Post on 17-Dec-2015
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
Donghui Zhang et al. Optimal Location Query 2
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.
Donghui Zhang et al. Optimal Location Query 3
Who will be interested?Who will be interested?
• Corporations– Chained restaurants (e.g. McDonald’s, Burger
King, Starbucks)– Supermarkets (e.g. Wal-Mart, Costco, Stop &
Shop)– Location-based service providers (e.g. Verizon,
AT&T)
• Computer Scientists especially in– Databases– Computational Geometry– Algorithms
Donghui Zhang et al. Optimal Location Query 4
min-dist OLmin-dist OL
• Without any new site: AD = (200+200+600+600)/4 = 400.
200
200
600
600
Donghui Zhang et al. Optimal Location Query 5
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
Donghui Zhang et al. Optimal Location Query 6
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
Donghui Zhang et al. Optimal Location Query 7
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)(
• “Solution”: compute all AD(l). But…
Donghui Zhang et al. Optimal Location Query 8
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?
Donghui Zhang et al. Optimal Location Query 9
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.
Donghui Zhang et al. Optimal Location Query 10
L1 DistanceL1 Distance
• d(o, s) = |o.x – s.x|+|o.y – s.y|
Donghui Zhang et al. Optimal Location Query 11
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)
Donghui Zhang et al. Optimal Location Query 12
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
Donghui Zhang et al. Optimal Location Query 13
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)
Donghui Zhang et al. Optimal Location Query 14
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)
Donghui Zhang et al. Optimal Location Query 15
How to compute RNN(How to compute RNN(ll)?)?
• This is an implementation detail, dealing with computational geometry and spatial databases.
• Naïve solution: o O , compare with all sites and l.
• More efficient: 1. Compute Voronoi cell of l.2. Retrieve objects inside the Voronoi cell
using a range search on R-tree.
Donghui Zhang et al. Optimal Location Query 16
How to compute RNN(How to compute RNN(ll)?)?(1) Compute Voronoi cell(1) Compute Voronoi cell
• Remember: RNN(l) is the set of objects close to l than to any existing site in S.
• Consider all sites. Draw a spatial region close to l than to any site.
l
Donghui Zhang et al. Optimal Location Query 17
How to compute RNN(How to compute RNN(ll)?)?(2) Retrieve objects(2) Retrieve objects
• Standard range search.• Any spatial access methods, e.g. R-
tree.
Donghui Zhang et al. Optimal Location Query 18
20 4 6 8 10
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Range query: find the objects in a given range.E.g. find all hotels in Boston.
No index: scan through all objects. NOT EFFICIENT!
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20 4 6 8 10
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aE3
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Minimum Bounding Rectangle (MBR)
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Donghui Zhang et al. Optimal Location Query 24
20 4 6 8 10
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y axis
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E1 E2
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Donghui Zhang et al. Optimal Location Query 25
2. Limit #candidates2. Limit #candidates
• Theorem: within the X/Y range of Q, draw grid lines crossing objects. Only need to consider intersections!
Q
Donghui Zhang et al. Optimal Location Query 26
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
Donghui Zhang et al. Optimal Location Query 27
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.
Donghui Zhang et al. Optimal Location Query 28
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.
Q
Donghui Zhang et al. Optimal Location Query 29
2. Further Limit #candidates2. Further Limit #candidates
• Only consider objects in VCU(Q).
5x6=30 candidates
Donghui Zhang et al. Optimal Location Query 30
2. Further Limit #candidates2. Further Limit #candidates
5x6=30 candidates
• Only consider objects in VCU(Q).
Donghui Zhang et al. Optimal Location Query 31
2. Further Limit #candidates2. Further Limit #candidates
4x4=16 candidates
• Only consider objects in VCU(Q).
Donghui Zhang et al. Optimal Location Query 32
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…
Donghui Zhang et al. Optimal Location Query 33
Progressive IdeaProgressive Idea
• Treat Q as a cell and consider its corners.
Donghui Zhang et al. Optimal Location Query 34
Progressive IdeaProgressive Idea
• Divide the cell.
Donghui Zhang et al. Optimal Location Query 35
Progressive IdeaProgressive Idea
• Divide the cell.
Donghui Zhang et al. Optimal Location Query 36
Progressive IdeaProgressive Idea
• Recursively divide a sub-cell.
Donghui Zhang et al. Optimal Location Query 37
Progressive IdeaProgressive Idea
• Recursively divide a sub-cell.
• Able to check all candidates.
Donghui Zhang et al. Optimal Location Query 38
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
Donghui Zhang et al. Optimal Location Query 39
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
Donghui Zhang et al. Optimal Location Query 40
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
Donghui Zhang et al. Optimal Location Query 41
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.
Donghui Zhang et al. Optimal Location Query 42
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.
Donghui Zhang et al. Optimal Location Query 43
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
Donghui Zhang et al. Optimal Location Query 44
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
Donghui Zhang et al. Optimal Location Query 45
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.
Donghui Zhang et al. Optimal Location Query 46
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.
Donghui Zhang et al. Optimal Location Query 47
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.
Donghui Zhang et al. Optimal Location Query 48
4x4=16 candidates
• Only consider objects in VCU(Q).
2. Further Limit #candidates2. Further Limit #candidates
Donghui Zhang et al. Optimal Location Query 49
Effect of VCU ComputationEffect of VCU Computation
Donghui Zhang et al. Optimal Location Query 50
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
Donghui Zhang et al. Optimal Location Query 51
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.
Donghui Zhang et al. Optimal Location Query 52
Comparison of Lower BoundsComparison of Lower Bounds
Donghui Zhang et al. Optimal Location Query 53
Effect of Batch PartitioningEffect of Batch Partitioning
Donghui Zhang et al. Optimal Location Query 54
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.
Donghui Zhang et al. Optimal Location Query 55
ProgressivenessProgressiveness
•Each step: partition a cell to 40 sub-cells.•After 200 steps, accurate answer.•After 20 steps, answer is 1% away from optimal.
Donghui Zhang et al. Optimal Location Query 56
ConclusionsConclusions
• Introduced the min-dist optimal-location query.
• Proved theorems to limit the number of candidates.
• Presented lower-bound estimators.• Proposed a progressive algorithm.