restaurants within 3 miles
  from my location?
• Let me know if I am near to a restaurant while
  any of my friends are there
• Send E-coupons to all customers within 3 miles of
  my stores
• Get me the list of all customers that I am
  considered their nearest restaurant

5
Location-based Database Servers
Layered Approach
6
Variety of Location-aware Queries

• Query Stationary
• Object Moving

7
Tutorial Outline

• Location-aware Environments
• Location-aware Snapshot Query Processing
• Snapshot Past Queries
• Snapshot Present Queries
• Snapshot Future Queries
• Spatio-temporal Access Methods
• Location-aware Continuous Query Processing
• Scalable Execution of Continuous Queries
• Location-aware Query Optimization
• Uncertainty in Location-aware Query Processing
• Case Study
• Open Research Issues

8
Location-aware Snapshot Query Processing
Querying the Past

• Examples
• Querying Along the Temporal Dimension What was
  the location of a certain object from 700 AM to
  1000 AM yesterday?
• Querying Along the Spatial Dimension Find all
  objects that were in a certain area at 700 AM
  yesterday
• Querying Along the Spatio-temporal Dimension
  Find all objects that were close to each other
  from 700 AM to 800 AM yesterday
• Features
• Large number of historical trajectories
• Persistent read-only data
• The ability to query the spatial and/or temporal
  dimensions

9
Location-aware Snapshot Query ProcessingIndexing
the Time Dimension

• Historical trajectories are represented by their
  three-dimensional Minimum Bounding Rectangle
  (MBR)

Time

• 3D-R-tree is used to index the MBRs

• Technique simple and easy to implement
• Does not scale well
• Does not provide efficient query support

10
Location-aware Snapshot Query ProcessingMulti-ver
sion Index Structures

• Maintain an R-tree for each time instance
• R-tree nodes that are not changed across
  consecutive time instances are linked together

Timestamp 1

• A multi-version R-tree can be combined with a
  3D-R-tree to support interval queries

11
Location-aware Snapshot Query ProcessingQuerying
the Present

• Time is always NOW
• Example Queries
• Find the number of objects in a certain area
• What is the current location of a certain object?
• Features
• Continuously changing data
• Real-time query support is required
• Index structures should be update-tolerant
• Present data is always accessed through
  continuous queries

12
Location-aware Snapshot Query ProcessingUpdating
Index Structures

• Traditional R-tree updates are top-down
• Updates translated to delete and insert
  transactions
• To support frequent updates
• Updates can be managed in space without the need
  for deletion or insertions
• Bottom-up approaches through auxiliary index
  structures to locate the object identifier

Hash based on OID
13
Location-aware Snapshot Query ProcessingUpdate
Memos

• Keep a memo with the R-tree
• The memo contains the recent updates to the
  existing R-tree
• The query answer returned from the R-tree should
  be passed through the memo
• The update memo is reflected to the R-tree once
  the relevant disk page is retrieved

Spatio-temporal Queries
Raw answer set
Final answer set
14
Location-aware Snapshot Query ProcessingQuerying
the Future

• Examples
• What will my nearest restaurant be after 30
  minutes?
• Does my path conflict with any other cars for the
  next hour?
• Features
• Predict the movement through a velocity vector
• Prediction could be valid for only a limited time
  horizon in the future

15
Location-aware Snapshot Query ProcessingDuality
Transformation

• A line (trajectory) in the two-dimensional space
  can be transformed into a point in another dual
  two-dimensional space
• Trajectory x(t) vt a ? Point (v,a)
• All queries will need to be transformed into the
  dual space
• Rectangular queries will be represented as
  polygons

16
Location-aware Snapshot Query ProcessingTime-Para
meterized Data Structures

• The Time-parameterized R-tree (TPR-tree) consists
  of
• Minimum bounding rectangles (MBR)
• Velocity bounding rectangles (VBR)

• A bounding rectangle with MBR VBR is guaranteed
  to contain all its moving objects as long as they
  maintain their velocity vector
• High degree of overlap when the velocity vector
  is not updated

17
Location-aware Snapshot Query ProcessingIndexing
Past, Present, and Future

• A unified index structure for both past, present,
  and future data
• Makes use of the partial-persistent R-tree for
  past data and the TPR-tree for current and future
  data
• Double Time-Parameterized Bounding rectangles are
  used to bound moving objects. Double TPBR has two
  components
• Tail MBR that starts at the time of the last
  update and extends to infinity. The tail is a
  regular TPBR of the TPR-tree
• Head MBR to bound the finite historical
  trajectories. The head is an optimized TPBR
• Querying is similar to regular PPR-tree search
  with the exception of redefining the intersection
  function to accommodate for the double TPBR

18
Spatio-temporal Access Methods
RPPF-tree
Red Future Blue Past Green Present Brown All
19
Tutorial Outline

• Location-aware Environments
• Location-aware Snapshot Query Processing
• Location-aware Continuous Query Processing
• Continuous Queries Vs. Snapshot Queries
• Approaches for Continuous Query Evaluation
• Scalable Execution of Continuous Queries
• Location-aware Query Optimizer
• Uncertainty in Location-aware Query Processing
• Case Study
• Open Research Issues

20
Snapshot vs. Continuous Query Processing

• Traditional (Snapshot) Queries

Data
21
Location-aware Continuous Query Processing
Approaches

• Straightforward Approach
• Abstract the continuous queries to a series of
  snapshot queries evaluated periodically
• Result Validation
• Result Caching
• Result Prediction
• Incremental Evaluation

22
Location-aware Continuous Query ProcessingResult
Validation

• Associate a validation condition with each query
  answer

• Valid time (t)
• The query answer is valid for the next t time
  units
• Valid region (R)
• The query answer is valid as long as you are
  within a region R

• It is challenging to maintain the computation of
  valid time/region for querying moving objects
• Once the associated validation condition expires,
  the query will be reevaluated

23
Location-aware Continuous Query
ProcessingCaching the Result

• Observation Consecutive evaluations of a
  continuous query yield very similar results
• Idea Upon evaluation of a continuous query,
  retrieve more data that can be used later

• K-NN query
• Initially, retrieve more than k
• Range query
• Evaluate the query with a larger range

• How much we need to pre-compute?
• How do we do re-caching?

24
Location-aware Continuous Query
ProcessingPredicting the Result

• Given a future trajectory movement, the query
  answer can be pre-computed in advance

• The trajectory movement is divided into N
  intervals, each with its own query answers Ai

Nearest-Neighbor Query

• The query is evaluated once (as a snapshot
  query). Yet, the answer is valid for longer time
  periods

• Once the trajectory changes, the query will be
  reevaluated

25
Location-aware Continuous Query
ProcessingIncremental Evaluation

• The query is evaluated only once. Then, only the
  updates of the query answer are evaluated
• There are two types of updates. Positive and
  Negative updates

Query Result

• Only the objects that cross the query boundary
  are taken into account

• Need to continuously listen for notifications
  that someone cross the query boundary

26
Tutorial Outline

• Location-aware Environments
• Location-aware Snapshot Query Processing
• Location-aware Continuous Query Processing
• Scalable Execution of Continuous Queries
• Location-aware Centralized Database Systems
• Location-aware Distributed Database Systems
• Location-aware Data Stream Management Systems
• Location-aware Query Optimizer
• Uncertainty in Location-aware Query Processing
• Case Study
• Open Research Issues

27
Scalability of Location-aware Continuous Queries
Motivation
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Scalability of Location-aware Continuous Queries
Main Concepts

• Continuous queries last for long times at the
  server side
• While a query is active in the server, other
  queries will be submitted
• Shared execution among multiple queries
• Should we index data OR queries?
• Data and queries may be stationary or moving
• Data and queries are of large size
• Data and queries arrive to the system with very
  high rates
• Treat data and queries similarly
• Queries are coming to data OR data are coming to
  queries?
• Both data and queries are subjected to each other
• Join data with queries

29
Scalability of Location-aware Continuous Queries
Main Concepts (Cont.)

• Evaluating a large number of concurrent
  continuous spatio-temporal queries is abstracted
  as a spatio-temporal join between moving objects
  and moving queries

30
Scalability of Location-aware Continuous Queries
Location-aware Centralized Database Systems

• Centralized index structures
• Index the queries instead of data

• Valid only for stationary queries

31
Scalability of Location-aware Continuous Queries
Location-aware Centralized Database Systems
(Cont.)

• To accommodate for the continuous movement of
  both data and queries
• Concurrent continuous queries share a grid
  structure
• Moving objects are hashed to the same grid
  structure as queries
• The spatio-temporal join is done by overlaying
  the two grid structures

32
Scalability of Location-aware Continuous Queries
Location-aware Distributed Database Systems

• Motivation Centralized location-aware servers
  will have a bottleneck at the server side
• Assumption Moving objects have devices with the
  capability of doing some computations
• Idea
• Server will ship some of its processing to the
  moving objects
• Server will act as a mediator among moving
  objects
• Implementation Moving objects should welcome
  cooperation in such environments

33
Scalability of Location-aware Continuous Queries
Location-aware Distributed Database Systems
(Cont.)

• Each moving object O maintains a list of the
  queries that O may be part of their answer
• It is the responsibility of the moving object O
  to report that O becomes part of the answer of a
  certain query
• Once a query updates its location, it sends the
  new location to the server, which will propagate
  the new location to the interested users
• The server is responsible in determining which
  objects will be interested in which queries

34
Scalability of Location-aware Continuous Queries
Location-aware Data Stream Management Systems

• Motivation Very high arrival rates that are
  beyond the system capability to store
• Idea Only store those objects that are likely to
  produce query results, i.e., only significant
  objects are stored, all other data are simply
  dropped
• Significant objects A moving object O is
  significant if there is at least one query that
  is interested in Os location
• Challenge Discovering that an object becomes
  insignificant

35
Scalability of Location-aware Continuous Queries
Location-aware Data Stream Management Systems
(Cont.)

• Only significant objects are stored in-memory
• An object is considered significant if it is
  either in the query area or the cache area

• Due to the query and object movements, a stored
  object may become insignificant at any time
• Larger cache area indicates more storage overhead
  and more accurate answer

36
Scalability of Location-aware Continuous Queries
Location-aware Data Stream Management Systems
(Cont.)

• The first k objects are considered an initial
  answer
• K-NN query is reduced to a circular range query

However, the query area may shrink or grow
K 3
37
Scalability of Location-aware Continuous Queries
Location-aware Data Stream Management Systems
(Cont.)
Each query is a single thread
One thread for all continuous queries
38
Scalability of Location-aware Continuous Queries
Location-aware Data Stream Management Systems
(Cont.)

• Query Load Shedding
• Reduce the cache area
• Possibly reduce the query area
• Immediately drop insignificant tuples
• Intuitive and simple to implement

• Object Load Shedding
• Objects that satisfy less than k queries are
  insignificant
• Lazily drop insignificant tuples
• Challenge I How to choose k?
• Challenge II How to provide a lower bound for
  the query accuracy?

39
Tutorial Outline

• Location-aware Environments
• Location-aware Snapshot Query Processing
• Location-aware Continuous Query Processing
• Scalable Execution of Continuous Queries
• Location-aware Query Optimization
• Uncertainty in Location-aware Query Processing
• Case Study
• Open Research Issues

40
Location-aware Query Optimization

• Spatio-temporal pipelinable query operators
• Range queries
• Nearest-neighbor queries
• Selectivity estimation for spatio-temporal
  queries/operators
• Spatio-temporal histograms
• Sampling
• Adaptive query optimization for continuous
  queries

41
Spatio-temporal Query Operators

• Existing Approaches are Built on Top of DBMS (at
  the Application Level)

Continuously report the trucks in this area
Scalar functions (Stored procedure)
SELECT O. ID FROM Objects O WHERE O.type
truck INSIDE Area A
42
Spatio-temporal Query Operators

• Continuously report the Avis cars in a certain
  area

SELECT M.ObjectID FROM MovingObjects M,
AvisCars A WHERE M.ID A.ID INSIDE RegionR
Spatio-temporal Operators
Scalar Function
/-
/-
INSIDE
/-
AvisCars
Moving Objects
AvisCars
Moving Objects
43
Spatio-temporal Selectivity Estimation

• Estimating the selectivity of spatio-temporal
  operators is crucial in determining the best plan
  for spatio-temporal queries

SELECT ObjectID FROM MovingObjects M WHERE
Type Truck INSIDE Region R
44
Spatio-temporal Histograms

• Moving objects in D-dimensional space are mapped
  to 2D-dimensional histogram buckets

x
t
45
Spatio-temporal Histograms with Query Feedback

• Estimating the selectivity of spatio-temporal
  operators is crucial in determining the best plan
  for spatio-temporal queries

10
Q1
Query
Query Optimizer
Query plan
Feedback
46
Adaptive Query Optimization

• Continuous queries last for long time (hours,
  days, weeks)
• Environment variables are likely to change
• The initial decision for building a query plan
  may not be valid after a while
• Need continuous optimization and ability to
  change the query plan
• Training period Spatio-temporal histogram,
  periodicity mining
• Online detection of changes

47
Tutorial Outline

• Location-aware Environments
• Location-aware Snapshot Query Processing
• Location-aware Continuous Query Processing
• Scalable Execution of Continuous Queries
• Location-aware Query Optimizer
• Uncertainty in Location-aware Query Processing
• Case Study
• Open Research Issues

48
Uncertainty in Moving Objects

• Location information from moving objects is
  inherently inaccurate
• Sources of uncertainty
• Sampling. A moving object sends its location
  information once every t time units. Within any
  two consecutive locations, we have no clue about
  the objects exact location
• Reading accuracy. Location-aware devices do not
  provide the exact location
• Object movement and network delay. By the time
  that a certain reading is received by the server,
  the moving object has already changed its location

49
Uncertainty in Moving Objects

• Historical data (Trajectories)

• Current data

T0?0
T0?1
T0?2
T0
T1
50
Uncertainty in Moving ObjectsError in Query
Answer

• Range Queries

• Nearest Neighbor Queries

51
Representing Uncertain Data usingEllipses

• Given
• Start point
• End point
• Maximum possible speed ? Maximum traveling
  distance S
• If S is greater than the distance between the two
  end points, then the moving object may have
  deviated from the given route

52
Representing Uncertain Data usingCylinders

• Given
• Start and end points
• Constraint
• An object would report its location only if it is
  deviated by a certain distance r from the
  predicted trajectory

r
53
Representing Uncertain Data in Road Networks

• Given
• Start and end points
• Constraints
• Deviation threshold r
• Speed threshold v

54
Querying Uncertain DataUncertain Keywords

• KEYWORDS
• Probability possibly, definitely
• Temporal sometimes, always
• Spatial somewhere, everywhere
• Examples
• What are the objects that are possibly sometimes
  within area R at time interval T?
• What are the objects that definitely passed
  through a certain region?
• Retrieve all the objects that are always inside a
  certain region
• Retrieve all the objects that are sometimes
  definitely inside region R

55
Querying Uncertain DataUncertain Keywords (Cont.)
O

• Object O is definitely always in Q1

• Object O is possibly always in Q2

• Object O is definitely sometimes in Q3

• Object O is possibly sometimes in Q4

56
Querying Uncertain DataProbabilistic Queries

• With each query answer, associate a probability
  that this answer is true
• The answer set of a query Q is represented as a
  set of tuples ltID, pgt where ID is the tuple
  identifier and p is the probability that the
  object ID belongs to the answer set of Q
• Assumptions
• Objects can lie anywhere uniformly within their
  uncertainty region

57
Querying Uncertain DataProbabilistic Range
Queries
E
A
C
D
F
B

• Query Answer
• (B, 50)
• (C, 90)
• D
• E
• (F, 30)

58
Querying Uncertain DataProbabilistic
Nearest-Neighbor Queries
E
A
C
D
F
B

• Query Answer (k1)
• (C, p1)
• (D, p2)
• (E, p3)

59
Tutorial Outline

• Location-aware Environments
• Location-aware Snapshot Query Processing
• Location-aware Continuous Query Processing
• Scalable Execution of Continuous Queries
• Location-aware Query Optimizer
• Uncertainty in Location-aware Query Processing
• Case Studies
• DOMINO
• SECONDO
• PLACE
• Open Research Issues

60
Case Study IDOMINO

• DOMINO Databases fOr MovINg Objects tracking
• Built on top of database management systems using
  a three-layers approach the DBMS layer, the GIS
  layer, and the DOMINO layer
• Utilize dynamic attributes for future predicted
  locations
• Manage uncertainty that is inherent in future
  motion plans
• Support various location models
• Exact point location
• An area in which the object is located in
• An approximate motion plan
• A complete motion plan

61
DOMINO Architecture
62
Uncertainty Management in DOMINO

• Uncertainty operators are implemented as
  user-defined functions (UDFs) in Oracle
• Uncertainty operators
• E.g., Always_Definitely_Inside,
  Sometime_Definitely_Inside, Possibly_Always_Inside
  , Possibly_Sometime_Inside
• Example
• SELECT oid
• FROM MovingObjects
• WHERE Possibly_Always_Inside (trajectory,
  region,
• time interval)

63
Case Study IISECONDO

• SECONDO An Extensible DBMS Architecture and
  Prototype
• A generic database system frame that can be
  filled with implementation of various data models
  (relational, object-oriented, or XML) and data
  types (spatial data, moving objects)
• A database is a set of SECONDO objects of the
  form (name, type, value), where type is one of
  the implemented algebras
• About 20 implemented algebras, e.g., standard
  algebra, relational algebra, R-Tree algebra, and
  spatial algebra
• Query optimizer includes optimization of
  conjunctive queries, selectivity estimation, and
  implementation of an SQL-like query language

64
SECONDO Architecture
Generic GUI independent of data models. The
interface includes command prompt and is
extensible by a set of different viewers
The core functionality is the optimization of
conjunctive queries, i.e., producing an efficient
query plan
On top of the query optimizer, there is a
SQL-like language in a notation adopted to PROLOG
SECONDO Kernel
Berkeley DB (C)
Built on top of Berkeley DB. Includes specific
data models, algebra modules, and query
processors over the implemented algebra.
65
Case Study III The PLACE Server

• PLACE Pervasive Location-Aware Computing
  Environments
• Scalable execution of continuous queries over
  spatio-temporal data streams
• Shared execution among concurrent continuous
  queries

• Built inside a database engine
• Incremental evaluation of continuous queries
• Spatio-temporal query operators

66
PLACE Architecture
DBMS
Query Parser
Query Processor
Relational Operators
Storage Engine
67
PLACE Architecture
PLACE
A Query Processor for Real-time Spatio-temporal
Data Streams
NILE

• Continuous Predicate-based Window Queries
• Moving Queries

A Query Processing Engine for Data Streams
Continuous time-based Sliding Window Queries
PREDATOR
INSIDE inside_clause
WINDOW window_clause
SQL Language
kNN knn_clause
W-Expire Operator
INSIDE Operator
Query processor
Negative Tuples
kNN Operator
Storage engine
Stream_Scan Operator
Stream of Moving Objects/Queries
Stream data types
Abstract data types
68
Extended SQL Syntax

• inside_clause
• Stationary query (x1,y1,x2,y2)
• Moving query (M,OID, width, length)
• knn_clause
• Stationary query (k,x,y)
• Moving query (M, OID, k)

69
Tutorial Outline

• Location-aware Environments
• Location-aware Snapshot Query Processing
• Location-aware Continuous Query Processing
• Scalable Execution of Continuous Queries
• Location-aware Query Optimizer
• Uncertainty in Location-aware Query Processing
• Case Study
• Open Research Issues

70
Open Research Issues Location Privacy
YOU ARE TRACKED!!!!
New technologies can pinpoint your location at
any time and place. They promise safety and
convenience but threaten privacy and security

Cover story, IEEE Spectrum, July 2003
71
Open Research Issues Spatio-temporal Data Mining

• Mining the history ? Predicting the future
• Online outlier detection for moving objects
• Suspicious movement in video surveillance
• Analysis of tsunami, hurricanes, or earthquakes
• Phenomena detection and tracking

72
Open Research Issues Reducing the Gap between ST
Databases and DBMSs/DSMSs

• What do Spatio-temporal researchers offer?
• 50 spatial index structure, 30 spatio-temporal
  indexing structure
• Wide variety of spatio-temporal query processing
  techniques
• What do DBMS designers want?
• Little disturbance to their code
• Large number of customers
• The result is
• DB2 and SQLServer do not support the R-tree (and
  may not be willing to)
• Oracle supports only R-tree and Quadtree
• Can we reduce this gap?
• YES. Think in the minimal additions to the engine
• Example I B-tree with SFC
• Example II GiST and SP-GiST
• Example III Add-in query operators

73
References

• Overview Papers
• Ouri Wolfson, Bo Xu, Sam Chamberlain, and Liqin
  Jiang. Moving Objects Databases Issues and
  Solutions. In Proceeding of the International
  Conference on Scientific and Statistical Database
  Management, SSDBM, pages 111-122, Capri, Italy,
  July 1998.
• Mohamed F. Mokbel, Walid G. Aref, Susanne E.
  Hambrusch, and Sunil Prabhakar. Towards Scalable
  Location-aware Services Requirements and
  Research Issues. In Proceeding of the ACM
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  November 2003.
• Christian S. Jensen. Database Aspects of
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• Dik Lun Lee, Manli Zhu, and Haibo Hu. When
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• Spatio-temporal Access Methods
• Mohamed F. Mokbel, Thanaa M. Ghanem, and Walid G.
  Aref. Spatio-temporal Access Methods. IEEE Data
  Engineering Bulletin, 26(2)40-49, June 2003.
• X. Xu, Jiawei Han, and W. Lu. RT-Tree An
  Improved R-Tree Indexing Structure for Temporal
  Spatial Databases. In Proceeding of the
  International Symposium on Spatial Data Handling,
  SSDH, pages 1040-1049, Zurich, Switzerland, July
  1990.
• Yannis Theodoridis, Michael Vazirgiannis, and
  Timos Sellis. Spatio-temporal Indexing for Large
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  IEEE Conference on Multimedia Computing and
  Systems, ICMCS, pages 441-448, Hiroshima, Japan,
  June 1996.
• Mario A. Nascimento and Jeerson R. O. Silva.
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• Jamel Tayeb, Ozgur Ulusoy, and Ouri Wolfson. A
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74
References

• Spatio-temporal Access Methods (Cont.)
• Dieter Pfoser, Christian S. Jensen, and Yannis
  Theodoridis. Novel Approaches in Query Processing
  for Moving Object Trajectories. In Proceeding of
  the International Conference on Very Large Data
  Bases, VLDB, pages 395-406, Cairo, Egypt,
  September 2000.
• Yufei Tao and Dimitris Papadias. MV3R-Tree A
  Spatio-temporal Access Method for Timestamp and
  Interval Queries. In Proceeding of the
  International Conference on Very Large Data
  Bases, VLDB, pages 431-440, Roma, Italy,
  September 2001.
• Yufei Tao and Dimitris Papadias. Efficient
  Historical R-Trees. In Proceeding of the
  International Conference on Scientific and
  Statistical Database Management, SSDBM, pages
  223-232, Fairfax, VA, July 2001.
• George Kollios, Vassilis J. Tsotras, Dimitrios
  Gunopulos, Alex Delis, and Marios
  Hadjieleftheriou. Indexing Animated Objects Using
  Spatiotemporal Access Methods. IEEE Transactions
  on Knowledge and Data Engineering, TKDE,
  13(5)758-777, 2001.
• Marios Hadjieleftheriou, George Kollios, Vassilis
  J. Tsotras, and Dimitrios Gunopulos. Efficient
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  of the International Conference on Extending
  Database Technology, EDBT, pages 251-268, Prague,
  Czech Republic, March 2002.
• Zhexuan Song and Nick Roussopoulos. SEB-Tree An
  Approach to Index Continuously Moving Objects. In
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  Melbourne, Australia, January 2003.
• Elias Frentzos. Indexing Objects Moving on Fixed
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  Greece, July 2003.
• V. Prasad Chakka, Adam Everspaugh, and Jignesh M.
  Patel. Indexing Large Trajectory Data Sets with
  SETI. In Proceeding of the International
  Conference on Innovative Data Systems Research,
  CIDR, Asilomar, CA, January 2003.
• Yuhan Cai and Raymond T. Ng. Indexing
  Spatio-temporal Trajectories with Chebyshev
  Polynomials. In Proceeding of the ACM
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75
References

• Spatio-temporal Access Methods (Cont.)
• Dieter Pfoser and Christian S. Jensen. Trajectory
  Indexing Using Movement Constraints.
  GeoInformatica, 9(2)93-115, June 2005.
• Jinfeng Ni and Chinya V. Ravishankar. PA-Tree A
  Parametric Indexing Scheme for Spatio-temporal
  Trajectories. In Proceeding of the International
  Symposium on Advances in Spatial and Temporal
  Databases, SSTD, pages 254-272, Angra dos Reis,
  Brazil, August 2005.
• Mario A. Nascimento, Jeerson R. O. Silva, and
  Yannis Theodoridis. Evaluation of Access
  Structures for Discretely Moving Points. In
  Proceeding of the International Workshop on
  Spatio-temporal Database Management, STDBM, pages
  171-188, Edinburgh, UK, September 1999.
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