Monitoring k-NN Queries over Moving Objects

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Monitoring k-NN Queries over Moving Objects

• Xiaohui Yu
• University of Toronto
• xhyu_at_cs.toronto.edu
• Joint work with Ken Pu and Nick Koudas

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k-Nearest Neighbors

• k-NN search Given a set of points, find the k
  points that are closest to the query point.
• We focus on k-NN for spatio-temporal data

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Problem

• A set of moving objects P on a 2D plane
• Monitoring the nearest neighbors of query points
  (Q) in a specified region over time

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Applications

• Location-based advertising
• E-flyers distribution identifying the customers
  closest to the store

• Location-based mixed reality games

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Outline

• Related work
• Object-Indexing
• Overhaul algorithm
• Incremental index update/query answering
• Query-Indexing
• Hierarchical Object-Indexing
• Experiments
• Conclusions

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Previous Work

• Most work has focused on predictive queries
• Who will be my NNs five minutes from now?
• Assumption the trajectories of the objects are
  fully predictable
• linear/non-linear/autoregressive functions
• very frequent updates/re-evaluations when the
  assumption does not hold Sun et al. 2004

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Previous Work

• The assumption is often violated in real
  applications where the objects movements are
  non-predictable.

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Our approach

• No assumptions on the motion of objects
  arbitrary speeds/directions

receive buffer
updates

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Grid-based index structures

• Residing in main memory
• Partition the space into NN grids
• Easy to maintain, supporting fast query
  processing

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Outline

• Related work
• Object-Indexing
• Overhaul algorithm
• Incremental index update/query answering
• Query-Indexing
• Hierarchical Object-Indexing
• Experiments
• Conclusions

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Object Indexing
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Example Finding 3-NN
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The algorithm

• Initial computation
• Progressively enlarge the search region
• Until k neighbors are found
• Calculate the critical region Rcrit (guaranteed
  to contain the querys k-NNs)
• Search in the region for the k-NNs.

The overhaul algorithm
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Overhaul algorithm Analysis

• Notation
• NP - number of objects
• NQ - number of queries
• Running time breakdown
• Tindex a0NP
• Tquery a1 NQ ( of cells in Rcrit) a2 NQ
  ( of objects in Rcrit)
• Optimal cell size to minimize Tquery (assuming
  uniformity)
• Proportional to

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Analysis non-uniform data
Measure of non-uniformity

• Reasonably skewed distributions Tquery
• Highly skewed Tquery

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Outline

• Related work
• Object-Indexing
• Overhaul algorithm
• Incremental index update/query answering
• Query-Indexing
• Hierarchical Object-Indexing
• Experiments
• Conclusions

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From overhaul to incremental

• Incremental update of the object-index
• Check if the new position falls in the same cell
  as in the previous cycle
• Yes do nothing
• No remove it from old cell, insert it into the
  new cell
• Incremental query answering
• Compute the critical region based its previous
  k-NN
• Search the critical region for the current k-NN

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Which one is better?

• Mobility is the key
• Index maintenance
• The probability of exiting the current cell is
  crucial
• Incremental query answering
• When mobility is low, the cost of query answering
  is
• Worst case O(NP )

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Outline

• Related work
• Object-Indexing
• Overhaul algorithm
• Incremental index update/query answering
• Query-Indexing
• Hierarchical Object-Indexing
• Experiments
• Conclusions

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Query-Indexing

• Cost of Object-Indexing dominated by indexing
  time when NQ is small
• Indexing queries instead of objects

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Query-Index
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Constructing a Query Index
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Query answering
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Query-Indexing algorithm

• Index-building
• Compute the critical region
• Insert a query into cells contained in its
  critical region
• Query-answering
• For each object
• Determine the cell it belongs to
• For each query registered with the cell, update
  its k-NN if necessary.

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Analysis

• indexing time query answering time
• Theoretically,
• QI suffers from less localized access to objects

• QI is preferable when NQ is small

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Outline

• Related work
• Object-Indexing
• Overhaul algorithm
• Incremental index update/query answering
• Query-Indexing
• Hierarchical Object-Indexing
• Experiments
• Conclusions

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Problem with one-level object indexing
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Hierarchical Object-Indexing

• Split the overly crowded cells into smaller cells

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• Refined approximation

q
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Outline

• Related work
• Object-Indexing
• Overhaul algorithm
• Incremental index update/query answering
• Query-Indexing
• Hierarchical Object-Indexing
• Experiments
• Conclusions

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Experiments

• Verify the analytical results
• NP, NQ, k, velocity, cell-size
• Compare the performance of the proposed
  structures with that of R-tree-based methods
• NP, NQ, k, velocity, skew

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Highlights

• Tindex and Tquery of Object-Indexing
• Optimal cell size
• Overhaul v.s. incremental computation as
  velocities of objects vary
• Object-Indexing v.s. Query-Indexing
• Comparison of grid-based algorithms with
  R-tree-based algorithms

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Performance of overhaul w.r.t. NP
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Effect of cell-size on performance
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Overhaul v.s. incremental index maintenance
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Object-Indexing v.s. Query Indexing
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Comparison with R-trees datasets
uniform
hi-skewed
skewed
Simulation using Illinois road network
(600km600km)
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Comparison with R-trees
NP 100,000, NQ 5,000, k 10
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Conclusions

• We proposed two solutions to monitor k-NN
• Object-Indexing
• Query-Indexing
• Extensions to handle skewed data
• Outperform R-tree-based solutions