2007/3/6 1 online chasing problems for regular n-gons hiroshi fujiwara* kazuo iwama kouki yonezawa
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Online Chasing Problemsfor Regular n-gons
Hiroshi Fujiwara*Kazuo Iwama
Kouki Yonezawa
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We consider 1-server Problem
• Before that…• Related Work: k-server problem
– Fundamental online problem introduced by Manasse, McGeoch, and Sleator [MMS90]
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Related Work: k-server Problem
Minimize: Total travel distance
Request 1
2
3
4
Input: Requests given onlineOutput: How to move servers
Server
Server
Server
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Related Work: k-server Problem
2
3
4
Minimize: Total travel distance
Request 1
Server
Input: Requests given onlineOutput: How to move servers
ALG
OPT (offline)
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Performance of algorithm:Competitive ratio of ALG is c, if for all request sequences
Related Work: k-server Problem
ALG c OPT Total travel distance
Optimal offline total travel distance
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Related Work: k-server Problem
• Lower Bound k [MMS90]• Upper Bound 2k-1 achieved by
Work Function Algorithm [KP95]
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We consider 1-server Problem
2
3
4
This is NOT k-server problem with a single server
Request 1
No choice!
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1-server Problem
2
3
4
Request := Region
1
Choice of next position!
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1-server Problem
Server may move like this…
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1-server Problem
2
3
41ALG
0A
1A
2A
3A
4A
Input: Request regionsOutput: How to chase
Minimize: Total travel distance 1i iALG A A
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Optimal Offline Algorithm
2
3
41
OPT
To solve optimal offline distance involves convex programming
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Performance of Algorithm
OPT
Competitive ratio of ALG is c, if for all request sequences
ALG c OPT
ALG
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ApplicationServer = Relay broadcasting car Requests = Events
RIVF
ALG
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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]
2
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Greedy Algorithm (GRD)
A
DA D
(i) (ii)
X
Previous position , present request region
(i) If , move to such that minimizes
(ii) If , do not move
A D
A D
X DA D( , )d A X
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Our ResultsTheorem: Competitive ratio of greedy algorithm for regular n-gons is
for odd n and for even n1
sin2n
1
sinn
2 1.41 3.24 2 (optimal)
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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 n1
sin2n
1
sinn
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Lower Bound
Zoom up
We found bad input like this:
(Case of hexagon)
fixed
sliding
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Lower Bound
2
1
GRD: Always vertical to side
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Lower Bound
OPT
Intersection of all requests
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Lower Bound1 3 5 7
2 4 6 8 GRD/OPT=2
6
3
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Lower Boundeven odd
2
n
n
1
sin2n
1
sinn
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Lower Bound
• No worse input• Next we prove upper bound of this value
1, odd ;
sin2
1, even
sin
n
n
n
n
Competitive ratio of GRD
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Upper Bound
1GRD
2GRD3GRD
4GRD
1OPT
2OPT
4OPT
Goal: Prove Basic idea: Compare for each request
GRD c OPT
?GRD c OPT
3OPT
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Upper BoundGoal: Prove Basic idea: Compare for each request
GRD c OPT
?GRD c OPT
Butis impossible to prove; and can happen at the same time
GRD c OPT
0GRD 0OPT
0GRD 0OPT
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Upper BoundGoal: Prove Basic idea: Compare for each request
GRD c OPT
?GRD c OPT
Therefore, we prove
instead0GRD 0OPT
GRD c OPT
To cancel
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Amortized Analysis
• Is called amortized analysis• Common technique for online
problems– For example, list accessing [ST85]
• is called potential function
Goal: ProveGRD c OPT
To proveis enough if
GRD c OPT const
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Amortized Analysis
• Then, choose potential function
Goal: ProveGRD c OPT
To proveis enough if
GRD c OPT const
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What is good ?Observation: Server of GRD always goes closer to server of OPT when 0OPT
So, some kind of distance between two servers works as potential function
GRD c OPT
0GRD 0OPT
should decrease, is canceledGRD
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What is good ?
• Euclidean distance does not work
• Manhattan distance does not work either
• Finally, we found– Extension of Manhattan
distance
/ 2 1
0
2 2( , ) sin cos
n
k
k kx y x y
n n
| | | |x y
2 2x y ( , )x y
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What is good ?/ 2 1
0
2 2( , ) sin cos
n
k
k kx y x y
n n
Sum of ‘s
( , )x y
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Worst Case for Upper Bound
6
1GRD 2OPT
?GRD c OPT 1
2c
(Case of hexagon)
0
3
3 2
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Upper Bound
Generally we have
1, odd ;
sin2
1, even
sin
n
nc
n
n
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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 n1
sin2n
1
sinn
Future Works