1. Minimum Back-Walk-Free Latency Problem with Multiple Servers Yaw-Ling Lin ( 林耀鈴 ) Dept Computer Sci. & Info. Management, Providence University, Taichung, Taiwan

2. Yaw-Ling Lin, Providence, Taiwan2 Minimum Latency Problem (MLP) Starts from s, sending goods to all other nodes. Traveling Salesperson Problem (TSP): Server oriented MLP: Client oriented MLP is also known as repairman problem or traveling repairman problem (TRP) s

3. Yaw-Ling Lin, Providence, Taiwan3 MLP: Formal Definition

4. Yaw-Ling Lin, Providence, Taiwan4 MLP vs. TSP TSP: minimizes the salesman’s total time. Server oriented, egoistic. –No contstant approximation algorithm for general case. –Christofides (1976): 3/2-approximation ratio for metric case; Arora (1992): metric TSP does not have PTAS unless P=NP. –Arora (1998 JACM): PTAS on Euclidean case. MLP: minimizes the customers’ total time. Clients oriented, altruistic. –Alias: deliveryman problem, traveling repairman problem (TRP). –Afrati (1986): MAX-SNP-hard for metric case. –Goeman (1996): 10.78-approximation ratio for metric case (with Garg, 1996FOCS, technique); 3.59-approximation ratio for trees. –Arora (1999 STOC): quasi-polynomial ( O(n O(log n) ) approximation scheme for trees and Euclidean space. –Sitters (2002, IPCO): MLP on trees is NP-complete; not known for caterpillars.

5. Yaw-Ling Lin, Providence, Taiwan5 MBLP: Back-Walk Free

6. Yaw-Ling Lin, Providence, Taiwan6 An Example

7. Yaw-Ling Lin, Providence, Taiwan7 Our Results COCOON2002, Singapore, single server MBLP( given a starting point of G ) –Trees : O(n log n ) time –k-path : O(n log k) ; path is O(n) time –DAG : NP-Hard (Reduce from 3-SAT) This talk (CMCT2003), multiple servers MBLP –k servers on paths : O(n 2 ) time –k servers on cycles : O(n 3 /k ) time –k origins on paths and cycles : O(n 3 log k ) time

8. Yaw-Ling Lin, Providence, Taiwan8 Properties

9. Yaw-Ling Lin, Providence, Taiwan9 Properties (contd’)

10. Yaw-Ling Lin, Providence, Taiwan10 Properties (contd’)

11. Yaw-Ling Lin, Providence, Taiwan11 Properties (contd’) is right-skew; is not. is decreasing right-skew partitioned.

12. Yaw-Ling Lin, Providence, Taiwan12 Properties (contd’)

13. Yaw-Ling Lin, Providence, Taiwan13 Path-Partition: Example

14. Yaw-Ling Lin, Providence, Taiwan14 Algorithm Path-Partition

15. Yaw-Ling Lin, Providence, Taiwan15 Main Result: k-MBLP on Paths 12 3 … n n1n1 n2n2 nknk …

16. Yaw-Ling Lin, Providence, Taiwan16 k-MBLP: Recurrence Scheme 12 3 … n n1n1 n2n2 nknk … one-server

17. Yaw-Ling Lin, Providence, Taiwan17 Base Cases Analysis

18. Yaw-Ling Lin, Providence, Taiwan18 k-MBLP on Cycles less than O(n/k) cuts

19. Yaw-Ling Lin, Providence, Taiwan19 k-server Origin Problems

20. Yaw-Ling Lin, Providence, Taiwan20 k-origins: Recurrence Scheme

21. Yaw-Ling Lin, Providence, Taiwan21 k-origins: Complexity Analysis

22. Yaw-Ling Lin, Providence, Taiwan22 k-origins on Cycles

23. Yaw-Ling Lin, Providence, Taiwan23 Future Research MLP on caterpillars. The binary encoding in k-origin setting could be further exploited. Multiple servers on trees, paths.