1. 2007/3/6 1 Online Chasing Problems for Regular n-gons Hiroshi Fujiwara* Kazuo Iwama Kouki Yonezawa

2. 2007/3/6 2 We consider 1-server Problem Before that… Related Work: k-server problem –Fundamental online problem introduced by Manasse, McGeoch, and Sleator [MMS90]

3. 2007/3/6 3 Related Work: k-server Problem Minimize: Total travel distance Request 1 2 3 4 Input: Requests given online Output: How to move servers Server

4. 2007/3/6 4 Related Work: k-server Problem 2 3 4 Minimize: Total travel distance Request 1 Server Input: Requests given online Output: How to move servers ALG OPT (offline)

5. 2007/3/6 5 Performance of algorithm: Competitive ratio of ALG is c, if for all request sequences Related Work: k-server Problem Total travel distance Optimal offline total travel distance

6. 2007/3/6 6 Related Work: k-server Problem Lower Bound k [MMS90] Upper Bound 2k-1 achieved by Work Function Algorithm [KP95]

7. 2007/3/6 7 We consider 1-server Problem 2 3 4 This is NOT k-server problem with a single server Request 1 No choice!

8. 2007/3/6 8 1-server Problem 2 3 4 Request := Region 1 Choice of next position!

9. 2007/3/6 9 1-server Problem Server may move like this…

10. 2007/3/6 10 1-server Problem 2 3 41 ALG Input: Request regions Output: How to chase Minimize: Total travel distance

11. 2007/3/6 11 Optimal Offline Algorithm 2 3 41 OPT To solve optimal offline distance involves convex programming

12. 2007/3/6 12 Performance of Algorithm OPT Competitive ratio of ALG is c, if for all request sequences ALG

13. 2007/3/6 13 Application Server = Relay broadcasting car Requests = Events RIVF ALG

14. 2007/3/6 14 Previous Works Convex region –Existence of competitive online algorithm [FN93] –Lower bound [FN93] –Offline problem (convex programming) is solvable in polynomial time [NN93] Non-convex set (more difficult) –E.g. CNN problem: Upper bound 879 [SS06]

15. 2007/3/6 15 Greedy Algorithm (GRD) (i)(ii) Previous position, present request region (i) If, move to such that minimizes (ii) If, do not move

16. 2007/3/6 16 Our Results Theorem: Competitive ratio of greedy algorithm for regular n-gons is for odd n and for even n 2 1.41 3.24 2 (optimal)

17. 2007/3/6 17 Our Results Tight analysis; Upper bound = Lower bound –Lower bound: Example of bad sequence –Upper bound: Amortized analysis Theorem: Competitive ratio of greedy algorithm for regular n-gons is for odd n and for even n

18. 2007/3/6 18 Lower Bound Zoom up We found bad input like this: (Case of hexagon) fixed sliding

19. 2007/3/6 19 Lower Bound 2 1 GRD: Always vertical to side

20. 2007/3/6 20 Lower Bound OPT Intersection of all requests

21. 2007/3/6 21 Lower Bound 1 3 5 7 2468 GRD/OPT=2

22. 2007/3/6 22 Lower Bound evenodd

23. 2007/3/6 23 Lower Bound No worse input Next we prove upper bound of this value Competitive ratio of GRD

24. 2007/3/6 24 Upper Bound Goal: Prove Basic idea: Compare for each request

25. 2007/3/6 25 Upper Bound Goal: Prove Basic idea: Compare for each request But is impossible to prove; and can happen at the same time

26. 2007/3/6 26 Upper Bound Goal: Prove Basic idea: Compare for each request Therefore, we prove instead To cancel

27. 2007/3/6 27 Amortized Analysis Is called amortized analysis Common technique for online problems –For example, list accessing [ST85] is called potential function Goal: Prove To prove is enough if

28. 2007/3/6 28 Amortized Analysis Then, choose potential function Goal: Prove To prove is enough if

29. 2007/3/6 29 What is good ? Observation: Server of GRD always goes closer to server of OPT when So, some kind of distance between two servers works as potential function should decrease, is canceled

30. 2007/3/6 30 What is good ? Euclidean distance does not work Manhattan distance does not work either Finally, we found –Extension of Manhattan distance

31. 2007/3/6 31 What is good ? Sum of ‘s

32. 2007/3/6 32 Worst Case for Upper Bound (Case of hexagon)

33. 2007/3/6 33 Upper Bound Generally we have

34. 2007/3/6 34 Conclusion Improvement for large n –Work Function Algorithm? Other shapes (esp. non-convex) With 2 or more servers Competitive ratio of GRD for regular n-gons is for odd n and for even n Future Works