CSIS7101 Advanced Database Technologies

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CSIS7101 Advanced Database Technologies

• Spatio-Temporal Data (Part 1)
• On Indexing Mobile Objects

Kwong Chi Ho Leo Wong Chi Kwong Simon Lui, Tak
Sing Arthur
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Content

• 1 Introduction
• 2 Indexing Mobile Objects
• 3 Indexing in 2 Dimensions
• 4 Summary
• 5 Reference

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1. Introduction

• Traditional Database Model assume that data
  stored in the database remain constant.
• Not appropriate for applications with
  continuously changing data.
• Better approach Object location as a function of
  time f(t), and update the database only when the
  parameters of f change.
• This approach introduces a novel problem since
  the database is not directly storing data values
  but function to compute these values.

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2. Indexing Mobile Objects

• 2.1 Partition the mobile objects into two
  categories
• Lets say objects are moving on an 1-dimensional
  plane.
• Object with low speed v ? 0.
• Objects with speed between v min and v max. (i.e.
  Moving Objects)
• 2.2 Space-Range Representation
• 2.3 Dual Space-Time Representation
• 2.4 Simplex Range Searching
• 2.5 Achieve Logarithmic Query Time

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2. Indexing Mobile Objects

• 2.2 Space-Range Representation
• Plot the trajectories y(t) v t a of the
  mobile objects as line in time-location (t,y)
  plane.
• The query is expressed as 2-dimensional interval
  (y1q, y2q), (t1q, t2q).

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2.2 Space-Range Representation

• Shortcomings
• Minimum Bounding Rectangle (MBR) much larger than
  a Line.
• Object retains trajectory until updated, all
  lines extend to infinity.
• Mapping a line segment as a point in 4
  dimensions, will not work.
• SAM can only access queries until end of the
  current session.

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2.3 Dual Space-Time Representation

• Map a line from primal plane (t, y) to a point in
  the dual plane.
• Hough-X plane
• y(t) v t a is represented by a point (v, a)
  in dual space.
• Hough-Y plane
• y(t) vt a is transformed to t 1/v y(t)
  a/v. In dual space, the coordinates are n 1/v
  and b -a/v.

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2.3 Dual Space-Time Representation

• Hough-X Dual Space
• Query is transformed in a polygon query in the
  dual space. This polygon uses a linear
  constraint polygon.

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2.3 Dual Space-Time Representation

• The 1-dimensional MOR query in dual Hough-X is
  expressed as
•  For v gt 0 the query is Q C1 ?C2 ?C3 ?C4,
  where
• C1 v ? v min,
• C2 v ? v max,
• C3 a t2qv gt y1q, and
• C4 a t1qv lt y2q.
•  For v lt 0. the query is Q D1 ?D2? D3 ?D4,
  where
• D1 v ? -v min,
• D2 v ? -v max,
• D3 a t1qv ? y1q, and
• D4 a t2qv ? y2q.

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2.3 Dual Space-Time Representation

• Problem range of a -v max x t now , y max -
  v min x t now
• Time is increasing, values of the intercepts are
  not bounded.
• V max is significant, values of intercept become
  very large, and potentially a problem (unbounded
  range of real numbers)
• To solve
• Assuming that when an object crosses a border it
  issues an update (deletion or reflection).
• With minimal speed, all objects updated their
  motion information at least once during a T
  period time instant,
• where T period y max / v min.

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2.3 Dual Space-Time Representation

• Two distinct index structures. Each object stores
  only once.
• 0, T period and T period, 2T period
• 1st index The intercept is computed using the
  line t 0.
• 2nd index The intercept is computed using the
  line t T period.
• With this, the intercept will always have the
  values between 0 and v max T period.
• Both indices are used in database query.

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2.3 Dual Space-Time Representation

• Keeping on that, after time 2T period, the first
  index is deleted and a new index is created. We
  then have
• T period, 2Tperiod and 2Tperiod, 3Tperiod
• With this method, intercept is bounded, and
  Performance of index structure remains
  asymptotically the same.

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2.4 Simplex range searching

• The dual space-time representation transforms the
  indexing mobile objects on a line to simplex
  range searching in two dimensions.
• Theorem 1 (Lower Bound)
• Simplex reporting in d-dimensions with a query
  time of O (n d k), where N is the number of
  points, n N/B, K is the number of the reported
  points, k K/B and 0 lt d 1, requires space O
  (n d(1- d) ? ) disk blocks, for any fixed ?.

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2.4 Simplex range searching

• Two approaches
• Point Access Methods
• Query Approximation Algorithm

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2.4.1 Point Access Method

• Point Access Method
• Using R-trees to answer simplex range queries,
  the 1-dimensional MOR query can be answered in
  the dual Hough-X space.

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2.4.2 Query Approximation Algorithm

• Query Approximation Algorithm
• Using Hough-Y dual plane

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2.4.2 Query Approximation Algorithm

• Since rectangular query is an efficient access
  method, the query is approximated by a
  rectangular one.
• The query rectangle
• The query area is enlarged by area E E1 E2,
  and

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2.4.2 Query Approximation Algorithm

• E is based on yr (where the b coordinate is
  computed) and the query interval (y1q, y2q),
  which is unknown.  
• If a constant c is use to keep equidistant yr, we
  can have observation index, y i y max /c i

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2.4.2 Query Approximation Algorithm

• A given 1-dimensional query MOR query will be
  forwarded to the index(es) that minimize E.
• Rectangle range search simple range search on
  the b coordinate axis.
• Each of the observation index can be a B tree.
• To process query interval y1q, y2q, two cases
  will be considered,
• Case 1 y2a y1q y max / c
• The area is bounded by

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2.4.2 Query Approximation Algorithm

• Case 2 y2q y1q gt y max / c
• The query is decomposed into collection of
  smaller sub-queries one sub-query per
  sub-terrain contained by the original querys
  endpoints. The sub-queries can be answered with
  bounded E using an appropriate observation
  index.
• Lemma 1
• The one dimensional MOR query can be answered in
  time O(log B n (K K) / B), where K is the
  approximation error. The space used is O(c n)
  where c is a small constant, and the update is
  O(c log B n).

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2.5 Achieve Logarithmic Query Time

• Restrict our queries to occur before the first
  time that a point overtakes another.
• To index mobile objects in a bounded time
  interval T in the future.
• Consider objects are moving on a line, Find all
  the objects that lie in the segment y l , y r
  at time t q (i.e. t q t1q t2q).

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2.5 Achieve Logarithmic Query Time

• Lemma 2
• If we have the relative ordering of all the
  objects at time t q, the position of the objects
  at time t c that corresponds to the closest
  crossing event before t q, and the speed of
  the objects, we can find the objects that are in
  y l, yr in O(log2N K) time.
• Consider objects p1, p2, . , p n, pi has
  relative position y i and a velocity vi. At time
  t q, the position pi is pi y i vi t q. The
  objects stored in the binary tree are sorted.
• In O(log2N) time we can find the positions of yl
  and yr relative the objects at time t q, and we
  report the objects that lie between.

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2.5 Achieve Logarithmic Query Time

• e.g. An object list p1, p2, p3, p4, p5, p6,
  p7, and at time t q and say, y l, yr p3,
  p6

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2.5 Achieve Logarithmic Query Time

• Lemma 3
• We can find all the crossings of objects in time
  O (Nlog2N Mlog2M), where M is the number of
  crosses in the time period 0,T.
• Consider N Objects p1, p2, , pN, pi has
  relative position yi and a velocity vi at time
  0. Another sorted order list pt(1) , pt(2) ,
  pt(N) at time T. Then objects i and j cross
  if and only if t(j) lt t(i).
• Keep the objects in a linked list, in the same
  order they were at time t 0. Scan the sorted
  lit of objects at time t T. All the crossings
  can be reported in O(N M). After all crossings
  are reported we can find when each occurs and
  sort them on their time attribute.
•  

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2.5 Achieve Logarithmic Query Time

• e.g.
• Ordered list p1, p2, p3, p4, p5, p6, p7 at time
  t 0
• Ordered list p1, p3, p2, p6, p4, p5, p7 at time
  t T
•  
• , p3, p2, p6, p4, p5, p7  
• , p3, , p6, p4, p5, p7 crossing
• , , , p6, p4, p5, p7
• , , , p6, , p5, p7 crossing
• , , , p6, , , p7 crossing
• , , , , , , p7 
• , , , , , ,
• 3 Crossings !
• M crossings define M ordered list of the N
  objects.

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2.5 Achieve Logarithmic Query Time

• Lemma 4
• We store the O(M) order lists of N objects in
  O(nm) blocks and perform a search on any list in
  O(log B(nm)) I/Os, where n N/B and m M/B.
• Embed the binary tree structure of the list of N
  objects inside a B-tree. Each B tree use O(n)
  nodes where each node hold B entries.
• A s node stores the evolution of trees B(0),
  B(t1), B(t2), ,
• B(t M).
• Post the entries of s as changes in the history
  of the parent node p. When a new copy of node s
  is created, a new record is added on the log of
  p that has the same position l, but a pointer to
  the new copy of s and the current time.

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2.5 Achieve Logarithmic Query Time

• Overall space O(nm), and
• Query time O(log B(nm))
• Theorem 2
• Given N objects and a time limit T, an one
  dimensional MOR1 query can be answered in time
  log B (n m) using space O(n m), where m M/B
  and M is the number of crosses of objects in the
  time limit T .

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3. Indexing in Two Dimensions

• The 1.5-dimensional problem
• Objects moving in the plane but their movement is
  restricted on using a given collection of routes
  (roads) on the finite terrain.
• Each predefined route can be represented as a
  sequence of connected (straight line) segments.
• Indexing the line segment by standard SAM.
• Indexing the object moving on the given route is
  an1-dimensional model.

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3. Indexing in Two Dimensions

• The 2-dimensional problem
• Decompose the motion of the object into two
  independent motions, one in the x-axis and the
  other in the y-axis.
• For each axis, use the methods for the
  1-dimensional case and answer two 1-dimensional
  MOR queries. The algorithms for the
  1-dimensional case can be used.
• Take the intersection of the two answers to find
  the answer to the initial query.

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4. Summary

• One-dimensional case,
• We have
• A dynamic, external memory algorithm with
  guaranteed worst case performance and linear
  space.
• A practical approximation algorithm also in the
  dynamic, external memory setting, which has
  linear space and expected logarithmic query time.
• An algorithmic query time for a restricted
  version of the problem.
• Two-dimensional case,
• Extending the techniques to 2-dimensional
  networks of 1-dimensional routes as well objects
  moving on a plane.

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5. Reference

• George Kollios, Dimitrios Gunopulos and Vassilis
  Tsotras. On Indexing Mobile Objects. ACM PODS,
  1999.

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On Indexing Mobile Objects

• The End
• Break or Move on to Presentation 2