Indexing the Positions of Continuously Moving Objects

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Indexing the Positions of Continuously Moving
Objects

• Saltenis, Jensen, Leutenegger and Lopez

2
Applications

• Mobile phones -gt wireless internet terminals
• Location aware service can be provided -gt
  improvement in QoS
• Vehicle navigation
• Monitoring positions of air,sea or land based
  equipments
• Airplanes, boats,trucks,

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Conventional Approach

• Data assumed constant - modification explicit
• Capture Continuous movement
• Very frequent updates
• Outdated, inaccurate data
• Requirement
• Capture movement directly
• Advancement of time -gt necessary explicit
  updates
• Solution
• Objects position function of time and store
• Updates(explicit) function parameters changes

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Indexing

• Indexing of history
• Indexing current and anticipated future positions
  (focus)
• Time-Parameterized R-tree(TPR-tree)
• Approach
• Index functional indefinitely optimized for
  specific time horizon- deteriorates as time
  progresses
• TPR , Bounding rectangles and the moving points
  are functions of time
• Intuitively bounding rectangles follow moving
  points or other rectangles as they move

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Problem Setting

• Objects position at time t X(t)
  (x1(t),x2(t),..xd(t))
• Position modeled as a linear fn of time 2
  params
• 1st position of object at tref i.e. X(tref)
• 2nd Velocity of the object V (v1,v2,.vd)
• Modeling as fn of time enables future prediction
  and solves frequent update problem
• Objects report positions and velocity vectors
  when they deviate from the current value (db
  value) by certain threshold

X(t)X(tref) V(tref)
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Moving objects position time 0
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R-Tree at time 0
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R-Tree at time t
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Better arrangement at time t
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Assignment

• From perspective of queries at time t, the last
  assignment of objects to MBRs is better
• This yields worst performance _at_ time 0
• Assignment of objects to MBRs must take into a/c
  when most queries will arrive

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Query types

• Retrieve all points with positions within
  specified regions.
• d-dimensional rectangle R spec d projections
• a1-,a1-,.ad-,ad-
• R, R1,R2 all d-dimensional rectangles
• t, t - lt t - 3 time values not less than
  current time

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Queries(contd)

• Time Slice query Q (R,t) specifies a hyper
  rectangle R located at time point t
• Window query Q(R,t-,t-) specifies a hyper
  rectangle R that covers t-,t-
• Retrieves all points with trajectories crossing
  (d1) hyper rectangle (a1-,a1-),, (ad-,ad-),
  t-,t-
• Moving query Q(R1,R2,t-,t-) specifies the
  (d1) dimensional trapezoid obtained by
  connecting R1 at time t- to R2 at time t-
• Window query generalises timeslice query
• Moving query generalises window query

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Value 40 30 20 10 0 -10 -20 -30 -40
o2
o1
Q1
Q0
Q2
o4
o3
Q3
1 2 3 4 5 time
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Query examples

• Q0 and Q1 are timeslice queries
• Q2 is window query and Q3 moving query
• Iss(Q) time when query issued
• Ref position and velocity depend on issue(Q)
  because objects update their parameters as time
  goes
• O1 movement desc by 1 trajectory for iss(Q) lt 1
• Another for 1lt iss(Q) lt 3 and another for iss(Q)
  gt 3
• Answer to query Q1 is o1 if iss(Q1) lt 1 and none
  if iss(Q1) gt1
• Queries in far future little value because
  positions predicted less accurate
• Real World expect queries concentrated in some
  limited time window extending from current time

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Problem Parameters

• 3 params affect indexing problem and qualities of
  TPR-tree
• Querying window(W) how far queries can look
  into the future
• Iss(Q) lt t lt Iss(Q) W for timeslice queries
• Iss(Q) lt t - lt t - lt Iss(Q) W for other
  queries
• Index Usage Time(U) time interval during which
  an index will be used for querying
• tl lt Iss(Q) lt tl U tl index creation time
• Time Horizon(H) length of the time interval
  from which t,t -, t - are drawn
• Time horizon for an index is index usage time
  plus the querying window

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• Newly created index must support queries that
  reach H units in future

HUW
W
Iss(Q)
t -
t -
tl
U
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Index Structure

• TPR tree is a balanced, multi way tree with
  structure of an R-tree
• Leaf nodes pairs of positions of a moving point
  and a pointer to the moving point
• Internal nodes pairs of pointer to subtree and
  a rectangle that bounds the positions of all
  moving points or other bounding rectangles in
  subtree
• Time parameterized d-dimensional bounding
  rectangles bound d-dimensional moving points or
  rectangles at all time not earlier than current
  time

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Conservative bounding rectangles

• Min at some time but possibly not at later times
• In 1d case lower bound of a conservative
  interval is set to move with the minimum speed of
  the enclosed points , upper bounds move with max
  speed of enclosed points
• Speeds ve or ve dep on direction
• Conservative bounding intervals never shrink
• Constant size when all points have same velocity
  vector
• It may move

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Querying

• A bounding interval (x -,x -,v -,v -)
  satisfies a query ((a -,a
  -),tq) iff
• a- lt x - v -(tq tl) a- gt x -
  v-(tq-tl)
• To answer window queries and moving queries we
  need to check if in (X,t) space the trapezoid of
  a query intersects with the trapezoid of a
  bounding rectangle that is b/n the start and end
  time times of the query
• Generic polyhedron-polyhedron intersection tests
  may be used