Loading in 5 sec....

Collection Depots Facility Location Problems in Trees R. Benkoczi, B. Bhattacharya, A. TamirPowerPoint Presentation

Collection Depots Facility Location Problems in Trees R. Benkoczi, B. Bhattacharya, A. Tamir

Download Presentation

Collection Depots Facility Location Problems in Trees R. Benkoczi, B. Bhattacharya, A. Tamir

Loading in 2 Seconds...

- 97 Views
- Uploaded on
- Presentation posted in: General

Collection Depots Facility Location Problems in Trees R. Benkoczi, B. Bhattacharya, A. Tamir

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Collection Depots Facility LocationProblems in TreesR. Benkoczi, B. Bhattacharya, A. Tamir

陳冠伶‧王湘叡‧李佳霖‧張經略

Jun 12, 2007

By 陳冠伶

Client (demand service)

Facility (service center)

Collection Depots

F

P2

D

P1

Service Cost

2(P1+P2)‧w(c)

C

Express Transportation

Garbage collection

- IN: given a tree and
- points of clients
- points of collection depots
- an integer k

- OUT
- Optimal placements of kfacilities
- that minimizes some global function of the service cost for all clients.

- Minimize the service cost of the most expensive client

C

C

D

D

D

C

D

F

D

C

C

1-center

Minimize the maximum distance to the facility

k-center

Minimize the maximum distance to the closest facility

- Minimize the total service cost

C

C

D

D

D

C

D

F

D

C

C

1-median

Minimize the average distance to the facility

k-median

Minimize the average distance to the closest facility

- Unrestricted 1-center problem
- O(n)

- Unrestricted median problems
- 1-median: O(nlogn)
- k-median: O(kn3)

- Restricted k-median problem
- NP-complete
- Facility setup costs are not identical

BY 王湘叡

- Every iteration, eliminate a fraction of impossible instances.
- Binary Search
- T(n)=T(n/2)+1
- T(n)=O(lg n)

- How about

- c(f)=max min r(f, vi)
- Service cost is non-decreasing when the facility goes away from the client.

- A linear time algorithm could determine!

T1

T2

Ti

Tk

depot

client

- Find the centroid and partition the tree into two parts

centroid

- S1 > 1/3 |T(i)|

- S2 > 1/3 |T(i)|

- Find the client Xmax with the largest service cost from the centroid.

Xmax

f

S2

S1

foptmust be in S1

Xmax

- Centroid is the optimal

Should be optimal

X’max

- Compute all depot distance
- Find the median δmed
- Separate all clients into two sets, K+ (red) and K- (blue)

δmed

- S2

- Consider f’ in S1, that depot distance δ(f’)< δmed

δ(f’)< δmed

f’

- S1

- Find all f’, they form trees T1, T2, …,Tn
- There are two cases, fopt is in ∪Tior not

f

T1

T2

T3

- If fopt in red, consider K+, δ(fopt)<δmed<δ(K+)
- For a facility F’ in S1 and a client in S2, δ(fopt, u) is in S1

δ(f’, u)

δ(f’, u)

fopt

f’

- If fopt is not in red, consider K-,
δ(K-)<δmed <δ(fopt)

- For a facility F’ in Sand a client in S2, δ(fopt, u) is in S2
- Similar to previous
case

- Only fopt in ∪Ti is considered.

δ(fopt, u)

f’

fopt

- Arbitrarily paired clients in K+
- For each pair (u, v), Compute tuv s.t. w(v)(tuv+d(c,v))=w(u).(tuv+d(c,u))
- Compare tmed and
d(fopt, c)+d((fopt,c),p(fopt, c))

δ(f’, u)

fopt

fopt

- consider tmed<tuv
- d(fopt, c)+d((fopt,c),p(fopt, c))<tmed<tuv

δ(f’, u)

fopt

fopt

- consider tmed>tuv
- d(fopt, c)+d((fopt,c),p(fopt, c))>tmed>tuv
- ¼ K+ can be removed

δ(f’, u)

fopt

fopt

BY 李佳霖

- Find a placement for facility to minimize the cost of all tours.
- i.e. minimize the sum of weighted distances of the facility to client, then to optimal depot, and return to facility.

- For the path of a facility to a client, the closest depot can be found efficiently.
- Brute Force: Ο(n2)
- Using Spine decomposition and pre-sorting: Ο(nlogn)

r0

5

3

3

3

2

r0

r0

r0

f

d

d2

v

dnew

d4

f

c2

d1

cj

d3

c1

c3

c4

- Construction for the SD has time complexity Ο(n) and space complexity Ο(n)
- Costs of the subtrees can be evaluated in constant timeonce j is determined.
- If we use binary search with dnew, we spend Ο(logn) time for every subtree. So Ο(log2n).
- Use the sequential search in sorted order. So Ο(logn).

- The 1-median collection depots problem in tree can be sloved in Ο(nlogn) time and Ο(n) space.

BY 張經略

- To minimize the sum of facility opening costs plus service costs for servicing the clients.

- We fixed an arbitrary optimal solution and explore its structure.

- Consider an arbitrary vertex v.
- xv: minimize the trip cost of serving v
- yv:be a closest facility to v.

yv

Assumed (for contradiction) servicing facility for client C

xv

v

Tleft

Tright

client C

Assumed (for contradiction) servicing facility for client C

yv

xv

v

client C

Tright

Tleft

- The blue part of the following graph is proven by symmetry.

yv

xv

v

Tleft

Tright

- The total cost can be partitioned into four categories: the red, yellow, blue cost and v.

yv

xv

v

Tleft

Tright

- The optimal solution has to be a combination of optimal substructures
- You have to be “optimal” in the red (to minimize the red cost) and the yellow (to minimize the yellow cost).
- This almost leads to Dynamic Programming already!

- Due to some complications, the final Dynamic Programming is much more complicated…
- But the proof requires no special technique beyond the “自給自足” property.
- The challenge is to devise the “right” recurrences to carry out the aforementioned intuitive approach.

- Easily verified to be polynomial.