1. Danny Z. Chen and Haitao Wang University of Notre Dame Indiana, USA
New Algorithms for 1-D Facility Location and Path Equipartition Problems
Danny Z. Chen and Haitao Wang
University of Notre Dame
Indiana, USA

2. Our Problems General model facility location L∞ path equipartition
uniform k-median
weighted k-median
k-coverage
linear model facility location
L∞ path equipartition
Ld path equipartition

3. The general model facility location
Input: A set of n sites/customers P={p1,…,pn} sorted on a line, an integer k≤n,
Goal: Choose at most k sites in P to set up facilities for serving all customers to minimize the sum of the facility opening cost and the service cost p1 pn
choose three sites to set up facilities

4. The general model facility location (cont.)
The facility opening cost
For each site pi, there is a cost ci for setting up a facility at pi
The service cost
The distance of any two points pi and pj is their Euclidean distance on the line
d(pi,pj)=|pi - pj|
Each customer pj is associated with a service cost function fj(d) for a distance d such that:
if we use a facility at pi to serve pj, the service cost for pj is fj(d(pi,pj))
fj(d) is a monotone increasing function with respect to d pj pi

5. The k-median No facility opening cost
ci=0, for each site pi
The service cost functions are linear
For each site pj, fj(d)=aj * d
the uniform k-median
all coefficients aj are the same
the weighted k-median
the coefficients aj are not necessarily the same

6. The k-coverage The facility opening cost The service cost ci≠0
For each site pj, it has a range rj, such that
fj(d)=0, if d≤rj
fj(d)=+∞, if d>rj pj rj rj

7. The linear model The facility opening cost
The service cost functions are linear
For each site pj, fj(d)=aj * d

8. Previous work and our results
T=min{n*sqrt{klogn}, n*2O(sqrt{logk loglogn})}
The general model: O(n2) time (Hassin and Tamir 91’)
k-median
Previous work: O(nk) time (Hassin and Tamir 91’; Auletta, Parente, and Persiano 98’)
Both uniform and weighted cases
Our result:
The uniform case: O(T) time, k=Ω(log n)
The weighted case: O(T logn) time, k=Ω(log3n)
k-coverage
Previous work: O(nk) time (Hassin and Tamir 91’; van Hoesel and Wagelmans 07’)
Our result: O(T) time, k=Ω(log n)
Linear model
Previous work: O(nk) time (Hassin and Tamir 91’)
Our result: O(T log2 n), k=Ω(log5n)

9. Result summary uniform k-median O(nk) O(T) k=Ω(log n) k-median
Problems
Previous
Our results
Ours is better when
uniform k-median
O(nk)
O(T)
k=Ω(log n)
k-median
O(T log n)
k=Ω(log3 n)
k-coverage
linear model
O(T log2n)
k=Ω(log5n)
T=min{n*sqrt{klogn}, n*2O(sqrt{logkloglogn})}

10. Our techniques
Model the general model as the k-link shortest path problem
Prove the concave Monge property holds
Apply two k-link shortest path algorithms as the high-level scheme for our problem
O(n*sqrt{klogn}) time, Aggarwal, Schieber, and Tokuyama, 94’
O(n*2O(sqrt{logk loglogn})) time, Schieber 98’
The problem modeling looks natural
Previous work does not use it

11. Our techniques (cont.)
The concave Monge property in this problem modeling is not obvious
Monge property is used in the previous work but only on the service cost
The Monge property in our problem modeling also involves the facility opening cost
There is an obvious monotone property on the service cost but not on the facility opening cost

12. The k-link shortest path problem
Input: A weighted DAG G with vertices v1,…,vn, and an integer k, 1≤k≤n
Output: Finding a path of k edges from v1 to vn such that the sum of the weights of the edges in the path is minimized
Concave Monge property:
For any i and j with 1<i+1<j<n,
w(i,j)+w(i+1,j+1) ≤ w(i,j+1)+w(i+1,j) v1 vi
vi+1 vj
vj+1 vn

13. Modeling the facility location problem as finding a k-link shortest path
Given the site set P={p1,…,pn} on a line, build a DAG G of n+1 vertices v0,…,vn v0 v1 v2 v3 v4 p1 p2 p3 p4 vn v0
a 3-link shortest path

14. Model the facility location problem as the k-link shortest path (cont
The weight w(i,j) of each edge from vi to vj is the minimum cost of opening one facility and serving all customers in the group defined by the edge
w(i,j) has two parts
The cost of opening one facility
The total service cost
We prove that the concave Monge property holds on w(i,j) vi vj

15. Proving the concave Monge property on w(i,j)
An observation: The positions of facilities satisfy a monotone property
wij vi vj
wij+1 vi
vj+1

16. The algorithms High-level scheme: The computational components
Apply the k-link shortest path algorithms
The computational components
Compute the edge weights w(i,j)
Running time O(T*W)
T: the k-link shortest path algorithms
W: time for computing each w(i,j)
k-median and k-coverage
Design data structures to answer w(i,j) queries
The linear model
Modify the k-link shortest path algorithms

17. Computing w(i,j) for the k-median
No facility opening cost, ci=0
The service cost functions are linear
For each site pj, fj(d)=aj * d
the uniform k-median: all coefficients aj are the same
The uniform case
The facility is set up at the median site in the group
O(1) time (O(n) time preprocessing)
The weighted case
The facility is the weighted median site in the group
O(log n) time by binary search (O(n) time preprocessing) vi vj
wij

18. Computing w(i,j) for the k-coverage
Facility opening cost, ci≠0
The service cost
For each site pj, it has a range rj, such that
fj(d)=0, if d≤rj
fj(d)=+∞, if d>rj pj rj rj

19. Computing w(i,j) for the k-coverage (cont.)
Reduce querying w(i,j) to the following points in intervals query problem
Given n points p1,…,pn sorted on a line, each pi is associated with a cost ci and an interval Ii that contains pi
Query: For each query (i,j) with i≤j, report a point pm with the smallest cost such that i ≤m≤ j and pm is contained in all intervals from Ii to Ij

20. Computing w(i,j) for the k-coverage (cont.)
Reduce querying w(i,j) to the following points in intervals query problem
Given n points p1,…,pn sorted on a line, each pi is associated with a cost ci and an interval Ii that contains pi
Query: For each query (i,j) with i≤j, report a point pm with the smallest cost such that i ≤m≤ j and pm is contained in all intervals from Ii to Ij j i

21. Computing w(i,j) for the k-coverage (cont.)
Our solution to the points in intervals query problem
Preprocessing: O(n) time
Query: O(1) time
Approach: shrink some intervals, range-minima j i

22. Computing w(i,j) for the linear model
The facility opening cost, ci≠0
The service cost functions are linear
For each site pj, fj(d)=aj * d
We do not build a query data structure
An observation: Compute a special group of O(n) edge weights in O(n log n) time
Modify the two k-link shortest path algorithms
Use a new row-minima algorithm instead of the linear time SMAWK algorithm
Overall running time: O(T log2n)

23. Conclusions uniform k-median O(nk) O(T) k=Ω(log n) k-median O(T log n)
Problems
Previous
Our results
Ours is better when
uniform k-median
O(nk)
O(T)
k=Ω(log n)
k-median
O(T log n)
k=Ω(log3 n)
k-coverage
linear model
O(T log2n)
k=Ω(log5n)
L∞ path partition
O(nk logk)
always
Ld path partition
T=min{n*sqrt{klogn}, n*2O(sqrt{logkloglogn})}

24. Thank you