Knowledge States: A Tool in Randomized Online Algorithms

Online Computation Knowledge States

Knowledge States: A Tool in Randomized Online Algorithms Wolfgang Bein Center for the Advanced Study of Algorithms School of Computer Science University of Nevada, Las Vegas

ADS 2007 coauthors: Lawrence L. Larmore, John Noga, Rudiger ¨ Reischuk supported by NSF grant CCR-0312093

Wolfgang Bein

Knowledge States: A Tool in Randomized Online Algorithms


Online Problems

offline: all input data is completely available before the algorithm starts.

Wolfgang Bein



Online Problems

offline: all input data is completely available before the algorithm starts. online: input data arrives a piece at a time the algorithm must make a decision without knowledge of the entire input.

Wolfgang Bein



Online Problems

offline: all input data is completely available before the algorithm starts. online: input data arrives a piece at a time the algorithm must make a decision without knowledge of the entire input. online problems: resource allocation in operating systems, network routing, robotics, data-structuring, distributed computing, scheduling...

Wolfgang Bein



Theme of the Talk It is difficult to construct good randomized online algorithms Use of work functions in the context of randomized online algorithms

Wolfgang Bein



Theme of the Talk It is difficult to construct good randomized online algorithms Use of work functions in the context of randomized online algorithms Give distributional descriptions of algorithms

Wolfgang Bein



Theme of the Talk It is difficult to construct good randomized online algorithms Use of work functions in the context of randomized online algorithms Give distributional descriptions of algorithms Use the concept of forgiveness

Wolfgang Bein



Theme of the Talk It is difficult to construct good randomized online algorithms Use of work functions in the context of randomized online algorithms Give distributional descriptions of algorithms Use the concept of forgiveness Give two new results: better than 2-comptitive 2-server algorithm in cross polytope spaces optimally competitive k-paging with only O(k) memory

Wolfgang Bein



Papers Equitable Revisited. Bein, Larmore, Noga, ESA 2007 A Randomized Algorithm for Two Servers in Cross Polytope Spaces. Bein, Iwama, Kawahara, Larmore, Oravec, WAOA 2007 Knowledge States: A Tool for Randomized Online Algorithms. Bein, Larmore, Reischuk, HICSS 2008 Knowledge State Algorithms: Randomization with Limited Information. Bein, Larmore, Reischuk, Arxiv 2007 Knowledge States for the Caching Problem in Shared Memory Multiprocessor Systems. Bein, Larmore, Reischuk, IJFCS, to appear Bein, Larmore. Trackless and Limited-Bookmark Algorithms for Paging. Bein, Larmore, ACM SIGACT News, 2004 Tutorial at www.cs.unlv.edu/∼bein/tutorial Wolfgang Bein



Paging Universe of pages in “slow” memory Fast memory can hold k pages Requests ̺ = r 1 r 2 ...r n for pages have to be served Hit (no cost) or Miss (cost 1) Fast Memory a b f h j k

Fast Memory x a b f j k Eject h Request x

Slow Memory

Slow Memory

Solution can be described as a sequence of configurations

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The CNN Problem

News crew in Manhattan (streets and avenues) “event” sequence ̺ = r 1r 2, . . . , r n

r1

Event can be “seen” either horizontally or vertically Solution can be described as a sequence configurations Goal: minimize total movement cost

Wolfgang Bein



The CNN Problem


r1


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The CNN Problem

News crew in Manhattan (streets and avenues) “event” sequence ̺ = r 1r 2, . . . , r n Event can be “seen” either horizontally or vertically Solution can be described as a sequence configurations

r2

Goal: minimize total movement cost

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The CNN Problem


r2


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The CNN Problem


r3


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The CNN Problem


r3


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The CNN Problem


r4


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The CNN Problem


r4


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The 2-Server Problem

Server 2

2 servers in a metric space M request sequence ̺ = r 1r 2, . . . , r n

Server 1

online: decision must be made before r i+1 is revealed Goal: minimize total movement cost

Wolfgang Bein




Server 2

2 servers in a metric space M request sequence ̺ = r 1r 2, . . . , r n

Server 1 r1

online: decision must be made before r i+1 is revealed Goal: minimize total movement cost

Wolfgang Bein




2 servers in a metric space M request sequence ̺ = r 1r 2, . . . , r n online: decision must be made before r i+1 is revealed

Server 1 r1 Server 2


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Wolfgang Bein







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Server 1

r3

Server 2


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r3

Server 2

Server 1


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2 servers in a metric space M r4

request sequence ̺ = r 1r 2, . . . , r n online: decision must be made before r i+1 is revealed Goal: minimize total movement cost

Wolfgang Bein

r3

Server 2

Server 1




2 servers in a metric space M r4 Server 2

request sequence ̺ = r 1r 2, . . . , r n online: decision must be made before r i+1 is revealed Goal: minimize total movement cost

Wolfgang Bein

Server 1



Configurations for the 2-Server Problem

the locations of the two servers is called a configuration solution can be described as a sequence of configurations

Server 2 r4

Server 1 r2

r1

r3

the movement cost is the transportation distance between configurations

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r4

Server 1 r2

r1 Server 2

r3


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Server 1

r4

r2 r3

Server 2


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the locations of the two servers is called a configuration

r4

solution can be described as a sequence of configurations the movement cost is the transportation distance between configurations

Wolfgang Bein

r3

Server 2

Server 1




the locations of the two servers is called a configuration

r4 Server 2

solution can be described as a sequence of configurations the movement cost is the transportation distance between configurations

Wolfgang Bein

Server 1



Online Algorithms

1

Algorithm A is at some initial configuration a0

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Online Algorithms

1 2

Algorithm A is at some initial configuration a0 Requests: ̺ = r 1 , . . . , r n .

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Online Algorithms

1 2 3


At time (t − 1), A is at configuration at−1 .

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Online Algorithms

1 2 3 4


At time (t − 1), A is at configuration at−1 . A has to serve r t not knowing r t+1 , . . .

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Online Algorithms

1 2 3 4 5


At time (t − 1), A is at configuration at−1 . A has to serve r t not knowing r t+1 , . . . A chooses a configuration at .

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Online Algorithms

1 2 3 4 5 6


At time (t − 1), A is at configuration at−1 . A has to serve r t not knowing r t+1 , . . . A chooses a configuration at . A incurs cost(at−1 , r t , at ).

Wolfgang Bein



Online Algorithms

1 2 3 4 5 6


At time (t − 1), A is at configuration at−1 . A has to serve r t not knowing r t+1 , . . . A chooses a configuration at . A incurs cost(at−1 , r t , at ).

If A uses randomization in bullet 5 then A is called a randomized online algorithm.

Wolfgang Bein



2-Server Example Example: k = 2 and ̺ = xyxyz y

2

z

1

x

2

x y z y

x

y z x

x y z

y

x

y z x

z y z x

cost = 7

“Work Function Algorithm” (WFA) Wolfgang Bein



The Adversary: Optimal Cost Function on configurations: Dynamic programming The optimal cost of being there then 0

y 2

y

z

1 2

x

x

2

z y

x

1

y

4

y z y

x

z

2

2

z

3

2

y

x

2

y

x

4

z z

x

optcost = 4 = min last workfunction

x

4

Given request sequence ρ ω ρ (a) = min cost of serving ρ and ending in configuration a ∈ X

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Support of a Work Functions

initial request x request y request x request y request z

ω({y , z}) 0 2 2 4 4 4

ω({x , z}) 1 1 3 3 4 4

ω({x , y }) 2 2 2 2 2 6

S ⊆ X supports ω if for any b ∈ X there exists some a ∈ S such that ω(b) = ω(a) + |a, b|.

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ω({y , z}) 0 2 2 4 4 4

ω({x , z}) 1 1 3 3 4 4

ω({x , y }) 2 2 2 2 2 6

S ⊆ X supports ω if for any b ∈ X there exists some a ∈ S such that ω(b) = ω(a) + |a, b|. A “reasonable algorithm” will move to configurations in the support.

Wolfgang Bein





ω({y , z}) 0 2 2 4 4 4

ω({x , z}) 1 1 3 3 4 4

ω({x , y }) 2 2 2 2 2 6

S ⊆ X supports ω if for any b ∈ X there exists some a ∈ S such that ω(b) = ω(a) + |a, b|. A “reasonable algorithm” will move to configurations in the support. WFA moves for request r from configuration a to configuration b such that |a, b| + ω(b) is minimized.

Wolfgang Bein



Competitiveness For request sequence ̺ = r 1 , r 2 , . . . consider costA (̺): the cost on ̺ achieved by A

costopt (̺): the cost on ̺ achieved by opt We say that A is C-competitive if for each sequence ̺ we have EcostA (̺) ≤ C · costopt (̺) + K

——————————————————Example: cost W FA (xyxyz) cost opt (xyxyz)

=

7 4

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WF A is 2-competitive



The Distributional Model A randomized algorithm can be viewed as a determistic algorithm on distributions. X = all configurations

π is a distribution on X . Algorithm A is at some initial configuration a0 .

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π is a distribution on X . Algorithm A is at some initial configuration a0 . Requests: ̺ = r 1 , . . . , r n .

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At time (t − 1), A is at distribution π t−1 .

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At time (t − 1), A is at distribution π t−1 . A has to serve r t not knowing r t+1 , . . .

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At time (t − 1), A is at distribution π t−1 . A has to serve r t not knowing r t+1 , . . .

A chooses deterministically a distribution π t .

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Cost in the Distributional Model y

1 2

y x

cost 0, 1/4transport z

1 4

z x

cost 2, 1/4 transport x

1 2

y x

y

z cost 2, 1/4 transport

z x

z

3 4

Total cost: 1

“Transportaion Cost” = 1 The cost incured by moving from one distribution to the next is calculated by moving mass along a transportaion problem. The transportation problem has the Monge property. Wolfgang Bein



Work Function “Guided”

_1 2

a 1 b

c _1 2

d

a b

_1 3

c

_1 3

_1 4

e

a

_1 3 b

_1 4

d

a _1 4

_1 4

b

c

Problem: The “support” grows without bound.

Wolfgang Bein



Forgiveness Lower the work function on selective configurations

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Forgiveness Lower the work function on selective configurations d _1 3

c

_1 3

_1 4

e

a

_1 3 b

_1 4

d

a _1 4

_1 4

b

c

e

a

1 b d c

Wolfgang Bein



Forgiveness Lower the work function on selective configurations d _1 3

c

_1 3

_1 4

e

a

_1 3 b

_1 4

d

a _1 4

_1 4

b

c

e

a

1 b d c

Work functions are now estimators Wolfgang Bein



Las Vegas

Algorithm is constructed using the “mixed model” of online computation _1 2

a 1 1 b

1

d

a

1 1

c _1 2

_1 3

b

c

_1 3

_1 3

a

b _ _2 3

0

_1 1 3

_1 3

_1 3

1

Wolfgang Bein

c

b

1

a c

c

a

a

d d

d 1

_1 3

b

b



The Randomized 2-Server Problem Best: RANDOM SLACK 2-competitive [Coppersmith, Doyle, Raghavan, Snir, 90]

Wolfgang Bein



The Randomized 2-Server Problem Best: RANDOM SLACK 2-competitive [Coppersmith, Doyle, Raghavan, Snir, 90] Not known to be best possible.

Wolfgang Bein



The Randomized 2-Server Problem Best: RANDOM SLACK 2-competitive [Coppersmith, Doyle, Raghavan, Snir, 90] Not known to be best possible. 1

Lower Bound: 1 + e− 2 ≈ 1.6065 [Chrobak, Larmore, Lund, Reingold, 97]

Wolfgang Bein



The Randomized 2-Server Problem Best: RANDOM SLACK 2-competitive [Coppersmith, Doyle, Raghavan, Snir, 90] Not known to be best possible. 1

Lower Bound: 1 + e− 2 ≈ 1.6065 [Chrobak, Larmore, Lund, Reingold, 97] Line: 155 78 ≈ 1.987 [Bartal, Chrobak, Larmore, 98]

Wolfgang Bein



2-Server Problem: M24 M24 consists of all metric spaces such that All distances are 1 or 2.

d (x, y) + d (x, z) + d (y, z) ≤ 4

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Why M24 ?

A step in the direction of the goal (better than 2-competitive randomized algorithm for the 2-server problem).

Wolfgang Bein



Why M24 ?

A step in the direction of the goal (better than 2-competitive randomized algorithm for the 2-server problem). Allows a simple example of the knowledge state method.

Wolfgang Bein



Why M24 ?

A step in the direction of the goal (better than 2-competitive randomized algorithm for the 2-server problem). Allows a simple example of the knowledge state method. An interesting class in its own right, generalizing the octahedron.

Wolfgang Bein



Exactly Three Infinite Families of Convex Regular Polytopes ¨ 1852) (Ludwig Schlafli, Infinite Family of Regular Polytopes

Graph Class

Metric Space Class

Regular Simplices

Complete Graphs

Uniform Spaces

Cross Polytopes = Orthoplices

Circulant Graphs Ci 1,...,n−1(2n)

M24

Hypercubes

Hypercubes

Hamming Metric Spaces

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3−d

4−d



What is a Knowledge State? Knowledge state k = (ω, π): ω : X → R is the estimator. π is a distribution on X .

0

7/8

x

z 1/8 1/2

y

π(x , y ) is the probability we are at {x , y }. ω(x , y ) is the estimated unpaid cost of the adversary if it is at {x , y }.

Wolfgang Bein



A Closer Look 0

7/8

x

1/8 10

z 0 1/8 1/2

1

y

Bxyz

The estimator and distribution are defined for all configurations but characterized by their values only on the

support

0 3/2

If a ∈ X − S, then π(a) = 0. ω(a) = minb∈S {ω(b) + ka, bk} Wolfgang Bein



Knowledge States for the 2-Server Problem 1_ 0 2 x 0 z 1_ 0 4 7 _ 6

Axz x

z 1_ 0 4

y 1_ 0 2 x

10 x

_7 12

z

_5 1 1_ 24 0 x _ 2 z Dxyz

1 z

Bxyz _5 12 z Exz

1_ 0 2 _ x Gxz

7_ 4

Cz

y _ _ 0 x 19 24

19 _ 24 0 _ z 29 _ 24

1_ 0 4 _ z

_3 0 8

3 _0 _ 8 x

_5 12

10

z x Fxz _5 0 24

z

x Hxz

Up to symmetry, there are 8 knowledge states of a algorithm for M2,4 . Wolfgang Bein

_ z

z _1 20

19 12 -competitive



There are Numerous Moves. Here is One. Intermediate State W

x

_ 11 _ 33

_1 0 4

x _ z

z y

_7 6

1_ 0 2

_ x

10 __ 9

13 2 _ _ 12 3

Dxyz _1 0 4

1 1_ _ 43

_ z

1_ 3

_ z

_1 0 2

_ x 1

_ Gzx

1_ 0 2

_ x

1_ 3

y z

11 _ _ 43 z

1 _ _y 1 6 3

10 _ 4

Wolfgang Bein

y

1 _ 20

_ x 7 _ 6

1_ 3

1 0 _ 4

_ Dyzx

10 _ 2

_ x

x z

1 _ 40

7 _ 6

x _ z

1 _0 4

__ Dyzx



One Move of the Algorithm

Start at a standard knowledge state over (x , y , z).

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Start at a standard knowledge state over (x , y , z). Read a request r .

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Start at a standard knowledge state over (x , y , z). Read a request r . Update the estimator.

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Start at a standard knowledge state over (x , y , z). Read a request r . Update the estimator. Move the distribution.

Wolfgang Bein




Start at a standard knowledge state over (x , y , z). Read a request r . Update the estimator. Move the distribution. Las Vegas. Randomly pick a subsequent.

Wolfgang Bein




Start at a standard knowledge state over (x , y , z). Read a request r . Update the estimator. Move the distribution. Las Vegas. Randomly pick a subsequent. We are at a standard knowledge state over (x , y , r ), (y , x , r ), (x , z, r ), (z, x , r ), (y , z, r ), or (z, y , r ).

Wolfgang Bein



Behavioral Version: The Wireframe Algorithm

dxyz

z

x

d yzx_ y

z _ x

y

z

_ 1 3

1_ 6 request x

x

_ z

_ z

y

1_ 3

1_ 6

d yzx_

x

_ z

_ x

Wolfgang Bein

x

z _ x

__ dyzx _ z y

x

__ dyzx y

z

_ z

_ x



Results for the 2-Server Problem in M2,4

2-Server Problem on M2,4 , C =

Wolfgang Bein

7 4




2-Server Problem on M2,4 , C = 2-Server Problem on M2,4 , C =

Wolfgang Bein

7 4 19 12

≈ 1.583




2-Server Problem on M2,4 , C = 2-Server Problem on M2,4 , C = This is optimal for M2,4 .

Wolfgang Bein

7 4 19 12

≈ 1.583





7 4 19 12

≈ 1.583

Uniform spaces i.e. paging, the optimal competitiveness is C = 1.5.

Wolfgang Bein





7 4 19 12

≈ 1.583

Uniform spaces i.e. paging, the optimal competitiveness is C = 1.5. Open: a better than 2-competitive randomized algorithm for 2 servers in general spaces.

Wolfgang Bein



Paging Paging: k-server problem in uniform spaces

CNN Amazon ..requesting NYTimes Yahoo UNLV Sandia

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History of k-paging [Fiat, Karp, Luby, McGeoch, Sleator, Young, 1991] Pk Lower bound, Hk = i=1 1/i (2Hk − 1)-competitive algorithm “RM ARK” RM ARK uses O(n) memory

Wolfgang Bein



History of k-paging [Fiat, Karp, Luby, McGeoch, Sleator, Young, 1991] Pk Lower bound, Hk = i=1 1/i (2Hk − 1)-competitive algorithm “RM ARK” RM ARK uses O(n) memory [McGeoch, Sleator, 1991] Hk -competitive algorithm PARTITION unbounded memory

Wolfgang Bein



History of k-paging [Fiat, Karp, Luby, McGeoch, Sleator, Young, 1991] Pk Lower bound, Hk = i=1 1/i (2Hk − 1)-competitive algorithm “RM ARK” RM ARK uses O(n) memory [McGeoch, Sleator, 1991] Hk -competitive algorithm PARTITION unbounded memory [Achlioptas, Chrobak, Noga, 2000] Hk -competitive algorithm E QUITABLE O(k 2 log k ) memory

Wolfgang Bein



History of k-paging [Fiat, Karp, Luby, McGeoch, Sleator, Young, 1991] Pk Lower bound, Hk = i=1 1/i (2Hk − 1)-competitive algorithm “RM ARK” RM ARK uses O(n) memory [McGeoch, Sleator, 1991] Hk -competitive algorithm PARTITION unbounded memory [Achlioptas, Chrobak, Noga, 2000] Hk -competitive algorithm E QUITABLE O(k 2 log k ) memory [Bein, Larmore, Noga, 2007] Hk -competitive algorithm E QUITABLE 2 O(k ) memory Wolfgang Bein



Bookmarks Aab

ab c 1_ 2

ca Bcab

_1 2

b

cb

b bookmarked

Acb

A trackless algorithm (i.e. an algorithm that does not use any bookmarks) cannot have optimal competitiveness. [Bein, Fleischer, Larmore, 00]

Wolfgang Bein



An Optimally Competitive Algorithm for 2-Paging

Algorithm K2 : a

a

a b b

c

1− 2

c a b

1− 2

c b a

a b c b

b a

b a

c

1 − 3

d a

2− 3

d b

c a

Competitiveness: CK2 =

Wolfgang Bein

d

3 2



Models of Online Computation Behavioral

Distributional a

a _1 2

c

b c

1 b

_1 2

c a

_1 2

c

a

c

b

a

b

a

c _1 2

b

a

1

a

1 1 a

a

c

c

b

Wolfgang Bein



Work Functions and Offset Functions a

1 0

c

c

b

a c 0 offset 1 c

c b

d

c 1

b

0

a

0

b

2

d

a

c

d

a

b

c

a

d

b

0 a

a

d offset 1 0

Wolfgang Bein

0

d a

2

2

2

a

0 c

b

c

b



Offset Function Guided Algorithm EQUITABLE [Achlioptas, Chrobak, Noga, 2000] The algorithm is descibed using the distributional model The algorithm’s distribution mass is only on the support of the work funtion

_1 2

a 1 b

c _1 2

d

a b

_1 3

c

_1 3

_1 4

e

a

_1 3 b

a

_1 4

_1 4

_1 4

d

b

c d _1 3

c

_1 3

_1 4

e

a

_1 3 b

_1 4

d

a _1 4

_1 4

b

c

e

a

1 b d c Wolfgang Bein



Las Vegas

Algorithm is constructed using the “mixed model” of online computation _1 2

a 1 1 b

1

d

a

1 1

c _1 2

_1 3

b

c

_1 3

_1 3

a

b _ _2 3

0

_1 1 3

_1 3

_1 3

1

Wolfgang Bein

c

b

1

a c

c

a

a

d d

d 1

_1 3

b

b



Behavioral Translation 1

d _2 3

b

c

_1 2

d

a

c

b

_1 3

b d

a

c

b

d

a

c

b

d

a

c

b

_1 3

a

c _1 2

a

_2 3

d

a

c

b

1

a

d

d

a

c

b

1 c

Wolfgang Bein

b



Knowledge State Description of K2 Work functions are denoted using the “bar notation”. A tuple T is in the support: at least i members of T are to the left of the i th bar. 0

A

a|b

1, 1 new (c)

1/2

B

A

0

d|a

c|ab new (d) 1/3, 1

1/3 C

d|abc

0

1/3

A

d|b

1/3 0

A

d|c Wolfgang Bein



The Case k = 3 A

a|b|c 0

Las Vegas Step 0

5/6

A

a|b|c c or d 0, 1/2 1/2

C

ab||cd e or d 0, 3/4

new 1, 1

1, 1 new

B

a|bcd|

1, 1 new

b, c or d 0, 1/3 c 0, 1/2 d or e 0, 3/4 ab||cde E 1

5/3

D

a|bcde|

a|b|d

1/10

1/10

a|e|f 0 new 0, 1

A0

1/10

a|bcdef|+6/5

−1/5, 1 new

b, c, d or e 0, 1/2

A

Q

Las Vegas Step

0

A 0 a|b|c

1

a|bc|def +1

C

1/2

C

1/2

C

1/2

ab||cd

c or b 0, 1/4

a|bc|de

new

F

1/ 6, 1

5/4

P

0

1/2

R

1/6 1/6

Las Vegas Step

a|bcd|ef+5/6 1/6

Wolfgang Bein

ab||cf

ac||df



Equitable, Equitable2

For general k: E QUITABLE forgives to a “cone” after O(k 2 log k) bookmarks

Wolfgang Bein



Equitable, Equitable2

For general k: E QUITABLE forgives to a “cone” after O(k 2 log k) bookmarks E QUITABLE forgives to a work function with small support (but not a cone) after O(k) bookmarks

Wolfgang Bein



Further Research

The 2-server problem for general spaces

Wolfgang Bein



Further Research

The 2-server problem for general spaces CNN:

√ Deterministic: lower bound: 6 + 17 [Koutsoupias, Taylor, 2005] Deterministic: upper bound: 105 , 879 [Sitters Stougie 2005] Randomized: Open

Wolfgang Bein


Knowledge States: A Tool in Randomized Online Algorithms

Knowledge States: A Tool in Randomized Online Algorithms

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