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Online Chasing Problems for Regular n-gons

Online Chasing Problems for Regular n-gons. Hiroshi Fujiwara* Kazuo Iwama Kouki Yonezawa. We consider 1-server Problem. Before that… Related Work: k-server problem Fundamental online problem introduced by Manasse, McGeoch, and Sleator [MMS90]. 3. Request 1. Server. 4. Server. Server.

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Online Chasing Problems for Regular n-gons

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  1. Online Chasing Problemsfor Regular n-gons Hiroshi Fujiwara* Kazuo Iwama Kouki Yonezawa

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

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

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

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

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

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

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

  9. 1-server Problem Server may move like this…

  10. Minimize: Total travel distance 3 1 4 ALG 2 1-server Problem Input: Request regions Output: How to chase

  11. OPT Optimal Offline Algorithm To solve optimal offline distance involves convex programming 3 1 4 2

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

  13. Application Server = Relay broadcasting car Requests = Events ALG RIVF

  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. Previous position , present request region • If , move to such that minimizes • If , do not move Greedy Algorithm (GRD) (i) (ii)

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

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

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

  19. Lower Bound 1 GRD: Always vertical to side 2

  20. Lower Bound Intersection of all requests OPT

  21. Lower Bound 1 3 5 7 GRD/OPT=2 2 4 6 8

  22. Lower Bound even odd

  23. Lower Bound • No worse input • Next we prove upper bound of this value Competitive ratio of GRD

  24. Goal: Prove Basic idea: Compare for each request Upper Bound

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

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

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

  28. To prove is enough if Goal: Prove Amortized Analysis • Then, choose potential function

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

  30. What is good ? • Euclidean distance does not work • Manhattan distance does not work either • Finally, we found • Extension of Manhattan distance

  31. Sum of ‘s What is good ?

  32. Worst Case for Upper Bound (Case of hexagon)

  33. Upper Bound Generally we have

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

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