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Analyzing the Competitive Ratio of Greedy Algorithms for One-Server Problems on Regular N-gons

In this paper, we explore the one-server problem within online algorithms, specifically focusing on the greedy algorithm's competitive ratio when applied to regular n-gons. We examine the constraints of the one-server problem, contrasting it with the k-server problem. Through our analysis, we establish competitive bounds, both upper and lower, and illustrate the performance of our algorithm relative to the optimal offline solution. We also discuss implications for various applications such as relay broadcasting and potential future research paths in non-convex environments with multiple servers.

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Analyzing the Competitive Ratio of Greedy Algorithms for One-Server Problems on 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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