R–LINE: A Better Randomized 2-ServerAlgorithm on the Line

Lucas Bang

University of Nevada Las Vegas

bang@unlv.nevada.edu

13 September 2012

2012 Workshop on Approximation and Online Algorithms (WAOA)Ljubljana, Slovenia

R–LINE UNLV

Abstract

Main Result

I A randomized on-line algorithm for the 2-server problem onthe line.

I Competitiveness ≤ 1.901 against the oblivious adversary.

I Previously best known competitiveness of 15578 ≈ 1.987.

R–LINE UNLV

Abstract

Main Result

I A randomized on-line algorithm for the 2-server problem onthe line.

I Competitiveness ≤ 1.901 against the oblivious adversary.

I Previously best known competitiveness of 15578 ≈ 1.987.

R–LINE UNLV

Abstract

Main Result

I A randomized on-line algorithm for the 2-server problem onthe line.

I Competitiveness ≤ 1.901 against the oblivious adversary.

I Previously best known competitiveness of 15578 ≈ 1.987.

R–LINE UNLV

Abstract

Main Result

I A randomized on-line algorithm for the 2-server problem onthe line.

I Competitiveness ≤ 1.901 against the oblivious adversary.

I Previously best known competitiveness of 15578 ≈ 1.987.

R–LINE UNLV

Background

The k-server Problem

I Introduced by Manasse, McGeoch, and Sleator, 1990.

I Given k mobile servers in a metric space M.

I Serve requests online in the metric space

I Move a single server to the request point.

I Goal: minimize total distance moved.

R–LINE UNLV

Background

The k-server Problem

I Introduced by Manasse, McGeoch, and Sleator, 1990.

I Given k mobile servers in a metric space M.

I Serve requests online in the metric space

I Move a single server to the request point.

I Goal: minimize total distance moved.

R–LINE UNLV

Background

The k-server Problem

I Introduced by Manasse, McGeoch, and Sleator, 1990.

I Given k mobile servers in a metric space M.

I Serve requests online in the metric space

I Move a single server to the request point.

I Goal: minimize total distance moved.

R–LINE UNLV

Background

The k-server Problem

I Introduced by Manasse, McGeoch, and Sleator, 1990.

I Given k mobile servers in a metric space M.

I Serve requests online in the metric space

I Move a single server to the request point.

I Goal: minimize total distance moved.

R–LINE UNLV

Background

The k-server Problem

I Introduced by Manasse, McGeoch, and Sleator, 1990.

I Given k mobile servers in a metric space M.

I Serve requests online in the metric space

I Move a single server to the request point.

I Goal: minimize total distance moved.

R–LINE UNLV

Background

4-server Problem

R–LINE UNLV

Background

4-server Problem

s

ss

s

R–LINE UNLV

Background

4-server Problem

s

ss

s

r1

R–LINE UNLV

Background

4-server Problem

s

ss

s

r1

R–LINE UNLV

Background

4-server Problem

s

s

s

s

R–LINE UNLV

Background

4-server Problem

s

s

s

sr 2

R–LINE UNLV

Background

4-server Problem

s

s

s

sr 2

R–LINE UNLV

Background

4-server Problem

s

s

s

s

R–LINE UNLV

Background

4-server Problem

s

s

s

sr 3

R–LINE UNLV

Background

4-server Problem

s

s

s

sr 3

R–LINE UNLV

Background

4-server Problem

s

s

s

s

R–LINE UNLV

Background

4-server Problem

s

s

s

sr 4

R–LINE UNLV

Background

4-server Problem

s

s

s

sr 4

R–LINE UNLV

Background

4-server Problem

s

s

s

s

R–LINE UNLV

Background

4-server Problem

s

s

s

s

R–LINE UNLV

Background

4-server Problem

s

s

s

s

R–LINE UNLV

Background

The (m, n)-server Problem

I Generalization of the k-server problem.

I Given m mobile servers in a metric space M.

I Serve requests online in the metric space.

I Each request requires n servers to move to the request point

I Goal: minimize total distance moved.

R–LINE UNLV

Background

The (m, n)-server Problem

I Generalization of the k-server problem.

I Given m mobile servers in a metric space M.

I Serve requests online in the metric space.

I Each request requires n servers to move to the request point

I Goal: minimize total distance moved.

R–LINE UNLV

Background

The (m, n)-server Problem

I Generalization of the k-server problem.

I Given m mobile servers in a metric space M.

I Serve requests online in the metric space.

I Each request requires n servers to move to the request point

I Goal: minimize total distance moved.

R–LINE UNLV

Background

The (m, n)-server Problem

I Generalization of the k-server problem.

I Given m mobile servers in a metric space M.

I Serve requests online in the metric space.

I Each request requires n servers to move to the request point

I Goal: minimize total distance moved.

R–LINE UNLV

Background

The (m, n)-server Problem

I Generalization of the k-server problem.

I Given m mobile servers in a metric space M.

I Serve requests online in the metric space.

I Each request requires n servers to move to the request point

I Goal: minimize total distance moved.

R–LINE UNLV

Background

The (m, n)-server Problem

I Generalization of the k-server problem.

I Given m mobile servers in a metric space M.

I Serve requests online in the metric space.

I Each request requires n servers to move to the request point

I Goal: minimize total distance moved.

R–LINE UNLV

Background

(4, 2)-server Problem

R–LINE UNLV

Background

(4, 2)-server Problem

s

ss

s

R–LINE UNLV

Background

(4, 2)-server Problem

s

ss

s

r1

R–LINE UNLV

Background

(4, 2)-server Problem

s

ss

s

r1

R–LINE UNLV

Background

(4, 2)-server Problem

s

s

ss

R–LINE UNLV

Background

(4, 2)-server Problem

s

s

ss

r2

R–LINE UNLV

Background

(4, 2)-server Problem

s

s

ss

r2

R–LINE UNLV

Background

(4, 2)-server Problem

sss

s

R–LINE UNLV

Background

(4, 2)-server Problem

sss

s

r3

R–LINE UNLV

Background

(4, 2)-server Problem

sss

s

r3

R–LINE UNLV

Background

(4, 2)-server Problem

s

s

s

s

R–LINE UNLV

Background

(4, 2)-server Problem

s

s

s

s

R–LINE UNLV

Background

(4, 2)-server Problem

s

s

s

s

R–LINE UNLV

Competitive Analysis

Definition of Competitive Ratio

I If A is a randomized online algorithm and OPT is an optimaloffline algorithm, then for any request sequence σ thecompetitive ratio, C , satisfies

E (costA(σ)) ≤ C · costOPT (σ) + K

where K is a constant, and E denotes the expected value.

R–LINE UNLV

Competitive Analysis

Definition of Competitive Ratio

I If A is a randomized online algorithm and OPT is an optimaloffline algorithm, then for any request sequence σ thecompetitive ratio, C , satisfies

E (costA(σ)) ≤ C · costOPT (σ) + K

where K is a constant, and E denotes the expected value.

R–LINE UNLV

Competitive Analysis

Definition of Competitive Ratio

I If A is a randomized online algorithm and OPT is an optimaloffline algorithm, then for any request sequence σ thecompetitive ratio, C , satisfies

E (costA(σ)) ≤ C · costOPT (σ) + K

where K is a constant, and E denotes the expected value.

R–LINE UNLV

Previous Related Work

Randomized 2-server AlgorithmsAlgorithm C Features

RANDOM–SLACK 2 General Spaces

HARMONIC 3 General Spaces

UNIFORM 32 Uniform Spaces

DOUBLE-COVER-2 15578 ≈ 1.987 Line

CROSS-POLYTOPE 1912 Cross Polytope Spaces

R–LINE ≤ 1.901 Lazy

R–TREE ≤ 1.901 (?) In Progress

SPLIT–DECOMPOSABLE < 2 (?) In Progress

General Spaces (?????) In Progress

R–LINE UNLV

Useful Theorems

TheoremGiven any C-competitive online algorithm for the (2n, n)-serverproblem, we can derive a randomized online algorithm for the2-server problem that is C-competitive.

I Therefore, it is sufficient to find a competitive algorithm forthe (2n, n)-server problem.

TheoremAny optimal offline strategy for the (2n, n) server problem keepsthe servers in two blocks of n each, assuming that the servers aretogether in two blocks in the initial configuration.

I Consequently, we may assume the optimal algorithm uses 2servers that move with cost n · d(s, r).

R–LINE UNLV

Useful Theorems

TheoremGiven any C-competitive online algorithm for the (2n, n)-serverproblem, we can derive a randomized online algorithm for the2-server problem that is C-competitive.

I Therefore, it is sufficient to find a competitive algorithm forthe (2n, n)-server problem.

TheoremAny optimal offline strategy for the (2n, n) server problem keepsthe servers in two blocks of n each, assuming that the servers aretogether in two blocks in the initial configuration.

I Consequently, we may assume the optimal algorithm uses 2servers that move with cost n · d(s, r).

R–LINE UNLV

Useful Theorems

TheoremGiven any C-competitive online algorithm for the (2n, n)-serverproblem, we can derive a randomized online algorithm for the2-server problem that is C-competitive.

I Therefore, it is sufficient to find a competitive algorithm forthe (2n, n)-server problem.

TheoremAny optimal offline strategy for the (2n, n) server problem keepsthe servers in two blocks of n each, assuming that the servers aretogether in two blocks in the initial configuration.

I Consequently, we may assume the optimal algorithm uses 2servers that move with cost n · d(s, r).

R–LINE UNLV

Useful Theorems

TheoremGiven any C-competitive online algorithm for the (2n, n)-serverproblem, we can derive a randomized online algorithm for the2-server problem that is C-competitive.

I Therefore, it is sufficient to find a competitive algorithm forthe (2n, n)-server problem.

TheoremAny optimal offline strategy for the (2n, n) server problem keepsthe servers in two blocks of n each, assuming that the servers aretogether in two blocks in the initial configuration.

I Consequently, we may assume the optimal algorithm uses 2servers that move with cost n · d(s, r).

R–LINE UNLV

Useful Theorems

TheoremGiven any C-competitive online algorithm for the (2n, n)-serverproblem, we can derive a randomized online algorithm for the2-server problem that is C-competitive.

I Therefore, it is sufficient to find a competitive algorithm forthe (2n, n)-server problem.

TheoremAny optimal offline strategy for the (2n, n) server problem keepsthe servers in two blocks of n each, assuming that the servers aretogether in two blocks in the initial configuration.

I Consequently, we may assume the optimal algorithm uses 2servers that move with cost n · d(s, r).

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

Given a request point and 2 servers with d(s1, r) ≤ d(s2, r)

r

s

s

1

2

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

Server s1 moves deterministically...

r

s

s

1

2

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

all the way to r .

r

s

s

1

2

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

Server s2 also moves toward r ...

r

s

s

1

2

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

but only part of the way.

r

s

s

1

2

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

The distanced moved by s2 is the slack .

r

s

s

1

2

slack

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

slack = 12(d(s1, s2) + d(s1, r)− d(s2, r))

r

s

s

1

2

slack

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

Comments on SLACK–COVERAGE

I Bartal’s Algorithm is 3-competitive.

I Appears ad-hoc.

I What motivated the formula?

I slack = 12(d(s1, s2) + d(s1, r)− d(s2, r))

I What is the underlying theory?

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

Comments on SLACK–COVERAGE

I Bartal’s Algorithm is 3-competitive.

I Appears ad-hoc.

I What motivated the formula?

I slack = 12(d(s1, s2) + d(s1, r)− d(s2, r))

I What is the underlying theory?

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

Comments on SLACK–COVERAGE

I Bartal’s Algorithm is 3-competitive.

I Appears ad-hoc.

I What motivated the formula?

I slack = 12(d(s1, s2) + d(s1, r)− d(s2, r))

I What is the underlying theory?

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

Comments on SLACK–COVERAGE

I Bartal’s Algorithm is 3-competitive.

I Appears ad-hoc.

I What motivated the formula?

I slack = 12(d(s1, s2) + d(s1, r)− d(s2, r))

I What is the underlying theory?

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

Comments on SLACK–COVERAGE

I Bartal’s Algorithm is 3-competitive.

I Appears ad-hoc.

I What motivated the formula?

I slack = 12(d(s1, s2) + d(s1, r)− d(s2, r))

I What is the underlying theory?

R–LINE UNLV

Bartal’s SLACK–COVERAGE Algorithm for (Rn,L2)

Comments on SLACK–COVERAGE

I Bartal’s Algorithm is 3-competitive.

I Appears ad-hoc.

I What motivated the formula?

I slack = 12(d(s1, s2) + d(s1, r)− d(s2, r))

I What is the underlying theory?

R–LINE UNLV

T-theory for Server Problems

3 points

R–LINE UNLV

T-theory for Server Problems

3 points

R–LINE UNLV

T-theory for Server Problems

3 points

z

x

y

R–LINE UNLV

T-theory for Server Problems

3 points

z

x

y

m

R–LINE UNLV

T-theory for Server Problems

3 points

z

x

y

m

R–LINE UNLV

T-theory for Server Problems

3 points

z

x

y

m

ab

c

R–LINE UNLV

T-theory for Server Problems

3 points

z

x

y

m

ab

c

The tight span of 3 points.

R–LINE UNLV

T-theory for Server Problems

The Tight Span of (M , d)

I Let T (M) = {f : M → R} such that

1. ∀x , y ∈ M, f (x) + f (y) ≥ d(x , y)2. ∀x ∈ M ∃y ∈ M : f (x) + f (y) = d(x , y)

I For f , g ∈ T (M), δ(f , g) = max |f (x)− g(x)|

The tight span of M is the metric space (T (M), δ).

R–LINE UNLV

T-theory for Server Problems

The Tight Span of (M , d)

I Let T (M) = {f : M → R} such that

1. ∀x , y ∈ M, f (x) + f (y) ≥ d(x , y)2. ∀x ∈ M ∃y ∈ M : f (x) + f (y) = d(x , y)

I For f , g ∈ T (M), δ(f , g) = max |f (x)− g(x)|

The tight span of M is the metric space (T (M), δ).

R–LINE UNLV

T-theory for Server Problems

The Tight Span of (M , d)

I Let T (M) = {f : M → R} such that

1. ∀x , y ∈ M, f (x) + f (y) ≥ d(x , y)

2. ∀x ∈ M ∃y ∈ M : f (x) + f (y) = d(x , y)

I For f , g ∈ T (M), δ(f , g) = max |f (x)− g(x)|

The tight span of M is the metric space (T (M), δ).

R–LINE UNLV

T-theory for Server Problems

The Tight Span of (M , d)

I Let T (M) = {f : M → R} such that

1. ∀x , y ∈ M, f (x) + f (y) ≥ d(x , y)2. ∀x ∈ M ∃y ∈ M : f (x) + f (y) = d(x , y)

I For f , g ∈ T (M), δ(f , g) = max |f (x)− g(x)|

The tight span of M is the metric space (T (M), δ).

R–LINE UNLV

T-theory for Server Problems

The Tight Span of (M , d)

I Let T (M) = {f : M → R} such that

1. ∀x , y ∈ M, f (x) + f (y) ≥ d(x , y)2. ∀x ∈ M ∃y ∈ M : f (x) + f (y) = d(x , y)

I For f , g ∈ T (M), δ(f , g) = max |f (x)− g(x)|

The tight span of M is the metric space (T (M), δ).

R–LINE UNLV

T-theory for Server Problems

The Tight Span of (M , d)

I Let T (M) = {f : M → R} such that

1. ∀x , y ∈ M, f (x) + f (y) ≥ d(x , y)2. ∀x ∈ M ∃y ∈ M : f (x) + f (y) = d(x , y)

I For f , g ∈ T (M), δ(f , g) = max |f (x)− g(x)|

The tight span of M is the metric space (T (M), δ).

R–LINE UNLV

T-theory for Server Problems

z

x

y

m

ab

c

a

b

c xyz

xzy

yzx

R–LINE UNLV

T-theory for Server Problems

z

x

y

m

ab

c

a

b

c xyz

xzy

yzx

R–LINE UNLV

T-theory for Server Problems

For any non trivial partition of the metric space, M = S t S ′ definethe split metric:

δS ,S ′(p, q) =

{1 p ∈ S , q ∈ S ′

0 otherwise

Notation: We sometimes omit the complement when it is clearfrom context. That is we write δS rather than δS,S ′

1 yzx

δx ,yz(x , y) = 1δx ,yz(x , z) = 1δx ,yz(y , z) = 0

R–LINE UNLV

T-theory for Server Problems

For any non trivial partition of the metric space, M = S t S ′ definethe split metric:

δS ,S ′(p, q) =

{1 p ∈ S , q ∈ S ′

0 otherwise

Notation: We sometimes omit the complement when it is clearfrom context. That is we write δS rather than δS,S ′

1 yzx

δx ,yz(x , y) = 1δx ,yz(x , z) = 1δx ,yz(y , z) = 0

R–LINE UNLV

T-theory for Server Problems

For any non trivial partition of the metric space, M = S t S ′ definethe split metric:

δS ,S ′(p, q) =

{1 p ∈ S , q ∈ S ′

0 otherwise

Notation: We sometimes omit the complement when it is clearfrom context. That is we write δS rather than δS,S ′

1 yzx

δx ,yz(x , y) = 1δx ,yz(x , z) = 1δx ,yz(y , z) = 0

R–LINE UNLV

T-theory for Server Problems

For any non trivial partition of the metric space, M = S t S ′ definethe split metric:

δS ,S ′(p, q) =

{1 p ∈ S , q ∈ S ′

0 otherwise

Notation: We sometimes omit the complement when it is clearfrom context. That is we write δS rather than δS,S ′

1 yzx

δx ,yz(x , y) = 1

δx ,yz(x , z) = 1δx ,yz(y , z) = 0

R–LINE UNLV

T-theory for Server Problems

For any non trivial partition of the metric space, M = S t S ′ definethe split metric:

δS ,S ′(p, q) =

{1 p ∈ S , q ∈ S ′

0 otherwise

Notation: We sometimes omit the complement when it is clearfrom context. That is we write δS rather than δS,S ′

1 yzx

δx ,yz(x , y) = 1δx ,yz(x , z) = 1

δx ,yz(y , z) = 0

R–LINE UNLV

T-theory for Server Problems

For any non trivial partition of the metric space, M = S t S ′ definethe split metric:

δS ,S ′(p, q) =

{1 p ∈ S , q ∈ S ′

0 otherwise

Notation: We sometimes omit the complement when it is clearfrom context. That is we write δS rather than δS,S ′

1 yzx

δx ,yz(x , y) = 1δx ,yz(x , z) = 1δx ,yz(y , z) = 0

R–LINE UNLV

T-theory for Server Problems

For any non trivial partition of the metric space, M = S t S ′ definethe split metric:

δS ,S ′(p, q) =

{1 p ∈ S , q ∈ S ′

0 otherwise

Notation: We sometimes omit the complement when it is clearfrom context. That is we write δS rather than δS,S ′

1 yzx

δx ,yz(x , y) = 1δx ,yz(x , z) = 1δx ,yz(y , z) = 0

R–LINE UNLV

T-theory for Server Problems

We can decompose the original metric into the sum of threemetrics:

ab

c

x

y

z

a

b

c xyz

xzy

yzx

=+

+

d = a · δx + b · δy + c · δz

R–LINE UNLV

T-theory for Server Problems

We can decompose the original metric into the sum of threemetrics:

ab

c

x

y

z

a

b

c xyz

xzy

yzx

=+

+

d = a · δx + b · δy + c · δz

R–LINE UNLV

T-theory for Server Problems

We can decompose the original metric into the sum of threemetrics:

ab

c

x

y

z

a

b

c xyz

xzy

yzx

=+

+

d = a · δx + b · δy + c · δz

R–LINE UNLV

T-theory for Server Problems

We can decompose the original metric into the sum of threemetrics:

ab

c

x

y

z

a

b

c xyz

xzy

yzx

=+

+

d = a · δx + b · δy + c · δz

R–LINE UNLV

T-theory for Server Problems

In general we say a finite metric, (M, d) , is split-decomposable ifwe can write:

d = d1 + d2 + ...+ dn

wheredi = αSi ,S

′i· δSi ,S ′i

andαSi ,S

′i

are constants, the isolation indices.

R–LINE UNLV

T-theory for Server Problems

In general we say a finite metric, (M, d) , is split-decomposable ifwe can write:

d = d1 + d2 + ...+ dn

wheredi = αSi ,S

′i· δSi ,S ′i

andαSi ,S

′i

are constants, the isolation indices.

R–LINE UNLV

T-theory for Server Problems

In general we say a finite metric, (M, d) , is split-decomposable ifwe can write:

d = d1 + d2 + ...+ dn

wheredi = αSi ,S

′i· δSi ,S ′i

andαSi ,S

′i

are constants, the isolation indices.

R–LINE UNLV

T-theory for Server Problems

In general we say a finite metric, (M, d) , is split-decomposable ifwe can write:

d = d1 + d2 + ...+ dn

wheredi = αSi ,S

′i· δSi ,S ′i

andαSi ,S

′i

are constants, the isolation indices.

R–LINE UNLV

T-theory for Server Problems

Isolation Indices, αFor s, s ′, t, t ′ ∈ M, define the point-wise isolation index

αs,s′;t,t′ = max

d(s, t) + d(s ′, t ′)d(s, t ′) + d(s, t ′)d(s, s ′) + d(t, t ′)

− d(s, s ′)− d(t, t ′)

Now, for any sets S and T , define

αS ,T = mins,s′∈St,t′∈T

{αs,s′;t,t′}

R–LINE UNLV

T-theory for Server Problems

Isolation Indices, αFor s, s ′, t, t ′ ∈ M, define the point-wise isolation index

αs,s′;t,t′ = max

d(s, t) + d(s ′, t ′)d(s, t ′) + d(s, t ′)d(s, s ′) + d(t, t ′)

− d(s, s ′)− d(t, t ′)

Now, for any sets S and T , define

αS ,T = mins,s′∈St,t′∈T

{αs,s′;t,t′}

R–LINE UNLV

T-theory for Server Problems

Isolation Indices, αFor s, s ′, t, t ′ ∈ M, define the point-wise isolation index

αs,s′;t,t′ = max

d(s, t) + d(s ′, t ′)d(s, t ′) + d(s, t ′)d(s, s ′) + d(t, t ′)

− d(s, s ′)− d(t, t ′)

Now, for any sets S and T , define

αS ,T = mins,s′∈St,t′∈T

{αs,s′;t,t′}

R–LINE UNLV

T-theory for Server Problems

4 points

R–LINE UNLV

T-theory for Server Problems

4 points

z

xy

w

R–LINE UNLV

T-theory for Server Problems

4 points

z

xy

w

R–LINE UNLV

T-theory for Server Problems

4 points

z

xy

w

R–LINE UNLV

T-theory for Server Problems

4 points

z

xy

w

R–LINE UNLV

T-theory for Server Problems

4 points

z

xy

wd c

ba

R–LINE UNLV

T-theory for Server Problems

4 points

z

xya b

cw

d

e

f

R–LINE UNLV

T-theory for Server Problems

4 points

z

xya b

cw

d

e

f

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

r

s

s

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

r

s

s

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

r

s

s

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

r

s

s

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

rs

s

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

rs

s

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

rs

s

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

rs

s

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

rs

s

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

rs

s

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

r

s

s

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

r

s

s

1

2αs1

slack

R–LINE UNLV

T-theory for Server Problems

Bartal’s SLACK-COVERAGE as T-theory

αs1 = mins,s′∈{S1}t,t′∈{s2,r}

{αs,s′;t,t′} =1

2(d(s1, s2)+d(s1, r)−d(s2, r)) = slack

r

s

s

1

2αs1

slack

R–LINE UNLV

T-theory for Server Problems

RANDOM-SLACK as T-theory

R–LINE UNLV

T-theory for Server Problems

RANDOM-SLACK as T-theory

r

s

s1

2

ab

c

R–LINE UNLV

T-theory for Server Problems

RANDOM-SLACK as T-theory

Move s1 with probability ba+b .

r

s

s1

2

ab

c

R–LINE UNLV

T-theory for Server Problems

RANDOM-SLACK as T-theoryMove s2 with probability a

a+b

r

s

s1

2

ab

c

R–LINE UNLV

T-theory for Server Problems

RANDOM-SLACK as T-theoryRLINE is inspired by this idea, and uses the same potential withslight modification.

r

s

s1

2

ab

c

R–LINE UNLV

T-theory for Server Problems

RANDOM-SLACK as T-theory

I RANDOM–SLACK competitiveness C = 2 proved usingCoppersmith-Doyle-Raghavan-Snir potential.

φ =k−1∑i=1

k∑j=i+1

d(si , sj) + k ·MinMatch

I The CDRS potential can be defined in terms of isolationindices as well.

φ =∑

ηi ,j · αi ,j

I R–LINE uses same potential making small adjustments to thecoefficients so that C < 2.

R–LINE UNLV

T-theory for Server Problems

RANDOM-SLACK as T-theory

I RANDOM–SLACK competitiveness C = 2 proved usingCoppersmith-Doyle-Raghavan-Snir potential.

φ =k−1∑i=1

k∑j=i+1

d(si , sj) + k ·MinMatch

I The CDRS potential can be defined in terms of isolationindices as well.

φ =∑

ηi ,j · αi ,j

I R–LINE uses same potential making small adjustments to thecoefficients so that C < 2.

R–LINE UNLV

T-theory for Server Problems

RANDOM-SLACK as T-theory

I RANDOM–SLACK competitiveness C = 2 proved usingCoppersmith-Doyle-Raghavan-Snir potential.

φ =k−1∑i=1

k∑j=i+1

d(si , sj) + k ·MinMatch

I The CDRS potential can be defined in terms of isolationindices as well.

φ =∑

ηi ,j · αi ,j

I R–LINE uses same potential making small adjustments to thecoefficients so that C < 2.

R–LINE UNLV

T-theory for Server Problems

RANDOM-SLACK as T-theory

I RANDOM–SLACK competitiveness C = 2 proved usingCoppersmith-Doyle-Raghavan-Snir potential.

φ =k−1∑i=1

k∑j=i+1

d(si , sj) + k ·MinMatch

I The CDRS potential can be defined in terms of isolationindices as well.

φ =∑

ηi ,j · αi ,j

I R–LINE uses same potential making small adjustments to thecoefficients so that C < 2.

R–LINE UNLV

T-theory for Server Problems

RANDOM-SLACK as T-theory

I RANDOM–SLACK competitiveness C = 2 proved usingCoppersmith-Doyle-Raghavan-Snir potential.

φ =k−1∑i=1

k∑j=i+1

d(si , sj) + k ·MinMatch

I The CDRS potential can be defined in terms of isolationindices as well.

φ =∑

ηi ,j · αi ,j

I R–LINE uses same potential making small adjustments to thecoefficients so that C < 2.

R–LINE UNLV

T-theory for Server Problems

RANDOM-SLACK as T-theory

I RANDOM–SLACK competitiveness C = 2 proved usingCoppersmith-Doyle-Raghavan-Snir potential.

φ =k−1∑i=1

k∑j=i+1

d(si , sj) + k ·MinMatch

I The CDRS potential can be defined in terms of isolationindices as well.

φ =∑

ηi ,j · αi ,j

I R–LINE uses same potential making small adjustments to thecoefficients so that C < 2.

R–LINE UNLV

T-theory for Server Problems

RANDOM-SLACK as T-theory

I RANDOM–SLACK competitiveness C = 2 proved usingCoppersmith-Doyle-Raghavan-Snir potential.

φ =k−1∑i=1

k∑j=i+1

d(si , sj) + k ·MinMatch

I The CDRS potential can be defined in terms of isolationindices as well.

φ =∑

ηi ,j · αi ,j

I R–LINE uses same potential making small adjustments to thecoefficients so that C < 2.

R–LINE UNLV

T-theory for Server Problems

Infinite Spaces

An infinite metric is split-decomposable ifevery finite subspace is split-decomposable.

Split-decomposable metrics include:

I All 4 point spaces

I The line

I Any tree

I The Manhattan plane

I The circle

R–LINE UNLV

T-theory for Server Problems

Infinite Spaces

An infinite metric is split-decomposable if

every finite subspace is split-decomposable.

Split-decomposable metrics include:

I All 4 point spaces

I The line

I Any tree

I The Manhattan plane

I The circle

R–LINE UNLV

T-theory for Server Problems

Infinite Spaces

An infinite metric is split-decomposable ifevery finite subspace is split-decomposable.

Split-decomposable metrics include:

I All 4 point spaces

I The line

I Any tree

I The Manhattan plane

I The circle

R–LINE UNLV

T-theory for Server Problems

Infinite Spaces

An infinite metric is split-decomposable ifevery finite subspace is split-decomposable.

Split-decomposable metrics include:

I All 4 point spaces

I The line

I Any tree

I The Manhattan plane

I The circle

R–LINE UNLV

T-theory for Server Problems

Infinite Spaces

An infinite metric is split-decomposable ifevery finite subspace is split-decomposable.

Split-decomposable metrics include:

I All 4 point spaces

I The line

I Any tree

I The Manhattan plane

I The circle

R–LINE UNLV

T-theory for Server Problems

Infinite Spaces

An infinite metric is split-decomposable ifevery finite subspace is split-decomposable.

Split-decomposable metrics include:

I All 4 point spaces

I The line

I Any tree

I The Manhattan plane

I The circle

R–LINE UNLV

T-theory for Server Problems

Infinite Spaces

An infinite metric is split-decomposable ifevery finite subspace is split-decomposable.

Split-decomposable metrics include:

I All 4 point spaces

I The line

I Any tree

I The Manhattan plane

I The circle

R–LINE UNLV

T-theory for Server Problems

Infinite Spaces

An infinite metric is split-decomposable ifevery finite subspace is split-decomposable.

Split-decomposable metrics include:

I All 4 point spaces

I The line

I Any tree

I The Manhattan plane

I The circle

R–LINE UNLV

T-theory for Server Problems

Infinite Spaces

An infinite metric is split-decomposable ifevery finite subspace is split-decomposable.

Split-decomposable metrics include:

I All 4 point spaces

I The line

I Any tree

I The Manhattan plane

I The circle

R–LINE UNLV

T-theory for Server Problems

T-Theory on the Line

I For all finite subspaces of the real line, isolation indices areinterval lengths with endpoints in the subspace.

I For R–LINE, we have our algorithms servers, s1, s2, ..., s2n, and

I Two adversary servers, a1 and a2 for a total of 2n + 2 points.

I Define αi ,j , the (i , j)th isolation index of a configuration, to bethe length of the longest interval that has exactly i algorithmservers to the left and exactly j adversary servers to the left.

I Formally,

αi ,j = max{0,min{si+1, aj+1} −max{si , aj}}

I s0 = a0 = −∞ and s2n+1 = a3 =∞

R–LINE UNLV

T-theory for Server Problems

T-Theory on the Line

I For all finite subspaces of the real line, isolation indices areinterval lengths with endpoints in the subspace.

I For R–LINE, we have our algorithms servers, s1, s2, ..., s2n, and

I Two adversary servers, a1 and a2 for a total of 2n + 2 points.

I Define αi ,j , the (i , j)th isolation index of a configuration, to bethe length of the longest interval that has exactly i algorithmservers to the left and exactly j adversary servers to the left.

I Formally,

αi ,j = max{0,min{si+1, aj+1} −max{si , aj}}

I s0 = a0 = −∞ and s2n+1 = a3 =∞

R–LINE UNLV

T-theory for Server Problems

T-Theory on the Line

I For all finite subspaces of the real line, isolation indices areinterval lengths with endpoints in the subspace.

I For R–LINE, we have our algorithms servers, s1, s2, ..., s2n, and

I Two adversary servers, a1 and a2 for a total of 2n + 2 points.

I Define αi ,j , the (i , j)th isolation index of a configuration, to bethe length of the longest interval that has exactly i algorithmservers to the left and exactly j adversary servers to the left.

I Formally,

αi ,j = max{0,min{si+1, aj+1} −max{si , aj}}

I s0 = a0 = −∞ and s2n+1 = a3 =∞

R–LINE UNLV

T-theory for Server Problems

T-Theory on the Line

I For all finite subspaces of the real line, isolation indices areinterval lengths with endpoints in the subspace.

I For R–LINE, we have our algorithms servers, s1, s2, ..., s2n, and

I Two adversary servers, a1 and a2 for a total of 2n + 2 points.

I Define αi ,j , the (i , j)th isolation index of a configuration, to bethe length of the longest interval that has exactly i algorithmservers to the left and exactly j adversary servers to the left.

I Formally,

αi ,j = max{0,min{si+1, aj+1} −max{si , aj}}

I s0 = a0 = −∞ and s2n+1 = a3 =∞

R–LINE UNLV

T-theory for Server Problems

T-Theory on the Line

I For all finite subspaces of the real line, isolation indices areinterval lengths with endpoints in the subspace.

I For R–LINE, we have our algorithms servers, s1, s2, ..., s2n, and

I Two adversary servers, a1 and a2 for a total of 2n + 2 points.

I Define αi ,j , the (i , j)th isolation index of a configuration, to bethe length of the longest interval that has exactly i algorithmservers to the left and exactly j adversary servers to the left.

I Formally,

αi ,j = max{0,min{si+1, aj+1} −max{si , aj}}

I s0 = a0 = −∞ and s2n+1 = a3 =∞

R–LINE UNLV

T-theory for Server Problems

T-Theory on the Line

I For all finite subspaces of the real line, isolation indices areinterval lengths with endpoints in the subspace.

I For R–LINE, we have our algorithms servers, s1, s2, ..., s2n, and

I Two adversary servers, a1 and a2 for a total of 2n + 2 points.

I Define αi ,j , the (i , j)th isolation index of a configuration, to bethe length of the longest interval that has exactly i algorithmservers to the left and exactly j adversary servers to the left.

I Formally,

αi ,j = max{0,min{si+1, aj+1} −max{si , aj}}

I s0 = a0 = −∞ and s2n+1 = a3 =∞

R–LINE UNLV

T-theory for Server Problems

T-Theory on the Line

I For all finite subspaces of the real line, isolation indices areinterval lengths with endpoints in the subspace.

I For R–LINE, we have our algorithms servers, s1, s2, ..., s2n, and

I Two adversary servers, a1 and a2 for a total of 2n + 2 points.

I Define αi ,j , the (i , j)th isolation index of a configuration, to bethe length of the longest interval that has exactly i algorithmservers to the left and exactly j adversary servers to the left.

I Formally,

αi ,j = max{0,min{si+1, aj+1} −max{si , aj}}

I s0 = a0 = −∞ and s2n+1 = a3 =∞

R–LINE UNLV

T-theory for Server Problems

T-Theory on the Line

I αi ,j is the length of the longest interval that has exactly ialgorithm servers to the left and exactly j adversary servers tothe left.

I αi ,j = max{0,min{si+1, aj+1} −max{si , aj}}I Example,

ssss

assa

α α α1,0 3,1 3,2

α3,0 = 0, ...

R–LINE UNLV

T-theory for Server Problems

T-Theory on the Line

I αi ,j is the length of the longest interval that has exactly ialgorithm servers to the left and exactly j adversary servers tothe left.

I αi ,j = max{0,min{si+1, aj+1} −max{si , aj}}I Example,

ssss

assa

α α α1,0 3,1 3,2

α3,0 = 0, ...

R–LINE UNLV

R–LINE: Overview of our method

Outline

I Define a randomized online algorithm for the (2n, n)-serverproblem.

I Use 2-person zero-sum game theory for randomized moves.

I Prove competitiveness by solving non-linear constrainedoptimization problem and a suitable potential.

I Derive randomized 2-server algorithm from (2n, n)-serveralgorithm, via Theorem 1.

I As n grows large, competitiveness decreases.

I For R–LINE, C ≤ 1.901

I R–LINE is lazy, but it is not memoryless.

R–LINE UNLV

R–LINE: Overview of our method

Outline

I Define a randomized online algorithm for the (2n, n)-serverproblem.

I Use 2-person zero-sum game theory for randomized moves.

I Prove competitiveness by solving non-linear constrainedoptimization problem and a suitable potential.

I Derive randomized 2-server algorithm from (2n, n)-serveralgorithm, via Theorem 1.

I As n grows large, competitiveness decreases.

I For R–LINE, C ≤ 1.901

I R–LINE is lazy, but it is not memoryless.

R–LINE UNLV

R–LINE: Overview of our method

Outline

I Define a randomized online algorithm for the (2n, n)-serverproblem.

I Use 2-person zero-sum game theory for randomized moves.

I Prove competitiveness by solving non-linear constrainedoptimization problem and a suitable potential.

I Derive randomized 2-server algorithm from (2n, n)-serveralgorithm, via Theorem 1.

I As n grows large, competitiveness decreases.

I For R–LINE, C ≤ 1.901

I R–LINE is lazy, but it is not memoryless.

R–LINE UNLV

R–LINE: Overview of our method

Outline

I Define a randomized online algorithm for the (2n, n)-serverproblem.

I Use 2-person zero-sum game theory for randomized moves.

I Prove competitiveness by solving non-linear constrainedoptimization problem and a suitable potential.

I Derive randomized 2-server algorithm from (2n, n)-serveralgorithm, via Theorem 1.

I As n grows large, competitiveness decreases.

I For R–LINE, C ≤ 1.901

I R–LINE is lazy, but it is not memoryless.

R–LINE UNLV

R–LINE: Overview of our method

Outline

I Define a randomized online algorithm for the (2n, n)-serverproblem.

I Use 2-person zero-sum game theory for randomized moves.

I Prove competitiveness by solving non-linear constrainedoptimization problem and a suitable potential.

I Derive randomized 2-server algorithm from (2n, n)-serveralgorithm, via Theorem 1.

I As n grows large, competitiveness decreases.

I For R–LINE, C ≤ 1.901

I R–LINE is lazy, but it is not memoryless.

R–LINE UNLV

R–LINE: Overview of our method

Outline

I Define a randomized online algorithm for the (2n, n)-serverproblem.

I Use 2-person zero-sum game theory for randomized moves.

I Prove competitiveness by solving non-linear constrainedoptimization problem and a suitable potential.

I Derive randomized 2-server algorithm from (2n, n)-serveralgorithm, via Theorem 1.

I As n grows large, competitiveness decreases.

I For R–LINE, C ≤ 1.901

I R–LINE is lazy, but it is not memoryless.

R–LINE UNLV

R–LINE: Overview of our method

Outline

I Define a randomized online algorithm for the (2n, n)-serverproblem.

I Use 2-person zero-sum game theory for randomized moves.

I Prove competitiveness by solving non-linear constrainedoptimization problem and a suitable potential.

I Derive randomized 2-server algorithm from (2n, n)-serveralgorithm, via Theorem 1.

I As n grows large, competitiveness decreases.

I For R–LINE, C ≤ 1.901

I R–LINE is lazy, but it is not memoryless.

R–LINE UNLV

R–LINE Details

Isolation Index Coefficients

I Every isolation index, α, has an associated coefficient, η.

I R-LINE is defined in terms of these constants, η.

I Define constants ηi ,j , the (i , j)th isolation index coefficient.

I The isolation index coefficients satisfy a symmetry property,

ηi ,j = η2n−i ,2−j

I We also have η0,0 = η2n,n = 0.

R–LINE UNLV

R–LINE Details

Isolation Index Coefficients

I Every isolation index, α, has an associated coefficient, η.

I R-LINE is defined in terms of these constants, η.

I Define constants ηi ,j , the (i , j)th isolation index coefficient.

I The isolation index coefficients satisfy a symmetry property,

ηi ,j = η2n−i ,2−j

I We also have η0,0 = η2n,n = 0.

R–LINE UNLV

R–LINE Details

Isolation Index Coefficients

I Every isolation index, α, has an associated coefficient, η.

I R-LINE is defined in terms of these constants, η.

I Define constants ηi ,j , the (i , j)th isolation index coefficient.

I The isolation index coefficients satisfy a symmetry property,

ηi ,j = η2n−i ,2−j

I We also have η0,0 = η2n,n = 0.

R–LINE UNLV

R–LINE Details

Isolation Index Coefficients

I Every isolation index, α, has an associated coefficient, η.

I R-LINE is defined in terms of these constants, η.

I Define constants ηi ,j , the (i , j)th isolation index coefficient.

I The isolation index coefficients satisfy a symmetry property,

ηi ,j = η2n−i ,2−j

I We also have η0,0 = η2n,n = 0.

R–LINE UNLV

R–LINE Details

Isolation Index Coefficients

I Every isolation index, α, has an associated coefficient, η.

I R-LINE is defined in terms of these constants, η.

I Define constants ηi ,j , the (i , j)th isolation index coefficient.

I The isolation index coefficients satisfy a symmetry property,

ηi ,j = η2n−i ,2−j

I We also have η0,0 = η2n,n = 0.

R–LINE UNLV

R–LINE Details

Isolation Index Coefficients

I Every isolation index, α, has an associated coefficient, η.

I R-LINE is defined in terms of these constants, η.

I Define constants ηi ,j , the (i , j)th isolation index coefficient.

I The isolation index coefficients satisfy a symmetry property,

ηi ,j = η2n−i ,2−j

I We also have η0,0 = η2n,n = 0.

R–LINE UNLV

R–LINE Details

Definition of the Potential, φ

I For any configuration, the potential is defined as the sum ofall isolation indices multiplied by their associated coefficients.

I Formally,

φ =∑

ηi ,j · αi ,j

R–LINE UNLV

R–LINE Details

Definition of the Potential, φ

I For any configuration, the potential is defined as the sum ofall isolation indices multiplied by their associated coefficients.

I Formally,

φ =∑

ηi ,j · αi ,j

R–LINE UNLV

R–LINE Details

Definition of the Potential, φ

I For any configuration, the potential is defined as the sum ofall isolation indices multiplied by their associated coefficients.

I Formally,

φ =∑

ηi ,j · αi ,j

R–LINE UNLV

R–LINE Details

Configurations

Notation

I We refer to si as the i th server and also its location.

I We number the servers left to right.

I s1 ≤ s2 ≤ ... ≤ s2nI Current request is r .

I Previous request is r ′.

I WLOG r ′ < r .

I Servers do not pass each other.

R–LINE UNLV

R–LINE Details

Configurations

Notation

I We refer to si as the i th server and also its location.

I We number the servers left to right.

I s1 ≤ s2 ≤ ... ≤ s2nI Current request is r .

I Previous request is r ′.

I WLOG r ′ < r .

I Servers do not pass each other.

R–LINE UNLV

R–LINE Details

Configurations

Notation

I We refer to si as the i th server and also its location.

I We number the servers left to right.

I s1 ≤ s2 ≤ ... ≤ s2nI Current request is r .

I Previous request is r ′.

I WLOG r ′ < r .

I Servers do not pass each other.

R–LINE UNLV

R–LINE Details

Configurations

Notation

I We refer to si as the i th server and also its location.

I We number the servers left to right.

I s1 ≤ s2 ≤ ... ≤ s2nI Current request is r .

I Previous request is r ′.

I WLOG r ′ < r .

I Servers do not pass each other.

R–LINE UNLV

R–LINE Details

Configurations

Notation

I We refer to si as the i th server and also its location.

I We number the servers left to right.

I s1 ≤ s2 ≤ ... ≤ s2n

I Current request is r .

I Previous request is r ′.

I WLOG r ′ < r .

I Servers do not pass each other.

R–LINE UNLV

R–LINE Details

Configurations

Notation

I We refer to si as the i th server and also its location.

I We number the servers left to right.

I s1 ≤ s2 ≤ ... ≤ s2nI Current request is r .

I Previous request is r ′.

I WLOG r ′ < r .

I Servers do not pass each other.

R–LINE UNLV

R–LINE Details

Configurations

Notation

I We refer to si as the i th server and also its location.

I We number the servers left to right.

I s1 ≤ s2 ≤ ... ≤ s2nI Current request is r .

I Previous request is r ′.

I WLOG r ′ < r .

I Servers do not pass each other.

R–LINE UNLV

R–LINE Details

Configurations

Notation

I We refer to si as the i th server and also its location.

I We number the servers left to right.

I s1 ≤ s2 ≤ ... ≤ s2nI Current request is r .

I Previous request is r ′.

I WLOG r ′ < r .

I Servers do not pass each other.

R–LINE UNLV

R–LINE Details

Configurations

Notation

I We refer to si as the i th server and also its location.

I We number the servers left to right.

I s1 ≤ s2 ≤ ... ≤ s2nI Current request is r .

I Previous request is r ′.

I WLOG r ′ < r .

I Servers do not pass each other.

R–LINE UNLV

R–LINE Details

S-Configuration (Satisfying)

I There are n servers at the request point.

(6, 3) Example

r

sss

s ss 51 2

3

4

6

R–LINE UNLV

R–LINE Details

S-Configuration (Satisfying)

I There are n servers at the request point.

(6, 3) Example

r

sss

s ss 51 2

3

4

6

R–LINE UNLV

R–LINE Details

S-Configuration (Satisfying)

I There are n servers at the request point.

(6, 3) Example

r

sss

s ss 51 2

3

4

6

R–LINE UNLV

R–LINE Details

S-Configuration (Satisfying)

I There are n servers at the request point.

(6, 3) Example

r

sss

s ss 51 2

3

4

6

R–LINE UNLV

R–LINE Details

D-Configuration (Deterministic)

I 1. More than n algorithm servers either strictly to the left orstrictly to the right of r ; r > sn+1 or r < sn.

2. If there are fewer than n algorithm servers at r ′

2.1 No algorithm server strictly between r ′ and r2.2 At least n algorithm servers at the points r ′ and r combined.

(6, 3) Example

r'

sss

s ss

r

1

2

3

4 5 6

R–LINE UNLV

R–LINE Details

D-Configuration (Deterministic)

I 1. More than n algorithm servers either strictly to the left orstrictly to the right of r ; r > sn+1 or r < sn.

2. If there are fewer than n algorithm servers at r ′

2.1 No algorithm server strictly between r ′ and r2.2 At least n algorithm servers at the points r ′ and r combined.

(6, 3) Example

r'

sss

s ss

r

1

2

3

4 5 6

R–LINE UNLV

R–LINE Details

D-Configuration (Deterministic)

I 1. More than n algorithm servers either strictly to the left orstrictly to the right of r ; r > sn+1 or r < sn.

2. If fewer than n algorithm servers at r ′

2.1 No algorithm server strictly between r ′ and r2.2 At least n algorithm servers at the points r ′ and r combined.

(6, 3) Example

r'

ss

s s ss

r

1 2 3

4

5 6

R–LINE UNLV

R–LINE Details

D-Configuration Moves

1. Must be m servers to the left of r , for some m > n.

2. Move sn+1...sm to r .

(6, 3) Example

R–LINE UNLV

R–LINE Details

D-Configuration Moves

1. Must be m servers to the left of r , for some m > n.

2. Move sn+1...sm to r .

(6, 3) Example

R–LINE UNLV

R–LINE Details

D-Configuration Moves

1. Must be m servers to the left of r , for some m > n.

2. Move sn+1...sm to r .

(6, 3) Example

R–LINE UNLV

R–LINE Details

D-Configuration Moves

1. Must be m servers to the left of r , for some m > n.

2. Move sn+1...sm to r .

(6, 3) Example

r'

sss

s ss

r

1

2

3

4 5 6

R–LINE UNLV

R–LINE Details

D-Configuration Moves

1. Must be m servers to the left of r , for some m > n.

2. Move sn+1...sm to r .

(6, 3) Example

r'

sss

s ss

r

1

2

3

4 5 6

R–LINE UNLV

R–LINE Details

D-Configuration Moves

1. Must be m servers to the left of r , for some m > n.

2. Move sn+1...sm to r .

(6, 3) Example

r'

sss

s ss

r

1

2

3

4 5 6

R–LINE UNLV

R–LINE Details

D-Configuration Moves

1. Must be m servers to the left of r , for some m > n.

2. Move sn+1...sm to r .

(6, 3) Example

r'

ss

s s ss

r

1 2 3

4

5 6

R–LINE UNLV

R–LINE Details

D-Configuration Moves

1. Must be m servers to the left of r , for some m > n.

2. Move sn+1...sm to r .

(6, 3) Example

r'

ss

s s ss

r

1 2 3

4

5 6

R–LINE UNLV

R–LINE Details

D-Configuration Moves

1. Must be m servers to the left of r , for some m > n.

2. Move sn+1...sm to r .

(6, 3) Example

r'

ss

s s ss

r

1 2 3

4

5 6

R–LINE UNLV

R–LINE Details

R-Configurations (Randomized)

The adversary’s hidden server.

I The adversary has two servers.

I The current request, r .

I The other server’s location, a, is “hidden”.

I There are only two hidden server locations to consider.

(6, 3) Example

R–LINE UNLV

R–LINE Details

R-Configurations (Randomized)

The adversary’s hidden server.

I The adversary has two servers.

I The current request, r .

I The other server’s location, a, is “hidden”.

I There are only two hidden server locations to consider.

(6, 3) Example

R–LINE UNLV

R–LINE Details

R-Configurations (Randomized)

The adversary’s hidden server.

I The adversary has two servers.

I The current request, r .

I The other server’s location, a, is “hidden”.

I There are only two hidden server locations to consider.

(6, 3) Example

R–LINE UNLV

R–LINE Details

R-Configurations (Randomized)

The adversary’s hidden server.

I The adversary has two servers.

I The current request, r .

I The other server’s location, a, is “hidden”.

I There are only two hidden server locations to consider.

(6, 3) Example

R–LINE UNLV

R–LINE Details

R-Configurations (Randomized)

The adversary’s hidden server.

I The adversary has two servers.

I The current request, r .

I The other server’s location, a, is “hidden”.

I There are only two hidden server locations to consider.

(6, 3) Example

R–LINE UNLV

R–LINE Details

R-Configurations (Randomized)

The adversary’s hidden server.

I The adversary has two servers.

I The current request, r .

I The other server’s location, a, is “hidden”.

I There are only two hidden server locations to consider.

(6, 3) Example

R–LINE UNLV

R–LINE Details

R-Configurations (Randomized)

The adversary’s hidden server.

I The adversary has two servers.

I The current request, r .

I The other server’s location, a, is “hidden”.

I There are only two hidden server locations to consider.

(6, 3) Example

r

sss

s ss 51 2

3

4

6

a

R–LINE UNLV

R–LINE Details

R-Configurations (Randomized)

The adversary’s hidden server.

I The adversary has two servers.

I The current request, r .

I The other server’s location, a, is “hidden”.

I There are only two hidden server locations to consider.

(6, 3) Example

r

sss

s ss 51 2

3

4

6

a

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized)

1. Exactly n algorithm servers on the same side of r as r ′. Eitherr ′ = sn < r or r < r ′ = sn+1.

2. No algorithm server strictly between r ′ and r .

3. At least n algorithm servers at the points r ′ and r combined.

(6, 3) Example

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized)

1. Exactly n algorithm servers on the same side of r as r ′. Eitherr ′ = sn < r or r < r ′ = sn+1.

2. No algorithm server strictly between r ′ and r .

3. At least n algorithm servers at the points r ′ and r combined.

(6, 3) Example

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized)

1. Exactly n algorithm servers on the same side of r as r ′. Eitherr ′ = sn < r or r < r ′ = sn+1.

2. No algorithm server strictly between r ′ and r .

3. At least n algorithm servers at the points r ′ and r combined.

(6, 3) Example

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized)

1. Exactly n algorithm servers on the same side of r as r ′. Eitherr ′ = sn < r or r < r ′ = sn+1.

2. No algorithm server strictly between r ′ and r .

3. At least n algorithm servers at the points r ′ and r combined.

(6, 3) Example

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized)

1. Exactly n algorithm servers on the same side of r as r ′. Eitherr ′ = sn < r or r < r ′ = sn+1.

2. No algorithm server strictly between r ′ and r .

3. At least n algorithm servers at the points r ′ and r combined.

(6, 3) Example

r'

ss

s ss

r

51

2

4 6

s3

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized) Moves

I There are two possible moves.

1. Move a single server.2. Complete the request using the servers from r ′.

I Choose between the two alternatives using randomization, bysolving a 2-person zero-sum game.

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized) Moves

I There are two possible moves.

1. Move a single server.

2. Complete the request using the servers from r ′.

I Choose between the two alternatives using randomization, bysolving a 2-person zero-sum game.

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized) Moves

I There are two possible moves.

1. Move a single server.2. Complete the request using the servers from r ′.

I Choose between the two alternatives using randomization, bysolving a 2-person zero-sum game.

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized) Moves

I There are two possible moves.

1. Move a single server.2. Complete the request using the servers from r ′.

I Choose between the two alternatives using randomization, bysolving a 2-person zero-sum game.

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized) Moves

I There are two possible moves.

1. Move a single server.2. Complete the request using the servers from r ′.

I Choose between the two alternatives using randomization, bysolving a 2-person zero-sum game.

r'

ss

s ss

r

51

2

4 6

s3

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized) Moves

I There are two possible moves.

1. Move a single server.2. Complete the request using the servers from r ′.

I Choose between the two alternatives using randomization, bysolving a 2-person zero-sum game.

r'

ss

s ss

r

51

2

4 6

s3

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized) Moves

I There are two possible moves.

1. Move a single server.2. Complete the request using the servers from r ′.

I Choose between the two alternatives using randomization, bysolving a 2-person zero-sum game.

r'

ss

s ss

r

5

1

2

4 6

s3

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized) Moves

I There are two possible moves.

1. Move a single server.2. Complete the request using the servers from r ′.

I Choose between the two alternatives using randomization, bysolving a 2-person zero-sum game.

r'

ss

s ss

r

51

2

4 6

s3

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized) Moves

I There are two possible moves.

1. Move a single server.2. Complete the request using the servers from r ′.

I Choose between the two alternatives using randomization, bysolving a 2-person zero-sum game.

r'

ss

s ss

r

51

2

4 6

s3

R–LINE UNLV

R–LINE Details

R-Configuration (Randomized) Moves

I There are two possible moves.

1. Move a single server.2. Complete the request using the servers from r ′.

I Choose between the two alternatives using randomization, bysolving a 2-person zero-sum game.

r'

sss ss

r

51

2

4 6

s3

R–LINE UNLV

R–LINE Details

Configurations: R-Configurations (Randomized)

sss

sr s s1 sss

sr s s

sss

s

ra

s s

(∆φ+ cost)11 (∆φ+ cost)12

sss

s

r a

s s

(∆φ+ cost)21 (∆φ+ cost)22

R–LINE UNLV

R–LINE Details

Configurations: R-Configurations (Randomized)

sss

sr s s1 sss

sr s s

sss

s

ra

s s

(∆φ+ cost)11 (∆φ+ cost)12

sss

s

r a

s s

(∆φ+ cost)21 (∆φ+ cost)22

R–LINE UNLV

R–LINE Details

Configurations: R-Configurations (Randomized)

(∆φ+ cost)11 (∆φ+ cost)12(∆φ+ cost)21 (∆φ+ cost)22

I Entries game matrix computed in terms of the isolation indexcoefficients, η.

I If currently p servers located at r :

(∆φ+ cost)11 = (ηn+p+1,2 − ηn+p,2 + 1) · (sn+p+1 − r)(∆φ+ cost)12 = (ηp,1 − ηn,1 + n − p) · (r − sn)(∆φ+ cost)21 = (ηn+p+1,1 − ηn+p,1 + 1) · (sn+p+1 − r)(∆φ+ cost)22 = (ηp,0 − ηn,0 + n − p) · (r − sn)

R–LINE UNLV

R–LINE Details

Configurations: R-Configurations (Randomized)

(∆φ+ cost)11 (∆φ+ cost)12(∆φ+ cost)21 (∆φ+ cost)22

I Entries game matrix computed in terms of the isolation indexcoefficients, η.

I If currently p servers located at r :

(∆φ+ cost)11 = (ηn+p+1,2 − ηn+p,2 + 1) · (sn+p+1 − r)(∆φ+ cost)12 = (ηp,1 − ηn,1 + n − p) · (r − sn)(∆φ+ cost)21 = (ηn+p+1,1 − ηn+p,1 + 1) · (sn+p+1 − r)(∆φ+ cost)22 = (ηp,0 − ηn,0 + n − p) · (r − sn)

R–LINE UNLV

R–LINE Details

Configurations: R-Configurations (Randomized)

(∆φ+ cost)11 (∆φ+ cost)12(∆φ+ cost)21 (∆φ+ cost)22

I Entries game matrix computed in terms of the isolation indexcoefficients, η.

I If currently p servers located at r :

(∆φ+ cost)11 = (ηn+p+1,2 − ηn+p,2 + 1) · (sn+p+1 − r)(∆φ+ cost)12 = (ηp,1 − ηn,1 + n − p) · (r − sn)(∆φ+ cost)21 = (ηn+p+1,1 − ηn+p,1 + 1) · (sn+p+1 − r)(∆φ+ cost)22 = (ηp,0 − ηn,0 + n − p) · (r − sn)

R–LINE UNLV

R–LINE Details

Configurations: R-Configurations (Randomized)

(∆φ+ cost)11 (∆φ+ cost)12(∆φ+ cost)21 (∆φ+ cost)22

I Entries game matrix computed in terms of the isolation indexcoefficients, η.

I If currently p servers located at r :

(∆φ+ cost)11 = (ηn+p+1,2 − ηn+p,2 + 1) · (sn+p+1 − r)(∆φ+ cost)12 = (ηp,1 − ηn,1 + n − p) · (r − sn)(∆φ+ cost)21 = (ηn+p+1,1 − ηn+p,1 + 1) · (sn+p+1 − r)(∆φ+ cost)22 = (ηp,0 − ηn,0 + n − p) · (r − sn)

R–LINE UNLV

R–LINE Details

Configurations: R-Configurations (Randomized)

(∆φ+ cost)11 (∆φ+ cost)12(∆φ+ cost)21 (∆φ+ cost)22

I Entries game matrix computed in terms of the isolation indexcoefficients, η.

I If currently p servers located at r :

(∆φ+ cost)11 = (ηn+p+1,2 − ηn+p,2 + 1) · (sn+p+1 − r)(∆φ+ cost)12 = (ηp,1 − ηn,1 + n − p) · (r − sn)(∆φ+ cost)21 = (ηn+p+1,1 − ηn+p,1 + 1) · (sn+p+1 − r)(∆φ+ cost)22 = (ηp,0 − ηn,0 + n − p) · (r − sn)

R–LINE UNLV

R–LINE Details

The Algorithm R–LINE

1. Receive a request.

2. If D–Config, make deterministic moves.

2.1 If result is S–Config, done.2.2 Otherwise result is R–Config.

3. If R–Config, make randomized moves until S–Config.

Srequest

D

R

S

≤ n

(Detailed (6, 3)-server example given in the paper)

R–LINE UNLV

R–LINE Details

The Algorithm R–LINE

1. Receive a request.

2. If D–Config, make deterministic moves.

2.1 If result is S–Config, done.2.2 Otherwise result is R–Config.

3. If R–Config, make randomized moves until S–Config.

Srequest

D

R

S

≤ n

(Detailed (6, 3)-server example given in the paper)

R–LINE UNLV

R–LINE Details

The Algorithm R–LINE

1. Receive a request.

2. If D–Config, make deterministic moves.

2.1 If result is S–Config, done.2.2 Otherwise result is R–Config.

3. If R–Config, make randomized moves until S–Config.

Srequest

D

R

S

≤ n

(Detailed (6, 3)-server example given in the paper)

R–LINE UNLV

Proof of Competitiveness

Overview of Proof

1. Provide a system of inequalities, S, involving the isolationindex coefficients, ηi ,j , and the competitiveness, C .

2. Show that if there exists an assignment of values to every ηi ,jthat satisfies S, then R–LINE is C -competitive.

3. Use numeric methods to find a solution to S that minimizes C .

R–LINE UNLV

Proof of Competitiveness

Sufficient Inequalities, Sn

∀ 0 ≤ i ≤ 2n : |ηi,1 − ηi,0| ≤ n · C (1)

∀ 1 ≤ i ≤ n and ∀ 1 ≤ j ≤ 2 : ηi,j + 1 ≤ ηi−1,j (2)

∀ 1 ≤ i ≤ n and ∀ 1 ≤ j ≤ 2 : ηi−1,j−1 ≤ ηi,j−1 + 1 (3)

∀ 1 ≤ i ≤ n : (ηi−1,1 − ηi,1 + 1)(ηn−i,1 − ηn,1 + i) ≤ (ηi−1,0 − ηi,0 + 1)(ηn−i,0 − ηn,0 + i) (4)

R–LINE UNLV

Proof of Competitiveness

Overview of Proof Steps

Show that if the system of inequalities, Sn, is satisfied, then thefollowing properties hold:

1. For adversary moves: ∆φ ≤ C · costAdv .

2. For R–LINE deterministic moves: ∆φ+ cost ≤ 0.

3. WLOG, adversary’s hidden server is at one of at most twopossible locations.

4. For R–LINE randomized moves: E(∆φ+ cost) ≤ 0.

Adding all of the inequalities over a request round:

E (costA(σ)) ≤ C · costADV(σ) + K

R–LINE UNLV

Proof of Competitiveness

Overview of Proof Steps

Show that if the system of inequalities, Sn, is satisfied, then thefollowing properties hold:

1. For adversary moves: ∆φ ≤ C · costAdv .

2. For R–LINE deterministic moves: ∆φ+ cost ≤ 0.

3. WLOG, adversary’s hidden server is at one of at most twopossible locations.

4. For R–LINE randomized moves: E(∆φ+ cost) ≤ 0.

Adding all of the inequalities over a request round:

E (costA(σ)) ≤ C · costADV(σ) + K

R–LINE UNLV

Proof of Competitiveness

Overview of Proof Steps

Show that if the system of inequalities, Sn, is satisfied, then thefollowing properties hold:

1. For adversary moves: ∆φ ≤ C · costAdv .

2. For R–LINE deterministic moves: ∆φ+ cost ≤ 0.

3. WLOG, adversary’s hidden server is at one of at most twopossible locations.

4. For R–LINE randomized moves: E(∆φ+ cost) ≤ 0.

Adding all of the inequalities over a request round:

E (costA(σ)) ≤ C · costADV(σ) + K

R–LINE UNLV

Proof of Competitiveness

Overview of Proof Steps

Show that if the system of inequalities, Sn, is satisfied, then thefollowing properties hold:

1. For adversary moves: ∆φ ≤ C · costAdv .

2. For R–LINE deterministic moves: ∆φ+ cost ≤ 0.

3. WLOG, adversary’s hidden server is at one of at most twopossible locations.

4. For R–LINE randomized moves: E(∆φ+ cost) ≤ 0.

Adding all of the inequalities over a request round:

E (costA(σ)) ≤ C · costADV(σ) + K

R–LINE UNLV

Proof of Competitiveness

Overview of Proof Steps

Show that if the system of inequalities, Sn, is satisfied, then thefollowing properties hold:

1. For adversary moves: ∆φ ≤ C · costAdv .

2. For R–LINE deterministic moves: ∆φ+ cost ≤ 0.

3. WLOG, adversary’s hidden server is at one of at most twopossible locations.

4. For R–LINE randomized moves: E(∆φ+ cost) ≤ 0.

Adding all of the inequalities over a request round:

E (costA(σ)) ≤ C · costADV(σ) + K

R–LINE UNLV

Proof of Competitiveness

Overview of Proof Steps

Show that if the system of inequalities, Sn, is satisfied, then thefollowing properties hold:

1. For adversary moves: ∆φ ≤ C · costAdv .

2. For R–LINE deterministic moves: ∆φ+ cost ≤ 0.

3. WLOG, adversary’s hidden server is at one of at most twopossible locations.

4. For R–LINE randomized moves: E(∆φ+ cost) ≤ 0.

Adding all of the inequalities over a request round:

E (costA(σ)) ≤ C · costADV(σ) + K

R–LINE UNLV

Solving the Inequalities

Given the system Sn

I Find a solution to Sn that minimizes C .

I Transform Sn into a new system S′n to simplify the solution.

R–LINE UNLV

Solving the Inequalities

Given the system Sn

I Find a solution to Sn that minimizes C .

I Transform Sn into a new system S′n to simplify the solution.

R–LINE UNLV

Solving the Inequalities

Introduce variables εi and δi for all 0 ≤ i < n. Then S′n consists ofthe following constraints:

(2i + εn−i )(2− εi + εi−1) = 4i ∀ 0 < i < n(2n + ε0)(2 + εn−1) ≥ 4n

δ = −ε0/2C = (2n − δ)/nδi = εi + 2δ ∀ 0 ≤ i < n

ηi ,0 = 3i − δi ∀ 0 ≤ i < nηi ,0 = 2n + i − 2δ ∀ n ≤ i ≤ 2nηi ,1 = 2n − i − δ ∀ 0 ≤ i < nηi ,1 = i − δ ∀ n ≤ i ≤ 2nηi ,2 = η2n−i ,0 ∀ 0 ≤ i ≤ 2n

R–LINE UNLV

Solving the Inequalities

Introduce variables εi and δi for all 0 ≤ i < n. Then S′n consists ofthe following constraints:

(2i + εn−i )(2− εi + εi−1) = 4i ∀ 0 < i < n(2n + ε0)(2 + εn−1) ≥ 4n

δ = −ε0/2C = (2n − δ)/nδi = εi + 2δ ∀ 0 ≤ i < n

ηi ,0 = 3i − δi ∀ 0 ≤ i < nηi ,0 = 2n + i − 2δ ∀ n ≤ i ≤ 2nηi ,1 = 2n − i − δ ∀ 0 ≤ i < nηi ,1 = i − δ ∀ n ≤ i ≤ 2nηi ,2 = η2n−i ,0 ∀ 0 ≤ i ≤ 2n

R–LINE UNLV

Solving the Inequalities

The (6, 3) Case

I When n = 3 we can solve S′ analytically.

I The desired values come from the roots of a fourth degreepolynomial.

R–LINE UNLV

Solving the Inequalities

The (6, 3) Case

I When n = 3 we can solve S′ analytically.

I The desired values come from the roots of a fourth degreepolynomial.

R–LINE UNLV

Solving the Inequalities

The (6, 3) Case

I For n = 3, R–LINE has competitiveness

C = 1 +

√71+17

√17

12 ≈ 1.98985407

Constants

δ = 3−√

71+17√

174

≈ 0.030437789

δ1 = 2−√

7+√

172

≈ 0.332433987

δ2 = 4−

√79−7√

17

2≈ 0.459581218

ηi,j =

0 1 2

0 0 6− δ 12− 2δ1 3− δ1 5− δ 11− 2δ2 6− δ2 4− δ 10− 2δ3 9− 2δ 3− δ 9− 2δ4 10− 2δ 4− δ 6− δ25 11− 2δ 5− δ 3− δ16 12− 2δ 6− δ 0

R–LINE UNLV

Solving the Inequalities

The (6, 3) Case

I For n = 3, R–LINE has competitiveness

C = 1 +

√71+17

√17

12 ≈ 1.98985407

Constants

δ = 3−√

71+17√

174

≈ 0.030437789

δ1 = 2−√

7+√

172

≈ 0.332433987

δ2 = 4−

√79−7√

17

2≈ 0.459581218

ηi,j =

0 1 2

0 0 6− δ 12− 2δ1 3− δ1 5− δ 11− 2δ2 6− δ2 4− δ 10− 2δ3 9− 2δ 3− δ 9− 2δ4 10− 2δ 4− δ 6− δ25 11− 2δ 5− δ 3− δ16 12− 2δ 6− δ 0

R–LINE UNLV

Solving the Inequalities

The (6, 3) Case

I For n = 3, R–LINE has competitiveness

C = 1 +

√71+17

√17

12 ≈ 1.98985407

Constants

δ = 3−√

71+17√

174

≈ 0.030437789

δ1 = 2−√

7+√

172

≈ 0.332433987

δ2 = 4−

√79−7√

17

2≈ 0.459581218

ηi,j =

0 1 2

0 0 6− δ 12− 2δ1 3− δ1 5− δ 11− 2δ2 6− δ2 4− δ 10− 2δ3 9− 2δ 3− δ 9− 2δ4 10− 2δ 4− δ 6− δ25 11− 2δ 5− δ 3− δ16 12− 2δ 6− δ 0

R–LINE UNLV

Solving the Inequalities

Larger values of n

I It is much harder to find a solution to S′ analytically for valuesof n larger than 3.

I Use numeric methods to solve S′.

R–LINE UNLV

Solving the Inequalities

Larger values of n

I It is much harder to find a solution to S′ analytically for valuesof n larger than 3.

I Use numeric methods to solve S′.

R–LINE UNLV

Solving the Inequalities

The (2n, n) Case, n > 3

I We find that C ≈ 1.90098671 for n = 2000.

R–LINE UNLV

Solving the Inequalities, Numeric Method

1. Guess εbn/2c.

2. If n odd, solve for ε(n+1)/2:

(n + 1 + ε(n−1)/2) = (2− ε(n+1)/2 + ε(n−1)/2)

3. For all 0 < i < b n2c in decreasing order:

3.1 Solve for εi :(2(i + 1) + εn−i−1)(2− εi+1 + εi ) = 4(i + 1)

3.2 Solve the following equation for εn−i :

2(n − i) + εi 2− εn−i + εn−i−1 = 4(n − i)

4. Solve the following equation for δ:

(2 + εn−1)(2− ε1 − 2δ) = 4

5. Verify the following inequality:(2n − 2δ)(2 + εn−1) ≥ 4n

6. If our search interval is small enough, proceed to the last step. Otherwise, return to step 1.

7. C ← 2n−δn

.

R–LINE UNLV

Future Work

Other spaces

I Trees - Preliminary work done, strongly suggests C ≤ 1.901.

R–LINE UNLV

Future Work

Other spaces

I Trees - Preliminary work done, strongly suggests C ≤ 1.901.

R–LINE UNLV

Future Work

Other spaces

I Trees - Preliminary work done, strongly suggests C ≤ 1.901.

R–LINE UNLV

Future Work

Other spaces

I Trees - Preliminary work done, strongly suggests C ≤ 1.901.

R–LINE UNLV

Future Work

Other spaces

I Split-Decomposable Spaces

I 6-point spaces,339 generic cases, completely classified.

(Sturmfeldt and Yu)

R–LINE UNLV

Future Work

Other spaces

I Split-Decomposable Spaces

I 6-point spaces,339 generic cases, completely classified.

(Sturmfeldt and Yu)

R–LINE UNLV

Future Work

Other spaces

I Split-Decomposable Spaces

I 6-point spaces,339 generic cases, completely classified.

(Sturmfeldt and Yu)

R–LINE UNLV

Future Work

Other spaces

I Split-Decomposable Spaces

I 6-point spaces,339 generic cases, completely classified.

(Sturmfeldt and Yu)

R–LINE UNLV

R–LINE

The End.

R–LINE UNLV

R–LINE

The End.

R–LINE UNLV

R–LINE

The End.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 1

If S holds, then Property 1 holds: For any move by the adversary,∆φ ≤ C · costAdv

sas i i+1j a'jx y

I By inequality (1), |ηi ,j − ηi ,j−1| ≤ n · C for j = 1, 2.

I Adversary server aj moves to the right, from x to y , wherex < y , with si ≤ x and y ≤ si+1.

I Thus, αi ,j decreases by y − x and αi ,j−1 increases by y − x .

I The cost to the adversary of this move is n(y − x).

I By definition of the potential,∆φ = (ηi ,j − ηi ,j−1)(y − x) ≤ n · C · (y − x) ≤ C · costAdv .

R–LINE UNLV

Proof of Competitiveness

Proof of Property 1

If S holds, then Property 1 holds: For any move by the adversary,∆φ ≤ C · costAdv

sas i i+1j a'jx y

I By inequality (1), |ηi ,j − ηi ,j−1| ≤ n · C for j = 1, 2.

I Adversary server aj moves to the right, from x to y , wherex < y , with si ≤ x and y ≤ si+1.

I Thus, αi ,j decreases by y − x and αi ,j−1 increases by y − x .

I The cost to the adversary of this move is n(y − x).

I By definition of the potential,∆φ = (ηi ,j − ηi ,j−1)(y − x) ≤ n · C · (y − x) ≤ C · costAdv .

R–LINE UNLV

Proof of Competitiveness

Proof of Property 1

If S holds, then Property 1 holds: For any move by the adversary,∆φ ≤ C · costAdv

sas i i+1j a'jx y

I By inequality (1), |ηi ,j − ηi ,j−1| ≤ n · C for j = 1, 2.

I Adversary server aj moves to the right, from x to y , wherex < y , with si ≤ x and y ≤ si+1.

I Thus, αi ,j decreases by y − x and αi ,j−1 increases by y − x .

I The cost to the adversary of this move is n(y − x).

I By definition of the potential,∆φ = (ηi ,j − ηi ,j−1)(y − x) ≤ n · C · (y − x) ≤ C · costAdv .

R–LINE UNLV

Proof of Competitiveness

Proof of Property 1

If S holds, then Property 1 holds: For any move by the adversary,∆φ ≤ C · costAdv

sas i i+1j a'jx y

I By inequality (1), |ηi ,j − ηi ,j−1| ≤ n · C for j = 1, 2.

I Adversary server aj moves to the right, from x to y , wherex < y , with si ≤ x and y ≤ si+1.

I Thus, αi ,j decreases by y − x and αi ,j−1 increases by y − x .

I The cost to the adversary of this move is n(y − x).

I By definition of the potential,∆φ = (ηi ,j − ηi ,j−1)(y − x) ≤ n · C · (y − x) ≤ C · costAdv .

R–LINE UNLV

Proof of Competitiveness

Proof of Property 1

If S holds, then Property 1 holds: For any move by the adversary,∆φ ≤ C · costAdv

sas i i+1j a'jx y

I By inequality (1), |ηi ,j − ηi ,j−1| ≤ n · C for j = 1, 2.

I Adversary server aj moves to the right, from x to y , wherex < y , with si ≤ x and y ≤ si+1.

I Thus, αi ,j decreases by y − x and αi ,j−1 increases by y − x .

I The cost to the adversary of this move is n(y − x).

I By definition of the potential,∆φ = (ηi ,j − ηi ,j−1)(y − x) ≤ n · C · (y − x) ≤ C · costAdv .

R–LINE UNLV

Proof of Competitiveness

Proof of Property 1

If S holds, then Property 1 holds: For any move by the adversary,∆φ ≤ C · costAdv

sas i i+1j a'jx y

I By inequality (1), |ηi ,j − ηi ,j−1| ≤ n · C for j = 1, 2.

I Adversary server aj moves to the right, from x to y , wherex < y , with si ≤ x and y ≤ si+1.

I Thus, αi ,j decreases by y − x and αi ,j−1 increases by y − x .

I The cost to the adversary of this move is n(y − x).

I By definition of the potential,∆φ = (ηi ,j − ηi ,j−1)(y − x) ≤ n · C · (y − x) ≤ C · costAdv .

R–LINE UNLV

Proof of Competitiveness

Proof of Property 1

If S holds, then Property 1 holds: For any move by the adversary,∆φ ≤ C · costAdv

sas i i+1j a'jx y

I By inequality (1), |ηi ,j − ηi ,j−1| ≤ n · C for j = 1, 2.

I Adversary server aj moves to the right, from x to y , wherex < y , with si ≤ x and y ≤ si+1.

I Thus, αi ,j decreases by y − x and αi ,j−1 increases by y − x .

I The cost to the adversary of this move is n(y − x).

I By definition of the potential,∆φ = (ηi ,j − ηi ,j−1)(y − x) ≤ n · C · (y − x) ≤ C · costAdv .

R–LINE UNLV

Proof of Competitiveness

Proof of Property 2

If S holds, then Property 2 holds: for any deterministic move byR–LINE, ∆φ+ cost ≤ 0.

s i s'ix y

I si moves from x to y , where x < y .

I The algorithm cost of the step is y − x .

I The move causes αi ,j to decrease by y − x and αi−1,j toincrease by the same amount.

I By inequality (2), ηi ,j + 1 ≤ ηi−1,j , and the definition of thepotential: ∆φ+ costR−−LINE = (y − x)(ηi ,j − ηi−1,j + 1) ≤ 0.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 2

If S holds, then Property 2 holds: for any deterministic move byR–LINE, ∆φ+ cost ≤ 0.

s i s'ix y

I si moves from x to y , where x < y .

I The algorithm cost of the step is y − x .

I The move causes αi ,j to decrease by y − x and αi−1,j toincrease by the same amount.

I By inequality (2), ηi ,j + 1 ≤ ηi−1,j , and the definition of thepotential: ∆φ+ costR−−LINE = (y − x)(ηi ,j − ηi−1,j + 1) ≤ 0.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 2

If S holds, then Property 2 holds: for any deterministic move byR–LINE, ∆φ+ cost ≤ 0.

s i s'ix y

I si moves from x to y , where x < y .

I The algorithm cost of the step is y − x .

I The move causes αi ,j to decrease by y − x and αi−1,j toincrease by the same amount.

I By inequality (2), ηi ,j + 1 ≤ ηi−1,j , and the definition of thepotential: ∆φ+ costR−−LINE = (y − x)(ηi ,j − ηi−1,j + 1) ≤ 0.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 2

If S holds, then Property 2 holds: for any deterministic move byR–LINE, ∆φ+ cost ≤ 0.

s i s'ix y

I si moves from x to y , where x < y .

I The algorithm cost of the step is y − x .

I The move causes αi ,j to decrease by y − x and αi−1,j toincrease by the same amount.

I By inequality (2), ηi ,j + 1 ≤ ηi−1,j , and the definition of thepotential: ∆φ+ costR−−LINE = (y − x)(ηi ,j − ηi−1,j + 1) ≤ 0.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 2

If S holds, then Property 2 holds: for any deterministic move byR–LINE, ∆φ+ cost ≤ 0.

s i s'ix y

I si moves from x to y , where x < y .

I The algorithm cost of the step is y − x .

I The move causes αi ,j to decrease by y − x and αi−1,j toincrease by the same amount.

I By inequality (2), ηi ,j + 1 ≤ ηi−1,j , and the definition of thepotential: ∆φ+ costR−−LINE = (y − x)(ηi ,j − ηi−1,j + 1) ≤ 0.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 2

If S holds, then Property 2 holds: for any deterministic move byR–LINE, ∆φ+ cost ≤ 0.

s i s'ix y

I si moves from x to y , where x < y .

I The algorithm cost of the step is y − x .

I The move causes αi ,j to decrease by y − x and αi−1,j toincrease by the same amount.

I By inequality (2), ηi ,j + 1 ≤ ηi−1,j , and the definition of thepotential: ∆φ+ costR−−LINE = (y − x)(ηi ,j − ηi−1,j + 1) ≤ 0.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 3

If S holds, then Property (3) holds: We may assume the adversary’shidden server is at one of at most two possible locations.

r

sss

s ss 51 2

3

4

6

aa

I Since a could be any point on the line, the payoff matrix ofthe game has infinitely many rows.

I We prove that just two of those rows, namely a = sn anda = sn+p+1, dominate the others.

I By batching the row strategies, we illustrate the ∞× 2 payoffmatrix in the next slide.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 3

If S holds, then Property (3) holds: We may assume the adversary’shidden server is at one of at most two possible locations.

r

sss

s ss 51 2

3

4

6

aa

I Since a could be any point on the line, the payoff matrix ofthe game has infinitely many rows.

I We prove that just two of those rows, namely a = sn anda = sn+p+1, dominate the others.

I By batching the row strategies, we illustrate the ∞× 2 payoffmatrix in the next slide.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 3

If S holds, then Property (3) holds: We may assume the adversary’shidden server is at one of at most two possible locations.

r

sss

s ss 51 2

3

4

6

aa

I Since a could be any point on the line, the payoff matrix ofthe game has infinitely many rows.

I We prove that just two of those rows, namely a = sn anda = sn+p+1, dominate the others.

I By batching the row strategies, we illustrate the ∞× 2 payoffmatrix in the next slide.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 3

If S holds, then Property (3) holds: We may assume the adversary’shidden server is at one of at most two possible locations.

r

sss

s ss 51 2

3

4

6

aa

I Since a could be any point on the line, the payoff matrix ofthe game has infinitely many rows.

I We prove that just two of those rows, namely a = sn anda = sn+p+1, dominate the others.

I By batching the row strategies, we illustrate the ∞× 2 payoffmatrix in the next slide.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 3

If S holds, then Property (3) holds: We may assume the adversary’shidden server is at one of at most two possible locations.

r

sss

s ss 51 2

3

4

6

aa

I Since a could be any point on the line, the payoff matrix ofthe game has infinitely many rows.

I We prove that just two of those rows, namely a = sn anda = sn+p+1, dominate the others.

I By batching the row strategies, we illustrate the ∞× 2 payoffmatrix in the next slide.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 3

If S holds, then Property (3) holds: We may assume the adversary’shidden server is at one of at most two possible locations.

r

sss

s ss 51 2

3

4

6

aa

I Since a could be any point on the line, the payoff matrix ofthe game has infinitely many rows.

I We prove that just two of those rows, namely a = sn anda = sn+p+1, dominate the others.

I By batching the row strategies, we illustrate the ∞× 2 payoffmatrix in the next slide.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 3

Move sn+p+1 Move sp+1 . . . sn

I a ≤ sn(ηn+p+1,2 − ηn+p,2 + 1

)(sn+p+1 − r

) (ηp,1 − ηn,1 + n − p

)(r − sn)(

ηp,1 − ηn,1 + n − p)(r − a)

II sn ≤ a ≤ r(ηn+p+1,2 − ηn+p,2 + 1

)(sn+p+1 − r

)+(

ηp,0 − ηn,0 + n − p)(a − sn)(

ηn+p+1,2 − ηn+p,2 + 1)(

sn+p+1 − a)

III r ≤ a ≤ sn+p+1 +(ηp,0 − ηn,0 + n − p

)(r − sn)(

ηn+p+1,1 − ηn+p,1 + 1)(a − r)

IV a ≥ sn+p+1(ηn+p+1,1 − ηn+p,1 + 1

)(sn+p+1 − r

) (ηp,0 − ηn,0 + n − p

)(r − sn)

By inequalities (2) and (3), the rows a = sn and a = sn+p+1

dominate all other rows.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 3

Move sn+p+1 Move sp+1 . . . sn

a = sn(ηn+p+1,2 − ηn+p,2 + 1

)(sn+p+1 − r

) (ηp,1 − ηn,1 + n − p

)(r − sn)

a = sn+p+1(ηn+p+1,1 − ηn+p,1 + 1

)(sn+p+1 − r

) (ηp,0 − ηn,0 + n − p

)(r − sn)

By inequalities (2) and (3), the rows a = sn and a = sn+p+1

dominate all other rows.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 4

If S holds, then Property 4 holds: For any randomized move byR–LINE, E(∆φ+ cost) ≤ 0.

(∆φ+ cost)11 (∆φ+ cost)12(∆φ+ cost)21 (∆φ+ cost)22

I By S, the upper left and lower right entries are negative.

I The upper right and lower left entries are positive.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 4

If S holds, then Property 4 holds: For any randomized move byR–LINE, E(∆φ+ cost) ≤ 0.

(∆φ+ cost)11 (∆φ+ cost)12(∆φ+ cost)21 (∆φ+ cost)22

I By S, the upper left and lower right entries are negative.

I The upper right and lower left entries are positive.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 4

If S holds, then Property 4 holds: For any randomized move byR–LINE, E(∆φ+ cost) ≤ 0.

(∆φ+ cost)11 (∆φ+ cost)12(∆φ+ cost)21 (∆φ+ cost)22

I By S, the upper left and lower right entries are negative.

I The upper right and lower left entries are positive.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 4

If S holds, then Property 4 holds: For any randomized move byR–LINE, E(∆φ+ cost) ≤ 0.

(∆φ+ cost)11 (∆φ+ cost)12(∆φ+ cost)21 (∆φ+ cost)22

I By S, the upper left and lower right entries are negative.

I The upper right and lower left entries are positive.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 4

If S holds, then Property 4 holds: For any randomized move byR–LINE, E(∆φ+ cost) ≤ 0.

(∆φ+ cost)11 (∆φ+ cost)12(∆φ+ cost)21 (∆φ+ cost)22

I By S, the upper left and lower right entries are negative.

I The upper right and lower left entries are positive.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 4

If S holds, then Property 4 holds: For any randomized move byR–LINE, E(∆φ+ cost) ≤ 0.

I The value of our game is

det(G)

(ηn+p+1,2 + ηn+p+1 − ηn+p,2 − ηn+p+1,1) · (sn+p+1 − r) + (ηp,0 + ηn,1 − ηn,0 − ηp,1) · (r − sn)

I The numerator is non-negative by inequality 4. Thedenominator is negative, which we can prove by combininginequalities of S labeled (2) and (3).

I Thus, E (∆φ+ costR−−LINE ) = v(G ) ≤ 0.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 4

If S holds, then Property 4 holds: For any randomized move byR–LINE, E(∆φ+ cost) ≤ 0.

I The value of our game is

det(G)

(ηn+p+1,2 + ηn+p+1 − ηn+p,2 − ηn+p+1,1) · (sn+p+1 − r) + (ηp,0 + ηn,1 − ηn,0 − ηp,1) · (r − sn)

I The numerator is non-negative by inequality 4. Thedenominator is negative, which we can prove by combininginequalities of S labeled (2) and (3).

I Thus, E (∆φ+ costR−−LINE ) = v(G ) ≤ 0.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 4

If S holds, then Property 4 holds: For any randomized move byR–LINE, E(∆φ+ cost) ≤ 0.

I The value of our game is

det(G)

(ηn+p+1,2 + ηn+p+1 − ηn+p,2 − ηn+p+1,1) · (sn+p+1 − r) + (ηp,0 + ηn,1 − ηn,0 − ηp,1) · (r − sn)

I The numerator is non-negative by inequality 4. Thedenominator is negative, which we can prove by combininginequalities of S labeled (2) and (3).

I Thus, E (∆φ+ costR−−LINE ) = v(G ) ≤ 0.

R–LINE UNLV

Proof of Competitiveness

Proof of Property 4

If S holds, then Property 4 holds: For any randomized move byR–LINE, E(∆φ+ cost) ≤ 0.

I The value of our game is

det(G)

(ηn+p+1,2 + ηn+p+1 − ηn+p,2 − ηn+p+1,1) · (sn+p+1 − r) + (ηp,0 + ηn,1 − ηn,0 − ηp,1) · (r − sn)

I The numerator is non-negative by inequality 4. Thedenominator is negative, which we can prove by combininginequalities of S labeled (2) and (3).

I Thus, E (∆φ+ costR−−LINE ) = v(G ) ≤ 0.

R–LINE UNLV

Proof of Competitiveness

Thus, by properties (1), (2), (3), and (4), if S is satisfied thenR–LINE is C -competitive.

R–LINE UNLV