2007/3/6 1

Online Chasing Problemsfor Regular n-gons

Hiroshi Fujiwara*Kazuo Iwama

Kouki Yonezawa

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]

2007/3/6 3

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

2007/3/6 4

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)

2007/3/6 5

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

2007/3/6 6

Related Work: k-server Problem

• Lower Bound k [MMS90]• Upper Bound 2k-1 achieved by

Work Function Algorithm [KP95]

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!

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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…

2007/3/6 10

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

2007/3/6 12

Performance of Algorithm

OPT

Competitive ratio of ALG is c, if for all request sequences

ALG c OPT

ALG

2007/3/6 13

ApplicationServer = Relay broadcasting car Requests = Events

RIVF

ALG

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]

2

2007/3/6 15

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

2007/3/6 16

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

2007/3/6 18

Lower Bound

Zoom up

We found bad input like this:

(Case of hexagon)

fixed

sliding

2007/3/6 19

Lower Bound

2

1

GRD: Always vertical to side

2007/3/6 20

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

2007/3/6 24

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

2007/3/6 29

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

2007/3/6 31

What is good ?/ 2 1

0

2 2( , ) sin cos

n

k

k kx y x y

n n

Sum of ‘s

( , )x y

2007/3/6 32

Worst Case for Upper Bound

6

1GRD 2OPT

?GRD c OPT 1

2c

(Case of hexagon)

0

3

3 2

2007/3/6 33

Upper Bound

Generally we have

1, odd ;

sin2

1, even

sin

n

nc

n

n

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 n1

sin2n

1

sinn

Future Works