Randomized Competitive Analysis for Two Server Problems

Wolfgang Bein

Kazuo Iwama

Jun Kawahara

k-server problem

Goal: Minimize the total distance

k-server problem

k-server problem

k-server problem

k-server problem

k-server problem

k-server problem

k-server problem

………

Greedy does not work,

2-server 3-point problem

2-server 3-point problem

2-server 3-point problem

2-server 3-point problem

2-server 3-point problem

a

b c

Adversary (always malicious): cababacb……Opt: cababacb……

one move per two inputs 2.0CR

Algorithm exists whose CR = 2.0

k-server: Known Facts

• Introduction of the problem [Manasse, McGeoch, Sleator 90]

• Lower bound: k [MMS90]• General upper bound: 2k-1 [Koutsoupias,

Papadimitriou 94]• k-server conjecture

– true for 2-servers, line, trees, fixed k+1 or K+2 points, ……

– still open for 3 server 7 points

Randomized k-server

• Very little is known for general cases• Even for 2-servers (CR=2 for det. case):

– On the line [Bartal, Chrobak, Lamore 98]

– Cross polytope space [Bein et al. 08]– Specific 3 points: Can use LP to derive an

optimal algorithm (but nothing was given about the CR) [Lund, Reingold 94]

– Almost nothing is known about its CR for a general metric space

1551.987

78CR

19

12CR

Randomized 2-server 3-point

a

b c

Adversary is malicious: c……

: 2.0 1.5CR

Select a server (a or b) at random

Adversary’s attempt fails with prob. 0.5

Our Contribution

• For (general) 2-server 3-point problem, we prove that CR < 1.5897.

• Well below 2.0 (=the lower bound for the deterministic case): Superiority of randomization for the server problem

• Our approach is very brute force

The Idea

• We can assume a triangle in the plain wolg.

• For a specific triangle, its algorithm can be given as a (finite) state diagram, which can be derived by LP [LR94]

• Calculation of its CR is not hard.

• Just try many (different shaped) triangles, then…..

111541511154151323

1

C

L

C

L R

11R L

R

R

1 2

The Idea

• We can assume a triangle in the plain wolg.

• For a specific triangle, its algorithm can be given as a (finite) state diagram, which can be derived by LP [LR94]

• Calculation of its CR is not hard.

• Just try many (different shaped) triangles, then…..

Testing Many Triangles

……

CR=1.5 1.53 1.489 1.533

1.536

almost the same but CR=1.89

1.537

1.0

Approximation Lemma

Line Lemma

Approximation Lemma

1.0 1a

1b

1.02a

2b

r r 1 1

2 2

max ,a b

a b

1OPT 2OPT

1 2

1a a

1

1b

11.0

1OPT

12

OPT OPT

2 2 1 1

2 1 1 1

ALG ALG ALG ALG

OPT OPT OPT OPT

Proof

algorithm A

Line Lemma1 n

1n

algorithm s.

1 11

1

1t.

1

1n

n

nn n

CR C

n

1.58191

e

e

Using Approximation Lemma

a1

b

2

( , )a bd

d

a

b

a

b

algorithm :A CR r

:a

A CR ra d

1

1

Using Line Lemma

a1

b

2

( , 1)n n

ab

: nA CR C1 n

1 11

1

11 1

n

n

nn n

C

n

decreasing

:2 n

nA CR C

n

2

a1

b

2

0Proving CR r

0

00 0

0

( )2 n

nf S C r

n

0( ) ii i

i i

af S r r

a d

( , );i i ia b CR r

id

0S

iS

finite set ofsquares(triangles)

Our algorithm = algs for squares + alg for the line

a1

b

2

Proving Better CR

jS ( ) : Maximumjf S

Computer Program

a1

b

2

a1

b

2

Computer Program

Computer Program

a1

b

2+ some heuristics

Some Data

• Conjecture: 1.5819

• Current bound: 1.5897– 13,285 squares, d=1/256~1

• Small area [5/4, 7/4, 1/16]: 1.5784– 69 squares, d=1/64~1/128

• Small area [7/4, 9/4, 1/4]: 1.5825– 555 squares, d=1/2048~1/64

(5/4, 7/4)

1/16

Proof for Line Lemma1 n

1n

algorithm s.

1 11

1

1t.

1

1n

n

nn n

CR C

n

1.58191

e

e

Final Remarks

• Strong conjecture that the real CR is e/(e-1). Analytical proof?

• Extension of our approach to, say, the 4-point case.

• Many very interesting open problems in the online server problem.

Thanks!