A Trajectory Splitting Model for Efficient SpatioTemporal Indexing

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A Trajectory Splitting Model for Efficient
Spatio-Temporal Indexing

• Slobodan Rasetic
• University of Alberta

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Content

• Introduction
• Background
• Motivation
• Trajectory Splitting Cost Model
• Optimal Algorithm
• Linear Heuristic Algorithm
• Experiments
• References

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Introduction

• Spatiotemporal data consists of observations with
  their timestamps and spatial location.

Moving object trajectory Pfo00
Moving object trajectories Pfo00
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Introduction

• We want to efficiently process spatiotemporal
  queries.
• Straightforward solution is to use R-tree index
  where each trajectory is represented by an MBR
• The problem is that this approach introduces a
  lot of dead space into our index resulting in
  high number of disk I/Os at the data level.
• The solution is to split the trajectories to
  reduce the amount of dead space and
  corresponding number of disk I/Os
• We develop a trajectory splitting cost model that
  minimizes number of disk I/Os with respect to a
  given average query size.
• We use this cost model to develop optimal
  splitting algorithm and near optimal linear
  heuristic that we evaluate in our experiments.

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Background

• R-Tree based structures provide a good mechanism
  for supporting a wide range of query types for
  Spatiotemporal data.
• Spatial-temporal data, like spatial data can be
  approximated using a Minimum Bounding Rectangle
  (MBR).

Approximation of Spatial Data Gut84
A resulting R-Tree Structure Gut84
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Background

• Approximating trajectories using MBRs and using
  an R-Tree based structure for query support has
  the following benefits
• 1. Low storage overhead.
• 2. Low computational overhead to answer user
  queries.
• Approximating trajectories using R-Tree based
  structures has the drawback of introducing a
  great deal of dead space.
• This dead space can be reduced by dividing
  trajectories into smaller segments and
  approximating their sub components.

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Motivation

• Approximating smaller trajectory segments reduces
  dead space.
• Too much dead space leads to poor query
  performance (false hits).
• Average query size should be considered when
  splitting trajectories

Reducing the dead space occupied by a trajectory
Pfo00
Relation between query size and trajectory splits
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Trajectory Splitting Cost Model

• Probability that a range query intersects a MBR

• Query extended MBR

A query extended MBR
The volume of the extended MBRs using 1,2 and
three segments
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Trajectory Splitting Cost Model

• Expected number of disk I/Os for a single
  trajectory

• Expected number of disk I/Os for the whole set

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Optimal Algorithm

• Optimal Algorithm is based on dynamic programming
  approach that is based on the following equations

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Linear Heuristic Algorithm

• The area of extended MBR obtained in j increments
  can be expressed using the following formula

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Linear Heuristic Algorithm

• Now assume a trajectory that is approximated by
  several MBRs, where each MBR contains the same
  number of j1 points
• The estimated sum of the areas of the query
  extended MBRs can now be computed

where t is the total number of trajectory points
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Linear Heuristic Algorithm

• To minimize the estimated sum the first
  derivative is applied

where k1, k2 and k3 are positive constants

• Resulting in a following formula from which j
  can be found

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Linear Heuristic Algorithm

• Algorithm LinearSplit
• //first two points of a trajectory T
• p1 T1, 1 p2 T2, 2
• u 1, v 2
• //while there are points in T
• while (pTv1,v1)
• if find w for Tu,v using Equation 17
• if (w lt v-u)
• // extract w points from Tu,v
• insert Tu,uw into the index
• else
• collect more points from T if
  available to form a segment of
• length w and then put Tu,uw
• into the index if the end of T
• is found before, finish the last
• segment for T and exit
• vuw1
• uuw

• Algorithm LinearSplit
• //first two points of a trajectory T
• p1 T1, 1 p2 T2, 2
• u 1, v 2
• //while there are points in T
• while (pTv1,v1)
• if put Tu,v into the index
• v
• insert the possible remaining segment of T into
  the index

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Experiments

• Varying Database Size

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Experiments

• Varying Query Size

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Experiments

• Robustness Test

LinearSplit algorithm
Optimal Split algorithm
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References

• Gut84 GUTTMAN, A. R-trees a Dynamic Index
  Structure for Spatial Searching. In Proc. of
  ACM-SIGMOD Conference on the Management of Data,
  pp. 47-57, 1984.
• Pfo00 PFOSER, D., JENSEN, C. S., AND
  THEODORIDIS, Y.Novel Approaches in Query
  Processing for Moving Object Trajectories. In
  Proceedings of the 26st VLDB Conf. (Cairo,Egypt,
  September 2000), pp. 395406.