Progressive Computation of The Min-Dist Optimal-Location Query Donghui Zhang, Yang Du, Tian Xia,...

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.


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


min-dist OLmin-dist OL

• Without any new site: AD = (200+200+600+600)/4 = 400.

200

200

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.

30600

60030

l1



• 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


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…


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?


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.


L1 DistanceL1 Distance

• d(o, s) = |o.x – s.x|+|o.y – s.y|


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)



• Remember

• Define

OoSodNN

OAD ),(

||

1

OolSodNN

OlAD }){,(

||

1)(


l

RNN(l)={o7, o8}AD(l) < AD



• Remember

• Define

OoSodNN

OAD ),(

||

1

OolSodNN

OlAD }){,(

||

1)(

• AD(l)=AD - ?


Average savings for customers in RNN(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)


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.


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


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.


20 4 6 8 10

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x axis

y axis

b

c

a

d

e f

g h

i j

k

l

m

Range query: find the objects in a given range.E.g. find all hotels in Boston.

No index: scan through all objects. NOT EFFICIENT!


20 4 6 8 10

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4

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10

x axis

y axis

b

c

aE3

a b c d e

E1 E2

E3 E4 E5

Root

E1 E2

E3E4

f g h

E5

d

e f

g h

i j

k

l

m

l m

E7

i j k

E6

E6 E7

Minimum Bounding Rectangle (MBR)


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x axis

y axis

b

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aE3

d

e f

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i j

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E4

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a b c d e

E1 E2

E3 E4 E5

Root

E1 E2

E3E4

f g h

E5

l m

E7

i j k

E6

E6 E7


20 4 6 8 10

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4

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x axis

y axis

b

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a

E1d

e f

g h

i j

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E2

a b c d e

E1 E2

E3 E4 E5

Root

E1 E2

E3E4

f g h

E5

l m

E7

i j k

E6

E6 E7


2. Limit #candidates2. Limit #candidates

• Theorem: within the X/Y range of Q, draw grid lines crossing objects. Only need to consider intersections!

Q


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


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.


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


2. Further Limit #candidates2. Further Limit #candidates

• Only consider objects in VCU(Q).

5x6=30 candidates



5x6=30 candidates




4x4=16 candidates



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…


Progressive IdeaProgressive Idea

• Treat Q as a cell and consider its corners.



• Divide the cell.



• Recursively divide a sub-cell.



• Recursively divide a sub-cell.

• Able to check all candidates.


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


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



• 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



• 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.


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.


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




Time

AD(best candidate)

LB(Q)





Time

AD(best candidate)

Min{ LB(C) | C in heap }


• User may choose to terminate any time.


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.


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.


4x4=16 candidates




Effect of VCU ComputationEffect of VCU Computation



• 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



• 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.


Comparison of Lower BoundsComparison of Lower Bounds


Effect of Batch PartitioningEffect of Batch Partitioning




Time

AD(best candidate)

Min{ LB(C) | C in heap }


• User may choose to terminate any time.



•Each step: partition a cell to 40 sub-cells.•After 200 steps, accurate answer.•After 20 steps, answer is 1% away from optimal.


ConclusionsConclusions

• Introduced the min-dist optimal-location query.

• Proved theorems to limit the number of candidates.

• Presented lower-bound estimators.• Proposed a progressive algorithm.

Progressive Computation of The Min-Dist Optimal-Location Query Donghui Zhang, Yang Du, Tian Xia,...

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