On Computing Topt Influential Spatial Sites

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On Computing Top-t Influential Spatial Sites

• Authors T. Xia, D. Zhang, E. Kanoulas, Y.Du
• Northeastern University, USA
• Appeared in VLDB 2005
• Presenter Xiangyuan Dai

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Outline

• Problem Definition
• Related Work
• The New Metric minExistDNN
• Data Structures and Algorithm
• Experimental Results
• Conclusions

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Motivation

• Which candidate position of a new McDonalds in
  Cenrtal and Western is the most influential among
  residential buildings?
• Sites candidate positions of new McDonalds
• Objects residential buildings
• Weight people in a building
• Query region Central and Western District
• Which wireless station in Hong Kong is the most
  influential among mobile users?

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Problem Definition

• Given
• a set of sites S
• a set of weighted objects O
• a spatial region Q
• an integer t.
• Top-t most influential sites query
• find t sites in Q with the largest influences.
• influence of a site s total weight of objects
  that consider s as the nearest site.

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Example

• Suppose all objects have weight 1, Q is the
  whole space, and t 1.
• The most influential site is s1, with influence
  3.

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Example
o2
o4
s2
s3
o5
o1
s4
s1
o3
o6

• Now that Q is the shadowed rectangle and t 2.
• Top-2 most influential sites s4 and s2.

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Outline

• Problem Definition
• Related Work
• The New Metric minExistDNN
• Data Structures and Algorithm
• Experimental Results
• Conclusions

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Related Work

• Bi-chromatic RNN query considers two datasets,
  sites and objects.
• The RNNs of a site s ? S are the objects that
  consider s as the nearest site.

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Related Work

• Solutions to the RNN query based on
  pre-computation KM00, YL01.

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Related Work

• Solution to RNN query based on Voronoi diagram
  SRAE01.
• Compute the Voronoi cell of s a region enclosing
  the locations closer to s than to any other
  sites.
• Querying the object R-tree using the Voronoi cell.

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Related Work SRAE01
o2
o4
s2
s3
o5
o1
s4
s1
o3
o6
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Our Problem vs. RNN Query

• RNN query
• A single site as an input.
• Interested in the actual set of the RNNs.
• Top-t most influential sites query
• A spatial region as an input.
• Interested in the aggregate weight of RNNs.

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Straightforward Solution 1

• For each site, pre-compute its influence.
• At query time, find the sites in Q and return the
  t sites with max influences.
• Drawback 1 Costly maintenance upon updates.
• Drawback 2 binding a set of sites closely with a
  set of objects.

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Straightforward Solution 2

• An extension of the Voronoi diagram based
  solution to the RNN query.
• Find all sites in Q.
• For each such site, find its RNNs by using the
  Voronoi cell, and compute its influence.
• Return the t sites with max influences.

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Straightforward Solution 2

• Drawback 1 All sites in Q need to be retrieved
  from the leaf nodes.
• Drawback 2 The object R-tree and the site R-tree
  are browsed multiple times.
• For each site in Q, browse the site R-tree to
  compute the Voronoi Cell.
• For each such Voronoi Cell, browse the object
  R-tree to compute the influence.

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Features of Our Solution

• Systematically browse both trees once.
• Pruning techniques are provided based on a new
  metric, minExistDNN.
• No need to compute the influences for all sites
  in Q, or even to locate all sites in Q.

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Outline

• Problem Definition
• Related Work
• The New Metric minExistDNN
• Data Structures and Algorithm
• Experimental Results
• Conclusions

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Motivation

• Intuitively, if some object in Oi may consider
  some site in Sj as an NN, Oi affects Sj.
• To estimate the influences of all sites in a site
  MBR Sj, we need to know whether an object MBR Oi
  will affect Sj.

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maxDist A Loose Estimation

• If maxDist(O1, S1) lt minDist(O1, S2), O1 does not
  affect S2.
• Why not good enough?

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minMaxDist A Tight Estimation?

• An object o does not affect S2, if there exists
  S1 such that
• minMaxDist(o1, S1) lt minDist(o1, S2)

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minMaxDist A Tight Estimation?

• Not true for an object MBR O1.

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A Tight Estimation?

• A metric m(O1, S1) should
• guarantee that, each location in O1 is within
  m(O1, S1) of a site in S1,
• and be the smallest distance with this property.

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New Metric minExistDNNS1(O1)

• Definition minExistDNNS1(O1)
• max minMaxDist(l, S1) ? location l? O1
• O1 does not affect S2, if there exists S1, s.t.
  minExistDNNS1(O1) lt minDist(O1, S2).

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Examples of minExistDNNS1(O1)

• How to calculate it?

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Calculating minExistDNNS1(O1)

• Step 1 Space partitioning

Every location l in the same partition is
associated with the second closest corner of S1
the distance is minMaxDist(l, S1)!
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Space Partitioning

• O1 is divided into multiple sub-regions, one in
  each partition.

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Calculating minExistDNNS1(O1)

• Step 2 Choose up-to 8 locations on O1 border
  and compute the minMaxDists to S1.
• minExistDNN is the largest one!

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Outline

• Problem Definition
• Related Work
• The New Metric minExistDNN
• Data Structures and Algorithm
• Experimental Results
• Conclusions

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Data Structure

• Two R-trees S of sites, O of objects.
• Three queues
• queueSIN entries of S inside Q.
• queueSOUT entries of S outside Q.
• queueO entries of O.

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Data Structure (cont)

• queueSIN Sj Sj is a visited but not expanded
  entry in S, whose MBR is inside Q and whose
  maxInuencegt0
• queueO Oi Oi is a visited but not expanded
  entry in O, which affects some entry in queueSIN
• queueSOUT Sj Sj is a visited but not
  expanded entry in S, whose MBR is outside Q and
  which is affected by some entry in queueO

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Data Structure (cont)

• queueO only consists of entries from O that
  affect at least one entry in queueSIN
• queueSOUT only consists of entries from S (but
  outside Q) that are affected by at least one
  entry in queueO.

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Data Structure
S1
S2

• queueSIN
• queueO
• queueSOUT

O1
S3
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maxInfluence and minInfluence

• For each entry Sj in queueSIN,
• maxInfluence total weight of entries in queueO
  that affect Sj.
• minInfluence total weight of entries in queueO
  that ONLY affect Sj, divided by the number of
  objects in Sj.
• queueSIN is sorted in decreasing order of
  maxInfluence.

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Algorithm Overview

• Expand an entry from one of the three queues.
• Remove the entry from the queue.
• Retrieve the referenced node, and insert the
  (unpruned) entries into the same queue.
• Update maxInfluence and minInfluence if
  necessary.
• If top-t entries in queueSIN are sites, with
  minInfluences maxInfluences of all remaining
  entries, return.

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Example

• queueSIN S1
• queueO O1
• queueSOUT S3
• queueSIN S5, S7
• queueO O6
• queueSOUT S9

Q

• S6 is not affected by O1, prune S6.
• O5 does not affect S5 and S7, prune O5.
• S8 is not affected by O6, prune S8.

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A Pruning Case
S1
Expand S1

• S2 is pruned because of minExistDNNS3(O1) lt
  minDist(S2, O1)

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Outline

• Problem Definition
• Related Work
• The New Metric minExistDNN
• Data Structures and Algorithm
• Experimental Results
• Conclusions

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Experimental Setup

• Data sets
• 24,493 populated places in North America
• 9,203 cultural landmarks in North America
• R-tree page size 1 KB
• LRU buffer 128 disk pages.
• t 4.
• Comparing to the solution using Voronoi diagram.

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Selected Experimental Results

• sites objects 1 2.5

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Selected Experimental Results

• sites objects 2.5 1

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Outline

• Problem Definition
• Related Work
• The New Metric minExistDNN
• Data Structures and Algorithm
• Experimental Results
• Conclusions

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Conclusions

• We addressed a new problem Top-t most
  influential sites query.
• We proposed a new metric minExistDNN. It can be
  used to prune search space in NN/RNN related
  problems.
• We carefully designed an algorithm which
  systematically browses both R-trees once.
• Experiments showed more than an order of
  magnitude improvement.