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

Optimal Location Query 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. Donghui Zhang et al. Optimal Location Query

Optimal Location Query min-dist OL 600 200 200 600 Without any new site: AD = (200+200+600+600)/4 = 400. Donghui Zhang et al. Optimal Location Query

Optimal Location Query min-dist OL 600 30 l1 30 600 Without any new site: AD = (200+200+600+600)/4 = 400. With new site l1: AD(l1) = (30+30+600+600)/4 = 315. Donghui Zhang et al. Optimal Location Query

Optimal Location Query min-dist OL 200 30 l2 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. Donghui Zhang et al. Optimal Location Query

Optimal Location Query 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 distance between o and its nearest site Donghui Zhang et al. Optimal Location Query

Optimal Location Query L1 Distance d(o, s) = |o.x – s.x|+|o.y – s.y| Donghui Zhang et al. Optimal Location Query

Optimal Location Query Challenging There are infinite number of locations in Q. How to produce a finite set of candidates (yet keeping optimality)? How to avoid computing AD(l) for all candidates? Donghui Zhang et al. Optimal Location Query

Optimal Location Query Solution Highlights Algorithm to compute AD(l). Theorems to limit #candidates. Lower-bound of AD(l) for all locations l in a cell C. Progressive algorithm. Donghui Zhang et al. Optimal Location Query

Optimal Location Query 1. Compute AD(l) Remember Define 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

Optimal Location Query 1. Compute AD(l) Remember Define 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

Optimal Location Query 1. Compute AD(l) Remember Define 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) Donghui Zhang et al. Optimal Location Query

Optimal Location Query 1. Compute AD(l) Theorem 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) Donghui Zhang et al. Optimal Location Query

Optimal Location Query 2. 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

Optimal Location Query 2. 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 5x6=30 candidates

Optimal Location Query 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. Donghui Zhang et al. Optimal Location Query

Optimal Location Query 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. Donghui Zhang et al. Optimal Location Query

2. Further Limit #candidates Only consider objects in VCU(Q). 5x6=30 candidates Donghui Zhang et al. Optimal Location Query

2. Further Limit #candidates Only consider objects in VCU(Q). 5x6=30 candidates Donghui Zhang et al. Optimal Location Query

2. Further Limit #candidates Only consider objects in VCU(Q). 4x4=16 candidates Donghui Zhang et al. Optimal Location Query

Optimal Location Query 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… Donghui Zhang et al. Optimal Location Query

Optimal Location Query Progressive Idea Treat Q as a cell and consider its corners. Donghui Zhang et al. Optimal Location Query

Optimal Location Query Progressive Idea Divide the cell. Donghui Zhang et al. Optimal Location Query

Optimal Location Query Progressive Idea Divide the cell. Donghui Zhang et al. Optimal Location Query

Optimal Location Query Progressive Idea Recursively divide a sub-cell. Donghui Zhang et al. Optimal Location Query

Optimal Location Query Progressive Idea Recursively divide a sub-cell. Able to check all candidates. Donghui Zhang et al. Optimal Location Query

Optimal Location Query Progressive Idea Q: What do you save? A: Cell pruning, if its lower bound  AD(l0) of some candidate l0. AD(lo ) =50 C Suppose 60 is a lower bound for AD(l), l Donghui Zhang et al. Optimal Location Query

3. LB(C): lower bound for AD(l), lC AD(c1)=1000 AD(c2)=3000 c AD(c3)=4000 AD(c4)=2500 Donghui Zhang et al. Optimal Location Query

3. LB(C): lower bound for AD(l), lC AD(c1)=1000 AD(c2)=3000 c AD(c3)=4000 AD(c4)=2500 Theorem: is a lower bound, where p is perimeter. e.g. LB(C)=3500-p/4 Donghui Zhang et al. Optimal Location Query

3. LB(C): lower bound for AD(l), lC A better lower bound Theorem: Comparing with the previous lower bound: Higher quality since the lower bound is larger. More computation. Donghui Zhang et al. Optimal Location Query

4. The Progressive Algorithm Maintain a heap of cells ordered by LB(). Initially one cell: Q. Maintain the best candidate lopt Pick the cell with minimum LB() and partition it. Compute AD() for the corners of sub-cells. Compute LB() for the sub-cells. Insert sub-cell ci to heap if LB(ci)<AD(lopt) Goto 3. Donghui Zhang et al. Optimal Location Query

Optimal Location Query Progressiveness 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

Optimal Location Query 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 Donghui Zhang et al. Optimal Location Query

Optimal Location Query 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. Donghui Zhang et al. Optimal Location Query

Optimal Location Query 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. Donghui Zhang et al. Optimal Location Query

Optimal Location Query 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. Donghui Zhang et al. Optimal Location Query

2. Further Limit #candidates review slide 2. Further Limit #candidates Only consider objects in VCU(Q). 4x4=16 candidates Donghui Zhang et al. Optimal Location Query

Effect of VCU Computation Donghui Zhang et al. Optimal Location Query

3. LB(C): lower bound for AD(l), lC review slide 3. LB(C): lower bound for AD(l), lC AD(c1)=1000 AD(c2)=3000 c AD(c3)=4000 AD(c4)=2500 Theorem: is a lower bound, where p is perimeter. e.g. LB(C)=3500-p/4 Donghui Zhang et al. Optimal Location Query

3. LB(C): lower bound for AD(l), lC review slide 3. LB(C): lower bound for AD(l), lC A better lower bound Theorem: Comparing with the previous lower bound: Higher quality since the lower bound is larger. More computation. Donghui Zhang et al. Optimal Location Query

Comparison of Lower Bounds Donghui Zhang et al. Optimal Location Query

Effect of Batch Partitioning Donghui Zhang et al. Optimal Location Query

Optimal Location Query review slide Progressiveness 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

Optimal Location Query Progressiveness 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

Optimal Location Query Conclusions Introduced the min-dist optimal-location query. Proved theorems to limit the number of candidates. Presented lower-bound estimators. Proposed a progressive algorithm. Q & A... Donghui Zhang et al. Optimal Location Query