Recent Advances on Querying and Managing Trajectories

1
Recent Advances on Querying and Managing
Trajectories

• Vassilis J. Tsotras
• tsotras_at_cs.ucr.edu

2
Talk Outline

• 1. Motivation
• 2. Query Models/Languages
• 3. Complex Pattern Queries
• 4. Trajectory Indexing Approaches
• 5. Open Problems ?

3
Trajectories are everywhere!

• The combination of GPS and cellular technologies,
  has created many applications that
  collect/process trajectorial data.
• Better location accuracy
• Instead of traditional cell phone tower
  triangulation method, use AGPS (Assisted GPS)
• GPS-enabled phones are now common
• Market research prediction 25-50 of cellphones
  in 2010 will have GPS
• TeleNav, smart2go, MapQuestFindMe, TomTom
  (typically for directions, LBS etc)

4
Applications

• E911 (enhanced 911) service locate any cell
  phone that dialed 911.
• Monitoring/Intelligence applications
• New applications have also appeared
• Asset Tracking (MobiTrack/Fluensee, VehiclePath,
  etc.)
• Family Tracking (Gtrac,AccuTracking, etc.)
• Offender/Criminal Tracking (BI-Exacutrack,
  iSecuretrac, etc.)

5
Applications (cont.)

• Cell-phone companies track cell phones and can
  get interesting aggregations
• Hour by hour census. Why?
• selective advertising e.g., change the billboard
  advertising over a given intersection by time
  (using the hour by hour population
  characteristics for that intersection)
• emergency management
• city facility planning (where to locate mobile
  businesses, more effective bus routes, etc.)

6
Applications (cont.)

• Hour by hour census (cont.)
• resource planning (better allocation of employee
  timetables if we know what kind, when and how
  many customers we get)
• Social Networking (Loopt, Molologo, Benefon,
  etc.)
• locate nearby friends, get alerts, events of
  common interest, etc.
• Gaming (Groundspeak's Geocaching,etc.)
• Cyber-trajectories (sites visited/when etc.)

7
Applications (cont.)

• Future applications?
• Imagine your digital/video camera will have GPS
• Can index my pictures by time/location
• Can see related pictures from my friends that
  visited the same touristic spots, etc.
• Can have visual trajectories (location, time,
  picture/video)
• IPhone GPS (next year?)

8
The audience is here !

• DoD, Homeland Security, Commercial Applications
• But How well can we manage Trajectories?
• - Good, but not that great yet
• Much work done, but more still needed

9
Commercial Support?

• Existing relational DBMSs do not support
  trajectories as fist-level objects
• very complex objects, not really relational
• Spatial DBMSs (Oracles Spatial Cartidge, ESRIs
  ArcInfo, IBMs DB2 Spatial Extender) do not focus
  much on trajectories and have thus limited
  support
• few spatial types, limited sets of predicates
  (intersects, contains, etc.) and few functions
  (distance etc.)

10
Basics

• What is a trajectory?
• A time-ordered sequence of recorded spatial
  locations of an object.
• Hence each trajectory has a unique oid
• Ti oidi, (L1I1), (L2I2), , where
• L1,L2, are locations in d-dimensional space
• I1, I2, are non-intersecting time intervals and
  I1 comes before I2 etc.

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Trajectory Approximation

• We assume all recorded locations are stored
• What about the times in between the recorded
  locations?
• We may have a location function
• We may assume linear interpolation etc.

x
y
time
12
Talk Outline

• 1. Motivation
• 2. Query Models/Languages
• 3. Complex Pattern Queries
• 4. Trajectory Indexing Approaches
• 5. Open Problems ?

13
Talk Outline

• 1. Motivation
• 2. Query Models/Languages
• 3. Complex Pattern Queries
• 4. Trajectory Indexing Approaches
• 5. Open Problems ?

14
2. Query Models/Languages

• Looking at the Spatio-Temporal Database research,
  there are two main approaches
• Abstract Data Type approach
• Guting et al, 2003, Erwig Schneider, 2002
• Constraint Data Model approach
• Grumpach et al, 2003, Chen Zaniolo, 2000,
  Mokhtar Su, 2005

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The Abstract Data Type Approach

• A set of base spatial, temporal and
  spatiotemporal data types
• Spatial data types point, line, region
• Time is linear and continuous
• A type constructor named moving takes any type
  a and gives a mapping from time to a.

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The Abstract Data Type Approach (cont.)

• The basic spatial predicates e.g., disjoint,
  meet, overlap, inside, etc EgenhoferFranzosa,
  1991 are temporally-lifted to become
  spatio-temporal operators.
• Basic such operators are embedded into SQL
  through few extensions.
• An extension mechanism for more complex
  spatiotemporal predicates also exists.
• The result language is STQL ErwigSchneider,
  1999

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The Constraint Data Model approach

• Temporal and spatial objects are represented as
  infinite collections of points satisfying 1-order
  formulas.
• Moreover, queries are also expressed by
  constraints.
• Based on constraint databases Kanellakis et al.,
  1995
• Typical assumption linear constraints

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The Constraint Data Model approach (cont.)

• Mokhtar Su, 2005 present a query language
  (TQ) for expressing trajectory properties.
• Grumbach et al, 2003 adapt the DEDALE algebra,
  on top of the constraint model
• Chen Zaniolo, 2000 introduce SQLST using a
  point-based model to represent time and a
  directed-triangulation model to represent spatial
  data.

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Other approaches

• There are also languages/models dedicated for
  future states of moving objects
• Wolfson et al, 1999, proposed the
• Moving Object SpatioTemporal (MOST) model
• Future Temporal Logic (FTL) query language
• Work on uncertainty models for moving objects
• Trajcevski et al, 2004 (DOMINO system)
• The uncertain trajectory is modeled as a
  three-dimensional (3D) cylindrical body
• New operators (sometime_possibly_inside, etc.)

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Other approaches (cont.)

• Pfoser Jensen, 1999, models location
  uncertainties because of GPS impression and
  sampling errors (error ellipses)
• Cheng et al, 2004, discusses probabilistic
  answers to queries over uncertain moving object
  data
• Work on modeling in constrained networks
• Guting et al, 2006
• Recent work on querying trajectories as streams
• Gedik Liu, 2004 MobiEyes
• Mokbel et al, 2004 SINA
• Patroumpas Sellis, 2004 query operators and
  structures that apply to trajectories, etc.

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What kind of queries?

• The typical ones
• Range related queries
• Show me all trajectories that were inside area A
  at time instant t (or time interval I)
• Nearest neighbor queries
• Find the trajectory that was closest to point B
  at time instant t (or time interval I)

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What kind of queries? (cont.)

• It seems that the majority of the existing work
  has dealt with queries that can be described by
• Q (SP,TP) where
• SP spatial predicate
• TP temporal predicate (time instant t, or time
  interval I)
• Q is true if SP holds at (or during, etc.) TP
• E.g., if TP is an interval, we may ask for SP to
  be true during some time within the interval.

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What kind of queries? (cont.)

• In general we can identify two query groups
  according to the query spatial predicate SP
• Binary predicates (for each trajectory the answer
  is a Yes/No)
• Includes the typical topological predicates like
  inside, intersect, touches, outside, etc.
• Numerical predicates (each trajectory evaluates
  to a value and then we typically pick a min/max
  or top-K values)
• The variation is on the distance function used
  (Euclidean, Manhattan etc.)

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What kind of queries? (cont.)

• We can thus roughly categorize typical queries as
  following
• Q1 find all trajectories that were in area A at
  time t
• Q2 find the trajectory closest to point B at
  time t.
• Q3 find all trajectories within a cylinder of
  radius r from trajectory TQ
• Q4 find the trajectory most similar to
  trajectory TQ

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Similarity Queries

• Q3 and Q4 are examples of similarity queries
• Given is the query trajectory TQ
• Assume TQ lasts for time interval I
• Q3 defines a tube of radius r around TQ
• Like a range query (circle of radius r) at each
  time in I

TQ
r
I
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Similarity Queries (cont.)

• Q3 is a binary-based similarity query
• Q4 is a distance-based similarity query
• Find the minimum of the sum of the distances for
  each time in I
• Similarity queries have also appeared extensively
  in Time-series research
• We are different!
• Where you are and at what time are important.
• While in time-series
• there is no spatial component
• We typically start with normalization

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Observations

• Note either a single SP at a given instant t,
  or, the same SP at all instants of an interval I.
• Think of Q(SP,TP) as a query atom
• Do these queries capture all we can ask about
  trajectories?

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Motion Pattern Queries

• In my view, trajectories represent behavior over
  time they capture the evolution of a movement
• Can we query the behavior/motion of trajectories?
• Yes! We can use complex motion patterns 1
• Example Find objects that crossed through
  region A at time t1, came as close as possible to
  point B at a later time t2 and then stopped
  inside circle C during interval (t3, t4)
• 1 Hadjieleftheriou et al, 2005

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Talk Outline

• Query Models/Languages
• Complex Pattern Queries

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3. Complex Pattern Queries

• A Motion Pattern (MP) query is actually a
  time-ordered sequence of query atoms
• Qmp Q1 ? Q2 ? Q3 , where
• Qi (SPi,TPi)
• TPi is before TPi1 (and non-overlapping) etc.
• The time-ordering of the spatial predicates may
  be explicit or implicit
• Example of implicit ordering
• find trajectories that first went through area
  A, then by point B, etc.

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Categorization

• Where are Motion Pattern queries fall in our
• table categorization?

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MP queries are different

• They are not typical similarity queries
• In typical similarity queries the same predicate
  holds for the duration of an interval (which can
  be quite restrictive)
• They are not typical range/NN queries
• We can now choose separate predicates at
  different times
• Effectively the user can tailor/adapt the query
  to her/his needs

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MP queries are different (cont.)

• Do we need new techniques?
• What about solving MP queries one predicate at a
  time, then combine the results
• Surely, if the MP query has only a sequence of
  binary predicates (ranges etc.)
• But, it will not work if the MP query contains
  numerical predicates

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Related Work to MP Queries

• Are Motion Pattern queries new?
• Well, not really!!!
• There is work on sequence queries for time-series
  in relational systems
• SEQ Seshadri et al, 1995
• SQL-TS Sadri et al, 2001
• Represent sequences as strings and provide a
  pattern-matching algorithm based on the
  Knuth-Morris-Pratt algorithm

35
Related Work to MP Queries (cont.)

• What about research from the Spatio-temporal
  domain?
• Erwig Schneider, 1999
• discuss the notion of developments which
  characterizes how the topological relationship
  among two moving objects unfolds in time
• Qu et al, 2001
• Describe algorithms for identifying movement
  patterns (moved up then down etc.)

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Regular Expression Queries

• There are also works that introduce patterns as
  strings
• from some finite alphabet
• Then the queries become regular expressions
• more powerful query languages
• Example A/B//C etc.
• i.e., first at location A, subsequently at B,
  some time later at C

37
Related Work on Regular Expression Trajectory
Queries

• Mainly from the GIS community
• Djafri et al, 2002
• Evolution queries over consecutive historical
  snapshots
• Laube et al, 2005
• RElative MOtion (REMO) system
• Create one matrix per object variable over time
  (e.g., azimuth)
• Specify the query as a string and do pattern
  matching on the matrix using variations of KMP

38
Mobility Patterns

• Interesting idea restrict the space into a
  collection of non-overlapping zones (can be a
  grid, etc.)
• Then each trajectory becomes a sequence of
  (zone-label, time-interval) pairs
• (A,t1,t2), (C,t3,t4), (B,t5,t6),
• Queries (mobility patterns) are regular
  expressions on the zone alphabet (mainly for
  range/topological predicates)
• A//B etc.
• Use NFAs to find matches over continuous regular
  expression queries
• Mouza et al, 2005

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Discussion

• Using zones is interesting since it provides an
  alternative way to store trajectories
• Reducing the alphabet leads to a powerful query
  language (path expressions)

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Talk Outline

• Complex Pattern Queries
• Trajectory Indexing Approaches
• 4.1 Indexing in the Native Space
• 4.2 Indexing in the Parametric Space
• 4.3 Approaches for MP Queries

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Trajectory Indexing Approaches

• The storage/indexing method depends on the query
  environment
• Two cases
• Querying the past
• typically archived trajectories
• Continuous trajectory queries
• Trajectories arrive as streams

42
Indexing for archived trajectories

• Assume we have stored the recorded locations of a
  moving object over time
• Main Question how can we approximate a
  trajectory?
• We can then index the approximations

x
y
time
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Two approaches

• 1. Indexing in the Native Space
• Typically approximate using MBRs then index
  these MBRs
• Advantage
• we can use R-trees etc.
• can also index other moving objects (areas etc.)
• Disadvantage trajectories are lines thus MBRs
  add extensive empty space.

44
Two approaches (cont.)

• 2. Indexing in the Parametric Space
• approximate each trajectory by a function
  (typically a polynomial) then index the
  functions coefficients
• Advantage better approximation
• Disadvantage
• translate btw Native
• Parametric spaces
• better approximation means
• more coefficients

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4.1 Indexing in the Native Space

• Previous approach
• One MBR per trajectory
• Too much empty space

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Cutting MBRs

• Can we do any better?
• Well, the more MBRs we use the better the
  approximation
• Where can we cut for MBRs?

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Cutting MBRs (cont.)

• Assume time is discrete
• Can introduce one MBR per time-instant of the
  objects lifetime.
• Between successive trajectory points we can
  assume piecewise linear functions
• more general piecewise functions can also be
  used as long as the functions are known, the
  location of an object at every time-instant can
  be deduced.

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But

• There is a serious problem!
• Pagel et al, 1993
• Disk Access Cost of MBR-based index methods is
  proportional to (i) the number of indexed MBRs
  and (ii) the empty volume.
• Unfortunately, to reduce the empty space we want
  to add more cuts (MBRs) but at the same time we
  add more objects to the tree
• Intuitively, if we represent each trajectory with
  too many MBRs index quality will deteriorate
• too many objects!
• On the other hand, if we represent each
  trajectory using very few MBRs, index quality
  will deteriorate
• Too much empty volume!

49
Using MVR-tree

• Hence the split-for-better-approximation policy
  will not help much with an R-tree.
• We need a different strategy!
• Use a multi-version structure (MVR-tree) to index
  the trajectory approximations

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Why multi-versioning?

• A traditional R-tree considers time as another
  dimension
• for example x,y,t creates a 3D R-tree
• Instead, an MVR-tree effectively provides a
  separate 2D R-tree, indexing each time-slice

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MVR-tree

• A 3D MBR with time dimension (ts,te) can be
  perceived as a 2D MBR inserted at time ts and
  deleted at time ts
• i.e., two updates
• The MVR-tree conceptually stores the evolution of
  a simple 2D R-tree over time.
• Hence, all nodes are augmented with insertion and
  deletion time fields.
• Kumar et al, 1998, Kollios et al, 2001, Tao
  and Papadias, 2001.

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Multi-version R-tree Tutorial
53
Back to the approximation

• Why the MVR-tree will work?
• Key observation
• A split of an MBR at time t does not change the
  number of objects indexed at time-slice t
• For the MVR-tree, at time t
• An existing object (MBR a) was deleted
• a new object (MBR b) was added

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Lets recap

• Given N trajectories, approximate them using K
  MBRs (NltK)
• First, find the best approximation possible per
  trajectory, per number of MBRs.
• Then, find a way to approximate a group of
  trajectories given a total of K MBRs (e.g., as a
  constraint on storage space), so as to minimize
  the total volume of the representations.
• Finally, index the K MBRs using an MVR-tree

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The Cost

• The optimal algorithm uses dynamic programming
  and is expensive O(NK2)
• A Greedy algorithm O(K logN)
• Each trajectory starts with one MBR
• Sort all trajectories according to the gain when
  an extra MBR is assigned to them
• Assign the next MBR to the best trajectory, until
  all available MBRs have been assigned

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The Improvement

• The Greedy algorithm is suboptimal
• An improved greedy algorithm O(K logN)
• Look-Ahead-m Greedy
• i.e., see if up to m assignments will be better
  for a given trajectory

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Comparison of Assignment Algorithms
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Experimental dataset
150K-400K trajectories, 100 time-instants 1000
random queries, range 1, time 1-2 time
instants Total of 1,200,000 to 9,500,000
movement functions Exact index would store
40,000,000 MBRs Piecewise index would store
1,200,000 to 9,500,000 MBRs
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Drawbacks

• The MVR-tree is optimized for time-slice or small
  time-interval queries

This is because the splitting idea works per time
instant and the MVR-tree provides access to the
R-tree as of some time.
Moreover, the MVR was designed to support both
moving regions and points
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Other recent work

• Rasetic et al, 2005 use knowledge about the
  average query size to guide the MBR splitting on
  R-trees.
• Anagnostopoulos et al, 2006 minimize pair-wise
  distance btw all trajectories
• more applicable for mining/similarity
  applications.

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4.2 Indexing in the Parametric Space

• Each trajectory is a collection of functions
• Trajectory
• Trj(Oi) IDi t0, t1,, tn f1(t), f2(t), ,
  fn(t)
• Assumptions
• fi(t) linear function, traj. becomes a polyline.
• 2-d XY-space

62
Previous work on Parametric Indexing

• Parametric indexing scheme in Porkaew et al,
  2001
• Use the parameters for each function fi(t) as the
  keys in the index structure.
• Problem
• Hundreds of functions per trajectory.
• Large storage overhead.
• Outperformed by MBR approximation

63
Previous work on Parametric Indexing (cont.)

• Cai Ng, 2004 approximate each trajectory with
  a Chebyshev polynomial
• Easy to compute
• Almost identical to optimal minmax polynomial
• Use the coefficients for indexing.
• Focused on similarity queries
• Over entire trajectories of equal length
• Same degree polynomials for all trajectories

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For spatiotemporal queries

• Ni Ravishankar, 2005 proposed the Polynomial
  Approximation Tree (PA-tree)

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PA-tree (cont.)

• When indexing the coefficients
• May want different order of approximation for
  different trajectories.
• High order approximation leads to high
  dimensional space.
• PA-tree solution two level index structures
• First level linear approximations with leading
  two coefficients.
• Second level elaborate the leaf nodes with the
  additional coefficients.

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PA-tree (cont.)

• Similar to R-tree
• Each time interval I has a PA-tree.
• Top level
• index the two leading Chebyshev coefficients
• Each entry
• Bottom level
• Use additional coefficients
• Cluster multiple trajectories using upper/lower
  bound envelopes

67
Performance
68
Advantages/Disadvantages

• The Parametric approach provides better
  approximations
• less empty space
• Provides advantage for larger time interval
  queries
• Assumes trajectories are smooth
• i.e., relative few coefficients can describe the
  trajectory well enough
• How do we select the splitting interval?
• What if queries smaller/larger than this interval

69
4.3 Approaches for MP Queries

• Examples
• Find trajectories that crossed through region A
  at time t1, came as close as possible to point B
  at a later time t2 and then stopped inside region
  C some time during interval (t3, t4)
• Find trajectories that first crossed through
  region A, then passed as close as possible from
  point B and finally stopped inside region C
• In this query no explicit time is defined, simply
  relative order

70
How can we solve the relative order MP queries?

• Spatio-temporal indices cannot be used!
• Full scan of temporal dimension of the index
• Ideally, we need to find a method that will
  retrieve only the trajectories that satisfy the
  predicates in the specified order

71
Idea

• What about a predicate index
• i.e., index the spatial predicates instead of the
  trajectories
• hm much like an inverse index !
• But, too many predicates (ranges/NNs etc)
• Lets reduce the alphabet
• What if we use a grid on the space
• While time expands, space is fixed
• For each grid cell keep a list with all
  trajectories that passed through it (and when/how
  long)

72
Idea (cont.)

• Using the grid, MP queries can be transformed
  into path expression queries (on the grid-cell
  alphabet)
• A//B//C etc
• first through cell A, then later through cell B,
  then later through C
• Given a MP query, we then access only the lists
  involved in this query
• Each list is ordered
• first in ascending trajectory identifier order,
  then by time-instant

73
MP queries with ranges

• Assume for simplicity we have an MP query with
  three range predicates, ordered as A, B, C
• Assume also each range fits exactly to a cell
• If smaller we overestimate
• If larger, get all cells that cover it and create
  one list

74
Merge-Join among the query lists

• Given N ordered range predicates, consider the
  corresponding N trajectory lists
• Take list 1 and search for its first trajectory
  identifier in list 2.
• If it does not exist or the time-instants do not
  satisfy the ordering, prune and start all over
  with the next identifier in list 1.
• Continue with the 2nd, 3rd, etc lists.
• Remniscent of PathStack algorithm in XML !

75
MP queries with distance predicates

• Use the lists involved in the query and an
  incremental ranking algorithm
• Iteratively en-queue and examine cells that are
  adjacent to each query predicate
• Compute lower bounding distances
• Evaluate predicates in round-robin fashion. For
  every new cell added in a queue, first join it
  with the queues of all other predicates. Prune
  according to order (same concept with range
  predicates)

76
Performance
77
Discussion/thoughts

• The inverted index approach described uses a
  normal grid but other grids can be used
• Similar in concept to the zones in the Mobility
  Patterns
• Mouza Rigaux, 2005
• Avoids overlapping of MBRs
• Can still prune trajectories since it maintains
  the temporal order
• Substantial space savings
• Good high-level spatial approximation (zoom
  in/zoom out)
• Probably can use it as a first level index, then
  for each cell we can have more detailed index ?

78
Talk Outline

• Trajectory Indexing Approaches
• 4.1 Indexing in the Native Space
• 4.2 Indexing in the Parametric Space
• 4.3 Approaches for MP Queries

• Open Problems?

79
5. Open Problems?

• More complex pattern queries?
• E.g., //, , NOT, OR, etc.
• Find trajectories that went from area A to (area
  B or C) through another area and did not go
  through D
• Find trajectories that left an area then went to
  B and then came back to that area
• XML-like query processing approaches?
• Can we optimize such queries?
• Which one first, etc.

80
Open Problems (cont.)

• Continuous Motion Pattern Queries?
• The time constraints are relative to the ever
  increasing current time
• Report objects that () between 10 and 20 minutes
  ago
• Can we change the patterns on-the-fly (add/remove
  MP atoms) ?

81
Open Problems (cont.)

• Other queries? Trajectory Joins?
• Other ways to define joins?
• Maybe using RNN queries? Xia Zhang, 2006

82
Open Problems (cont.)

• Trajectory Density-based queries?
• E.g., identify leaders (at least k other
  trajectories follow within 10 minutes)
• Flocks, convergence, encounter, etc. van Kreveld
  et al et al, 2007

83
Open Problems (cont.)

• Better indexing for trajectories?
• Probably multi-granularity, multi-level
• What about a multiversion-grid?
• Zoom-in / zoom-out
• How is Spatial Anonymity/Privacy affecting the
  above methods?
• Kalnis et al, 2006 PRIVE
• Mokbel et al, 2006 New Casper
• Gedik Liu, 2005, etc.

84

• Thank you!
• This research has been partially supported by
  NSF under IIS-0534781 many thanks to Erik Hoel,
  Eamonn Keogh, Shashi Shekhar and Ouri Wolfson for
  helpful comments

85
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86

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