Progressive Computation of The Min-Dist Optimal-Location Query
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Progressive Computation Progressive Computation of The Min-Dist of The Min-Dist
Optimal-Location QueryOptimal-Location Query
Donghui ZhangDonghui Zhang, ,
Yang Du, Tian Xia, Yufei Tao*Yang Du, Tian Xia, Yufei Tao*
Northeastern UniversityNortheastern University
* Chinese University of Hong Kong* Chinese University of Hong Kong
VLDB’06, Seoul, Korea
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MotivationMotivation
• “ 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 OLmin-dist OL
• Without any new site: AD = (200+200+600+600)/4 = 400.
200
200
600
600
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min-dist OLmin-dist OL
• Without any new site: AD = (200+200+600+600)/4 = 400.• With new site l1: AD(l1) = (30+30+600+600)/4 = 315.
30600
60030
l1
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min-dist OLmin-dist OL
• 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.
3030
l2200
200
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Formal DefinitionFormal 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
distance between o and its nearest site
OolSodNN
OlAD }){,(
||
1)(
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L1 DistanceL1 Distance
• d(o, s) = |o.x – s.x|+|o.y – s.y|
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ChallengingChallenging
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 HighlightsSolution 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 1. Compute AD(l)AD(l)
• Remember
• Define
OoSodNN
OAD ),(
||
1
OolSodNN
OlAD }){,(
||
1)(
• 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 1. Compute AD(l)AD(l)
• Remember
• Define
OoSodNN
OAD ),(
||
1
OolSodNN
OlAD }){,(
||
1)(
• 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 1. Compute AD(l)AD(l)
• Remember
• Define
OoSodNN
OAD ),(
||
1
OolSodNN
OlAD }){,(
||
1)(
• AD(l)=AD - ?
• Let RNN(l) be the objects “attracted” by l.• AD(l)=AD if RNN(l)=
Average savings for customers in RNN(l)
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1. Compute 1. Compute AD(l)AD(l)
• Theorem
)()),(),((
||
1)(
lRNNolodSodNN
OADlAD
• S and O are “static” versus l.– AD can be pre-computed.– So is dNN(o, S)
• To compute AD(l):– Find RNN(l) oRNN(l), compute d(o, l)
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2. Limit #candidates2. 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 #candidates2. Limit #candidates
• Theorem: within the X/Y range of Q, draw grid lines crossing objects. Only need to consider intersections!
5x6=30 candidates
Q
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2. Limit #candidates2. Limit #candidates• Proof idea: suppose the OL is not, move it
will produce a better (or equal) result.
l
• Consider RNN(l).
δ
• Move to the right saves total dist.
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2. VCU(2. VCU(QQ))
• 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 #candidates2. Further Limit #candidates
• Only consider objects in VCU(Q).
5x6=30 candidates
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2. Further Limit #candidates2. Further Limit #candidates
5x6=30 candidates
• Only consider objects in VCU(Q).
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2. Further Limit #candidates2. Further Limit #candidates
4x4=16 candidates
• Only consider objects in VCU(Q).
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Naïve AlgorithmNaïve 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 IdeaProgressive Idea
• Treat Q as a cell and consider its corners.
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Progressive IdeaProgressive Idea
• Divide the cell.
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Progressive IdeaProgressive Idea
• Divide the cell.
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Progressive IdeaProgressive Idea
• Recursively divide a sub-cell.
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Progressive IdeaProgressive Idea
• Recursively divide a sub-cell.
• Able to check all candidates.
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Progressive IdeaProgressive 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(3. LB(CC): lower bound for ): lower bound for AD(AD(ll), ), llCC
AD(c1)=1000 AD(c2)=3000
AD(c3)=4000 AD(c4)=2500
c
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3. LB(3. LB(CC): lower bound for ): lower bound for AD(AD(ll), ), llCC
• Theorem: 4
}2
)()(,
2
)()(max{ 3241 pcADcADcADcAD
AD(c1)=1000 AD(c2)=3000
AD(c3)=4000 AD(c4)=2500
is a lower bound, where p is perimeter.
• e.g. LB(C)=3500-p/4
c
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3. LB(3. LB(CC): lower bound for ): lower bound for AD(AD(ll), ), llCC
• A better lower bound Theorem:
||
|)(|*
4}
2
)()(,
2
)()(max{ 3241
O
CVCUpcADcADcADcAD
• Comparing with the previous lower bound:• Higher quality since the lower bound is larger.• More computation.
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4. The Progressive Algorithm4. The Progressive Algorithm
1. Maintain a heap of cells ordered by LB(). Initially one cell: Q.
2. Maintain the best candidate lopt3. Pick the cell with minimum LB() and
partition it.4. Compute AD() for the corners of sub-cells.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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ProgressivenessProgressiveness
• The algorithm quickly reports a candidate OL with a confidence interval, and keeps refining.
Time
AD(best corner of Q)
LB(Q)
AD( real OL ) is inside the interval
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ProgressivenessProgressiveness
• The algorithm quickly reports a candidate OL with a confidence interval, and keeps refining.
Time
AD(best candidate)
LB(Q)
AD( real OL ) is inside the interval
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ProgressivenessProgressiveness
• The algorithm quickly reports a candidate OL with a confidence interval, and keeps refining.
Time
AD(best candidate)
Min{ LB(C) | C in heap }
AD( real OL ) is inside the interval
• User may choose to terminate any time.
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Batch PartitioningBatch 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 SetupPerformance 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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4x4=16 candidates
• Only consider objects in VCU(Q).
2. Further Limit #candidates2. Further Limit #candidates
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Effect of VCU ComputationEffect of VCU Computation
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3. LB(3. LB(CC): lower bound for ): lower bound for AD(AD(ll), ), llCC
• Theorem: 4
}2
)()(,
2
)()(max{ 3241 pcADcADcADcAD
AD(c1)=1000 AD(c2)=3000
AD(c3)=4000 AD(c4)=2500
is a lower bound, where p is perimeter.
• e.g. LB(C)=3500-p/4
c
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3. LB(3. LB(CC): lower bound for ): lower bound for AD(AD(ll), ), llCC
• A better lower bound Theorem:
||
|)(|*
4}
2
)()(,
2
)()(max{ 3241
O
CVCUpcADcADcADcAD
• Comparing with the previous lower bound:• Higher quality since the lower bound is larger.• More computation.
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Comparison of Lower BoundsComparison of Lower Bounds
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Effect of Batch PartitioningEffect of Batch Partitioning
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ProgressivenessProgressiveness
• The algorithm quickly reports a candidate OL with a confidence interval, and keeps refining.
Time
AD(best candidate)
Min{ LB(C) | C in heap }
AD( real OL ) is inside the interval
• User may choose to terminate any time.
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ProgressivenessProgressiveness
•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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ConclusionsConclusions
• Introduced the min-dist optimal-location query.
• Proved theorems to limit the number of candidates.
• Presented lower-bound estimators.• Proposed a progressive algorithm.