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(1)

A Randomized Online Algorithm for Bandwidth Utilization

Sanjeev Arora and Bo Brinkman Princeton University

A Randomized Online Algorithm for Bandwidth Utilization – p.1/17

(2)

Motivation: Computer Networks

Goal: Send data as quickly as possible without any central

authority distributing bandwidth

?

“Find” the unused bandwidth

Use dropped packets as implicit information about bandwidth availability

(3)

Motivation: Computer Networks

Goal: Send data as quickly as possible without any central authority distributing bandwidth

?

“Find” the unused bandwidth

Use dropped packets as implicit information about bandwidth availability

A Randomized Online Algorithm for Bandwidth Utilization – p.2/17

(4)

Motivation: Computer Networks

Goal: Send data as quickly as possible without any central authority distributing bandwidth

Bottleneck

“Find” the unused bandwidth Use dropped packets as implicit information about

bandwidth availability

(5)

Motivation: Computer Networks

Goal: Send data as quickly as possible without any central authority distributing bandwidth

Bottleneck

“Find” the unused bandwidth

Use dropped packets as implicit information about bandwidth availability

A Randomized Online Algorithm for Bandwidth Utilization – p.2/17

(6)

Previous Work

AIMD (Additive Increase, Multiplicative Decrease) promotes the social good

[Jacobson ’88] “Congestion avoidance and control”

[CJ ’89] “Analysis of increase and decrease algorithms for congestion avoidance in computer networks”

Developing theoretical models for congestion control problems [KKPS ’00] “Algorithmic problems in congestion control”

(7)

Previous Work

AIMD (Additive Increase, Multiplicative Decrease) promotes the social good

[Jacobson ’88] “Congestion avoidance and control”

[CJ ’89] “Analysis of increase and decrease algorithms for congestion avoidance in computer networks”

Developing theoretical models for congestion control problems [KKPS ’00] “Algorithmic problems in congestion control”

A Randomized Online Algorithm for Bandwidth Utilization – p.3/17

(8)

Main Model [Karp et al]

At each step

i

:

The available bandwidth on the network is

b

i

The algorithm, not knowing

b

i, attempts to send

x

i bytes If

x

i

b

i, the algorithm succeeds. Otherwise, no data is transmitted

The sequence

b

i is chosen by a non-adaptive adversary Total available bandwidth,

B

de f

∑b

i, is offline optimal (

opt

)

(9)

Main Model [Karp et al]

At each step

i

:

The available bandwidth on the network is

b

i

The algorithm, not knowing

b

i, attempts to send

x

i bytes If

x

i

b

i, the algorithm succeeds. Otherwise, no data is

transmitted

The sequence

b

i is chosen by a non-adaptive adversary Total available bandwidth,

B

de f

∑b

i, is offline optimal (

opt

)

A Randomized Online Algorithm for Bandwidth Utilization – p.4/17

(10)

Main Model [Karp et al]

At each step

i

:

The available bandwidth on the network is

b

i

The algorithm, not knowing

b

i, attempts to send

x

i bytes If

x

i

b

i, the algorithm succeeds. Otherwise, no data is transmitted

The sequence

b

i is chosen by a non-adaptive adversary Total available bandwidth,

B

de f

∑b

i, is offline optimal (

opt

)

(11)

Main Model [Karp et al]

At each step

i

:

The available bandwidth on the network is

b

i

The algorithm, not knowing

b

i, attempts to send

x

i bytes If

x

i

b

i, the algorithm succeeds. Otherwise, no data is transmitted

The sequence 

b

i  is chosen by a non-adaptive adversary Total available bandwidth,

B

de f

∑b

i, is offline optimal (

opt

)

A Randomized Online Algorithm for Bandwidth Utilization – p.4/17

(12)

Main Model [Karp et al]

At each step

i

:

The available bandwidth on the network is

b

i

The algorithm, not knowing

b

i, attempts to send

x

i bytes If

x

i

b

i, the algorithm succeeds. Otherwise, no data is transmitted

The sequence 

b

i  is chosen by a non-adaptive adversary Total available bandwidth,

B

de f

∑b

i, is offline optimal (

opt

)

(13)

Restricted Adversaries

Karp et al consider three restricted adversaries:



a



b

-fixed-range: Adversary must pick

b

i from a fixed range 

a



b



α β

-additive: For some constants

α β 0

, adversary

must pick

b

i

b

i 1

α b

i 1

α

, with

β

as a lower bound.

µ

-multiplicative: For some constant

µ 1

, adversary must pick

b

i bi 1µ

µb

i 1

A Randomized Online Algorithm for Bandwidth Utilization – p.5/17

(14)

Restricted Adversaries

Karp et al consider three restricted adversaries:



a



b

-fixed-range: Adversary must pick

b

i from a fixed range 

a



b





α



β

-additive: For some constants

α



β

0

, adversary

must pick

b

i 

b

i 1

α



b

i 1

α

, with

β

as a lower bound.

µ

-multiplicative: For some constant

µ 1

, adversary must pick

b

i bi 1µ

µb

i 1

(15)

Restricted Adversaries

Karp et al consider three restricted adversaries:



a



b

-fixed-range: Adversary must pick

b

i from a fixed range 

a



b





α



β

-additive: For some constants

α



β

0

, adversary

must pick

b

i 

b

i 1

α



b

i 1

α

, with

β

as a lower bound.

µ

-multiplicative: For some constant

µ

1

, adversary must pick

b

i biµ1 

µb

i 1 

A Randomized Online Algorithm for Bandwidth Utilization – p.5/17

(16)

Results

Theorem (Upper Bound)

A randomized algorithm exists which achieves competitive ratio

83

lgµ

16

against a

µ

-multiplicative adversary.

Theorem (Lower Bound)

No randomized algorithm can achieve competitive ratio better than

lnµ 1

against a

µ

-multiplicative adversary.

Theorem [Karp et al]

No randomized algorithm can achieve competitive ratio better than

ln

ba

1

against a

a b

-fixed-range adversary.

(17)

Results

Theorem (Upper Bound)

A randomized algorithm exists which achieves competitive ratio

83

lgµ

16

against a

µ

-multiplicative adversary.

Theorem (Lower Bound)

No randomized algorithm can achieve competitive ratio better than

lnµ

1

against a

µ

-multiplicative adversary.

Theorem [Karp et al]

No randomized algorithm can achieve competitive ratio better than

ln

ba

1

against a

a b

-fixed-range adversary.

A Randomized Online Algorithm for Bandwidth Utilization – p.6/17

(18)

Results

Theorem (Upper Bound)

A randomized algorithm exists which achieves competitive ratio

83

lgµ

16

against a

µ

-multiplicative adversary.

Theorem (Lower Bound)

No randomized algorithm can achieve competitive ratio better than

lnµ

1

against a

µ

-multiplicative adversary.

Theorem [Karp et al]

No randomized algorithm can achieve competitive ratio better than

ln

ba

1

against a 

a



b

-fixed-range adversary.

(19)

Lower Bound

Proof [Karp et al]

Randomized adversary picks

b

i 

a



b

.

P



y

  ya2

if y



b

ab

if y



b

E



b

i   b

a

ay

y

2

dy

b a b



a



ln b

a

1



A Randomized Online Algorithm for Bandwidth Utilization – p.7/17

(20)

Lower Bound contd.

Algorithm tries to send

x

i, expected throughput is:

x

i b

xi

a

y

2

dy

x

i

a b



x

i 

a

x

i

a b



x

i

a b



a



Competitive ratio is

ln

ba

1

independent of

x

i. Lower bound is a simple corollary: The adversary could restrict

itself to the fixed range

b

0

µb

0 .

(21)

Lower Bound contd.

Algorithm tries to send

x

i, expected throughput is:

x

i b

xi

a

y

2

dy

x

i

a b



x

i 

a

x

i

a b



x

i

a b



a



Competitive ratio is

ln

ba

1

independent of

x

i.

Lower bound is a simple corollary: The adversary could restrict itself to the fixed range 

b

0 

µb

0 .

A Randomized Online Algorithm for Bandwidth Utilization – p.8/17

(22)

Our (Barely) Randomized Algorithm

This version has competitive ratio

4lgµ

8

. Set

x

0 (amount of bandwidth requested during the first step)

to a uniformly random power of

2

in

1 2µ

When algorithm succeeds, multiply amount requested by

When it fails, divide by

2

(23)

Our (Barely) Randomized Algorithm

This version has competitive ratio

4lgµ

8

.

Set

x

0 (amount of bandwidth requested during the first step) to a uniformly random power of

2

in 

1





When algorithm succeeds, multiply amount requested by

When it fails, divide by

2

A Randomized Online Algorithm for Bandwidth Utilization – p.9/17

(24)

Our (Barely) Randomized Algorithm

This version has competitive ratio

4lgµ

8

.

Set

x

0 (amount of bandwidth requested during the first step) to a uniformly random power of

2

in 

1





When algorithm succeeds, multiply amount requested by

When it fails, divide by

2

(25)

Our (Barely) Randomized Algorithm

This version has competitive ratio

4lgµ

8

.

Set

x

0 (amount of bandwidth requested during the first step) to a uniformly random power of

2

in 

1





When algorithm succeeds, multiply amount requested by

When it fails, divide by

2

A Randomized Online Algorithm for Bandwidth Utilization – p.9/17

(26)

Bandwidth Constant

µ

width (bytes)

x 2x

band−

(27)

Bandwidth Constant

µ width

(bytes)

x

time (steps) 2xµ

lg band−

A Randomized Online Algorithm for Bandwidth Utilization – p.10/17

(28)

Brief Spaces

µ x band−

width (bytes)

x

(29)

Brief Spaces

time (steps) µ

x band−

width (bytes)

lgµ x

A Randomized Online Algorithm for Bandwidth Utilization – p.11/17

(30)

Brief Spaces

µ x band−

width (bytes)

x

(31)

Brief Spaces

time (steps) µ

x band−

width (bytes)

lgµ x

A Randomized Online Algorithm for Bandwidth Utilization – p.11/17

(32)

Analysis

Consider the

lgµ

2

algorithms as if they were run in parallel

requests)

1 2 4 8 16

step i step i+1 (bandwidth

requests)

2 4 8 16

1 2 4 8 16

Bandwidth Available

step i step i+1 (bandwidth

Algorithms stay in lock-step Bandwidth achieved

is at least 12 available amount

(33)

Analysis

Consider the

lgµ

2

algorithms as if they were run in parallel

requests)

1 2 4 8 16

step i step i+1 (bandwidth

requests)

2 4 8 16

1 2 4 8 16

Bandwidth Available

step i step i+1 (bandwidth

Algorithms stay in lock-step Bandwidth achieved

is at least 12 available amount

A Randomized Online Algorithm for Bandwidth Utilization – p.12/17

(34)

Analysis

Consider the

lgµ

2

algorithms as if they were run in parallel

requests)

2 4 8 16

1 2 4 8 16

Bandwidth Available

step i step i+1 (bandwidth

Algorithms stay in lock-step Bandwidth achieved

is at least 12 available amount

(35)

Analysis

Consider the

lgµ

2

algorithms as if they were run in parallel

requests)

2 4 8 16

1 2 4 8 16

Bandwidth Available

step i step i+1 (bandwidth

Algorithms stay in lock-step Bandwidth achieved

is at least 12 available amount

A Randomized Online Algorithm for Bandwidth Utilization – p.12/17

(36)

Analysis contd.

Bandwidth increasing or constant: Ensemble succeeds again

µ 1

step i+1 (bandwidth

requests)

2 4 8 16

1 4 8 16

x=2

step i

Bandwidth decreases

to less than

x

and ensemble fails In the next step if ensemble

misses again, bandwidth x2 Total missed bandwidth

is bounded by

x

2x 4x

2x

(37)

Analysis contd.

Bandwidth increasing or constant: Ensemble succeeds again

µ 1

step i+1 (bandwidth

requests)

2 4 8 16

1 4 8 16

x=2

step i

Bandwidth decreases

to less than

x

and ensemble fails In the next step if ensemble

misses again, bandwidth x2 Total missed bandwidth

is bounded by

x

2x 4x

2x

A Randomized Online Algorithm for Bandwidth Utilization – p.13/17

(38)

Analysis contd.

Bandwidth increasing or constant: Ensemble succeeds again

µ 1

step i+1 (bandwidth

requests)

2 4 8 16

1 4 8 16

x=2

step i

Bandwidth decreases

to less than

x

and ensemble fails In the next step if ensemble

misses again, bandwidth  x2 Total missed bandwidth

is bounded by

x

2x 4x

2x

(39)

Analysis contd.

Bandwidth increasing or constant: Ensemble succeeds again

µ 1

step i+1 (bandwidth

requests)

2 4 8 16

1 4 8 16

x=2

step i

Bandwidth decreases

to less than

x

and ensemble fails In the next step if ensemble

misses again, bandwidth  x2 Total missed bandwidth

is bounded by

x

x2 4x    

2x

A Randomized Online Algorithm for Bandwidth Utilization – p.13/17

(40)

Analysis contd.

Putting it together:

In step

i

ensemble sends

x

Total available in step

i

Total missed until next success

Finally, pick one of the algorithms from ensemble at random:

E

bandwidth found by random algorithm

opt

4 lgµ 2

(41)

Analysis contd.

Putting it together:

In step

i

ensemble sends

x

Total available in step

i



2x

Total missed until next success

Finally, pick one of the algorithms from ensemble at random:

E

bandwidth found by random algorithm

opt 4 lgµ 2

A Randomized Online Algorithm for Bandwidth Utilization – p.14/17

(42)

Analysis contd.

Putting it together:

In step

i

ensemble sends

x

Total available in step

i



2x

Total missed until next success 

2x

Finally, pick one of the algorithms from ensemble at random:

E

bandwidth found by random algorithm

opt

4 lgµ 2

(43)

Analysis contd.

Putting it together:

In step

i

ensemble sends

x

Total available in step

i



2x

Total missed until next success 

2x

Finally, pick one of the algorithms from ensemble at random:

E

bandwidth found by random algorithm 

opt

4



lgµ

2



A Randomized Online Algorithm for Bandwidth Utilization – p.14/17

(44)

Final Details

Remove assumption that

µ

is a power of

2

:

lgµ

3

algorithms, competitive ratio

4lgµ

12

Instead of powers of

2

, use powers of

c

: Competitive ratio

83

lgµ 16

Algorithm also works for additive adversaries: Lower bound on competitive ratio is

ln

α ββ

1

, algorithm achieves

83

lg

α ββ

16

. (Use algorithm with

µ

α ββ )

(45)

Final Details

Remove assumption that

µ

is a power of

2

:

lgµ

3

algorithms, competitive ratio

4lgµ

12

Instead of powers of

2

, use powers of

c

: Competitive ratio

83

lgµ

16

Algorithm also works for additive adversaries: Lower bound on competitive ratio is

ln

α ββ

1

, algorithm achieves

83

lg

α ββ

16

. (Use algorithm with

µ

α ββ )

A Randomized Online Algorithm for Bandwidth Utilization – p.15/17

(46)

Final Details

Remove assumption that

µ

is a power of

2

:

lgµ

3

algorithms, competitive ratio

4lgµ

12

Instead of powers of

2

, use powers of

c

: Competitive ratio

83

lgµ

16

Algorithm also works for additive adversaries: Lower bound on competitive ratio is

ln

 αββ 

1

, algorithm achieves

83

lg

 αββ 

16

. (Use algorithm with

µ

  αββ)

(47)

MIMD

Interesting fact: Our algorithm is Multiplicative Increase, Multiplicative Decrease

[CJ ’89] observe that Multiplicative Increase algorithms are efficient (dynamical systems argument)

Also show that MIMD algorithms do not converge to fairness Is this a case where the optimal behavior for individuals is anti-social?

A Randomized Online Algorithm for Bandwidth Utilization – p.16/17

(48)

MIMD

Interesting fact: Our algorithm is Multiplicative Increase, Multiplicative Decrease

[CJ ’89] observe that Multiplicative Increase algorithms are efficient (dynamical systems argument)

Also show that MIMD algorithms do not converge to fairness Is this a case where the optimal behavior for individuals is anti-social?

(49)

MIMD

Interesting fact: Our algorithm is Multiplicative Increase, Multiplicative Decrease

[CJ ’89] observe that Multiplicative Increase algorithms are efficient (dynamical systems argument)

Also show that MIMD algorithms do not converge to fairness Is this a case where the optimal behavior for individuals is

anti-social?

A Randomized Online Algorithm for Bandwidth Utilization – p.16/17

(50)

MIMD

Interesting fact: Our algorithm is Multiplicative Increase, Multiplicative Decrease

[CJ ’89] observe that Multiplicative Increase algorithms are efficient (dynamical systems argument)

Also show that MIMD algorithms do not converge to fairness Is this a case where the optimal behavior for individuals is anti-social?

(51)

Interesting Questions

How “predictable” is real network traffic?

Can we incorporate feedback effects into this model?

Is end-to-end congestion control sufficient for fair and efficient network usage?

If not, what should we do in routers? (Floyd and Fall, Shenker)

A Randomized Online Algorithm for Bandwidth Utilization – p.17/17

(52)

Interesting Questions

How “predictable” is real network traffic?

Can we incorporate feedback effects into this model?

Is end-to-end congestion control sufficient for fair and efficient network usage?

If not, what should we do in routers? (Floyd and Fall, Shenker)

(53)

Interesting Questions

How “predictable” is real network traffic?

Can we incorporate feedback effects into this model?

Is end-to-end congestion control sufficient for fair and efficient network usage?

If not, what should we do in routers? (Floyd and Fall, Shenker)

A Randomized Online Algorithm for Bandwidth Utilization – p.17/17

(54)

Interesting Questions

How “predictable” is real network traffic?

Can we incorporate feedback effects into this model?

Is end-to-end congestion control sufficient for fair and efficient network usage?

If not, what should we do in routers? (Floyd and Fall, Shenker)

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