Progressive Computation
of The Min-Dist
Optimal-Location Query
Donghui Zhang,
Yang Du, Tian Xia, Yufei Tao*
Northeastern University
* Chinese University of Hong Kong
VLDB’06, Seoul, Korea
Motivation
• “What is the optimal location in
Boston area to build a new
McDonald’s store?”
• Suppose a customer drives to the
closest McDonald’s.
• Optimality: Minimize AVG driving
distance.
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min-dist OL
200
600
200
600
• Without any new site:
AD = (200+200+600+600)/4 = 400.
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min-dist OL
30
30
l1
600
600
• Without any new site:
AD = (200+200+600+600)/4 = 400.
• With new site l1:
AD(l1) = (30+30+600+600)/4 = 315.
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min-dist OL
200
30
30
200
• Without any new site:
AD = (200+200+600+600)/4 = 400.
• With new site l1:
AD(l1) = (30+30+600+600)/4 = 315.
• With new site l2 :
AD(l2) = (200+200+30+30)/4 = 115.
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l2
5
Formal Definition
• Given a set S of sites, a set O of objects,
and a query range Q ,
• min-dist OL is a location l  Q which
minimizes
1
AD(l ) 
dNN (o, S  {l})

oO
|O|
distance between o and its nearest site
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L1 Distance
• d(o, s) = |o.x – s.x|+|o.y – s.y|
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Challenging
1. There are infinite number of
locations in Q. How to produce a
finite set of candidates (yet keeping
optimality)?
2. How to avoid computing AD(l) for
all candidates?
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Solution Highlights
1. Algorithm to compute AD(l).
2. Theorems to limit #candidates.
3. Lower-bound of AD(l) for all
locations l in a cell C.
4. Progressive algorithm.
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1. Compute AD(l)
1
• Remember AD(l )  | O | oO dNN (o, S  {l})
1
• Define
AD 
dNN (o, S )

oO
|O|
• Let RNN(l) be the objects “attracted” by l.
• AD(l)=AD if RNN(l)=
l
RNN(l)=
AD=AD(l)
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1. Compute AD(l)
1
• Remember AD(l )  | O | oO dNN (o, S  {l})
1
• Define
AD 
dNN (o, S )

oO
|O|
• Let RNN(l) be the objects “attracted” by l.
• AD(l)=AD if RNN(l)=
l
RNN(l)={o7, o8}
AD(l) < AD
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1. Compute AD(l)
1
• Remember AD(l )  | O | oO dNN (o, S  {l})
1
• Define
AD 
dNN (o, S )

oO
|O|
• Let RNN(l) be the objects “attracted” by l.
• AD(l)=AD if RNN(l)=
• AD(l)=AD - ?
Average savings for customers in RNN(l)
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1. Compute AD(l)
• Theorem
1
AD(l )  AD 
(dNN (o, S )  d (o, l ))

oRNN ( l )
|O|
• S and O are “static” versus l.
– AD can be pre-computed.
– So is dNN(o, S)
• To compute AD(l):
– Find RNN(l)
– oRNN(l), compute d(o, l)
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2. Limit #candidates
• Theorem: within the X/Y range of
Q, draw grid lines crossing objects.
Only need to consider intersections!
Q
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2. Limit #candidates
• Theorem: within the X/Y range of
Q, draw grid lines crossing objects.
Only need to consider intersections!
Q
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5x6=30 candidates
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2. Limit #candidates
• Proof idea: suppose the OL is not,
move it will produce a better (or
equal) result.
δ
l
• Consider RNN(l).
• Move to the right  saves total dist.
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2. VCU(Q)
• A spatial region, enclosing the
objects closer to Q than to sites in
S.
• It’s the Voronoi cell of Q versus
sites in S.
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2. Further Limit #candidates
• Only consider objects in VCU(Q).
5x6=30 candidates
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2. Further Limit #candidates
• Only consider objects in VCU(Q).
5x6=30 candidates
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2. Further Limit #candidates
• Only consider objects in VCU(Q).
4x4=16 candidates
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Naïve Algorithm
• Derive candidates.
• Compute AD(l) for each.
• Pick smallest.
• Not efficient! Too many candidates!
To compute AD(l) for each one, need:
• compute RNN(l)
• retrieve all these objects…
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Progressive Idea
• Treat Q as a cell and consider its
corners.
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Progressive Idea
• Divide the cell.
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Progressive Idea
• Divide the cell.
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Progressive Idea
• Recursively divide a sub-cell.
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Progressive Idea
• Recursively divide a sub-cell.
• Able to check all candidates.
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Progressive Idea
• Q: What do you save?
• A: Cell pruning, if its lower bound 
AD(l0) of some candidate l0.
AD(lo )
=50
Suppose 60 is a lower bound for AD(l), l C
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3. LB(C): lower bound for
AD(l), lC
AD(c1)=1000
AD(c2)=3000
c
AD(c3)=4000
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AD(c4)=2500
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3. LB(C): lower bound for
AD(l), lC
AD(c1)=1000
AD(c2)=3000
c
AD(c3)=4000
• Theorem: max{
AD(c4)=2500
AD(c1 )  AD(c4 ) AD(c2 )  AD(c3 )
p
,
}
2
2
4
is a lower bound, where p is perimeter.
• e.g. LB(C)=3500-p/4
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3. LB(C): lower bound for
AD(l), lC
• A better lower bound Theorem:
AD(c1 )  AD(c4 ) AD(c2 )  AD(c3 )
p | VCU (C ) |
max{
,
} *
2
2
4
|O|
• Comparing with the previous lower bound:
• Higher quality since the lower bound is larger.
• More computation.
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4. The Progressive Algorithm
1. Maintain a heap of cells ordered by LB().
Initially one cell: Q.
2. Maintain the best candidate lopt
3. Pick the cell with minimum LB() and
partition it.
4. Compute AD() for the corners of subcells.
5. Compute LB() for the sub-cells.
6. Insert sub-cell ci to heap if LB(ci)<AD(lopt)
7. Goto 3.
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Progressiveness
•
The algorithm quickly reports a
candidate OL with a confidence interval,
and keeps refining.
AD(best corner of Q)
AD( real OL ) is inside the interval
LB(Q)
Time
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Progressiveness
•
The algorithm quickly reports a
candidate OL with a confidence interval,
and keeps refining.
AD(best candidate)
AD( real OL ) is inside the interval
LB(Q)
Time
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Progressiveness
•
The algorithm quickly reports a
candidate OL with a confidence interval,
and keeps refining.
AD(best candidate)
AD( real OL ) is inside the interval
Min{ LB(C) | C in heap }
•
Time
User may choose to terminate any time.
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Batch Partitioning
•
•
•
To partition a cell, should partition into
multiple sub-cells.
Reason: to compute AD(l), need to
access the R*-tree of objects. When
access the R*-tree, want to compute
multiple AD(l).
Tradeoff: if partition too much: wasteful!
Since some candidates could be pruned.
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Performance Setup
• O: 123,593 postal addresses in
Northeastern part of US. Stored
using an R*-tree.
• S: randomly select 100 sites from O.
• Buffer: 128 pages.
• Dell Pentium IV 3.2GHz.
• Query size: 1% in each dimension.
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2. Further Limit #candidates
• Only consider objects in VCU(Q).
4x4=16 candidates
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Effect of VCU Computation
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3. LB(C): lower bound for
AD(l), lC
AD(c1)=1000
AD(c2)=3000
c
AD(c3)=4000
• Theorem: max{
AD(c4)=2500
AD(c1 )  AD(c4 ) AD(c2 )  AD(c3 )
p
,
}
2
2
4
is a lower bound, where p is perimeter.
• e.g. LB(C)=3500-p/4
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3. LB(C): lower bound for
AD(l), lC
• A better lower bound Theorem:
AD(c1 )  AD(c4 ) AD(c2 )  AD(c3 )
p | VCU (C ) |
max{
,
} *
2
2
4
|O|
• Comparing with the previous lower bound:
• Higher quality since the lower bound is larger.
• More computation.
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Comparison of Lower Bounds
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Effect of Batch Partitioning
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Progressiveness
•
The algorithm quickly reports a
candidate OL with a confidence interval,
and keeps refining.
AD(best candidate)
AD( real OL ) is inside the interval
Min{ LB(C) | C in heap }
•
Time
User may choose to terminate any time.
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Progressiveness
•Each step: partition a cell to 40 sub-cells.
•After 200 steps, accurate answer.
•After 20 steps, answer is 1% away from optimal.
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Conclusions
• Introduced the min-dist optimallocation query.
• Proved theorems to limit the number
of candidates.
• Presented lower-bound estimators.
• Proposed a progressive algorithm.
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