On Computing Topt Most Influential Spatial Sites

1
On Computing Top-t Most Influential Spatial Sites

• Tian Xia, Donghui Zhang, Evangelos Kanoulas, Yang
  Du
  
• Northeastern University
• Boston, USA

2
Outline

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

3
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.

4
Motivation

• Which supermarket in Boston is the most
  influential among residential buildings?
  
• Sites supermarkets
• Objects residential buildings
• Weight people in a building
• Query region Boston
• Which wireless station in Boston is the most
  influential among mobile users?

5
Example

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

6
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.

7
Outline

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

8
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.

9
Related Work

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

10
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.

11
Related Work SRAE01
o2
o4
s2
s3
o5
o1
s4
s1
o3
o6
12
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.

13
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.

14
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.

15
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.

16
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.

17
Outline

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

18
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.

19
maxDist A Loose Estimation

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

20
minMaxDist A Tight Estimation?

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

21
minMaxDist A Tight Estimation?

• Not true for an object MBR O1.

22
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.

23
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)

24
Examples of minExistDNNS1(O1)

• How to calculate it?

25
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)!
26
Space Partitioning

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

27
Calculating minExistDNNS1(O1)

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

28
Outline

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

29
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.

30
Data Structure

• queueSIN
• queueO
• queueSOUT

S1
S2
O1
S3
31
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.

32
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.

33
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.

34
A Pruning Case
S1
Expand S1

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

35
Choosing an Entry to Expand

• Expand top entries in queueSIN.
• Expand the most important Oi.
• Importance Oi affected entries area(Oi)
• Expand Sj that contains the most important Oi.

36
Choosing an Entry to Expand

• Estimate the probability of pruning Oi using some
  Sj in queueSOUT.

• After expanding S2, O1 is likely not to affect S1.

37
Outline

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

38
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.

39
Selected Experimental Results

• sites objects 1 2.5

40
Selected Experimental Results

• sites objects 2.5 1

41
Outline

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

42
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.

43
Thank you!

• Q A