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A Randomized Online Algorithm for Bandwidth Utilization

Sanjeev Arora and Bo Brinkman Princeton University

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

Motivation: Computer Networks

Goal: Send data as quickly as possible

?

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

Motivation: Computer Networks

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

?

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

Motivation: Computer Networks

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

Bottleneck

“Find” the unused bandwidth

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

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

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”

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

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

Main Model [Karp et al]

At each step i: The available bandwidth on the network is bi

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

Main Model [Karp et al]

At each step i: The available bandwidth on the network is bi The algorithm, not knowing bi, attempts to send xi bytes

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

Main Model [Karp et al]

At each step i: The available bandwidth on the network is bi The algorithm, not knowing bi, attempts to send xi bytes If xi

• bi, the algorithm succeeds. Otherwise, no data is

transmitted

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

Main Model [Karp et al]

At each step i: The available bandwidth on the network is bi The algorithm, not knowing bi, attempts to send xi bytes If xi

• bi, the algorithm succeeds. Otherwise, no data is

transmitted The sequence

bi

is chosen by a non-adaptive adversary

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

Main Model [Karp et al]

At each step i: The available bandwidth on the network is bi The algorithm, not knowing bi, attempts to send xi bytes If xi

• bi, the algorithm succeeds. Otherwise, no data is

transmitted The sequence

bi

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

def

∑bi, is offline optimal (opt)

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

Restricted Adversaries

Karp et al consider three restricted adversaries:

a

b

• fixed-range: Adversary must pick bi from a fixed

range

a

b

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

Restricted Adversaries

Karp et al consider three restricted adversaries:

a

b

• fixed-range: Adversary must pick bi from a fixed

range

a

b

α

β

• additive: For some constants α

β

0, adversary

must pick bi

bi

1

α

bi

1

α

, with β as a lower bound.

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

Restricted Adversaries

Karp et al consider three restricted adversaries:

a

b

• fixed-range: Adversary must pick bi from a fixed

range

a

b

α

β

• additive: For some constants α

β

0, adversary

must pick bi

bi

1

α

bi

1

α

, with β as a lower bound.

µ-multiplicative: For some constant µ

1, adversary must

pick bi

bi

1

µ

µbi

1

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

Results

Theorem (Upper Bound) A randomized algorithm exists which achieves competitive ratio

8 3 lgµ

16 against a µ-multiplicative adversary.

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

Results

Theorem (Upper Bound) A randomized algorithm exists which achieves competitive ratio

8 3 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.

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

Results

Theorem (Upper Bound) A randomized algorithm exists which achieves competitive ratio

8 3 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 b

a

1 against a

a

b

• fixed-range adversary.

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

Lower Bound

Proof [Karp et al] Randomized adversary picks bi

a

b

.

P

y

a y2

if y

b

a b

if y

b E

bi

b a

ay y2 dy

ba b

a

ln b a

1

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

Lower Bound contd.

Algorithm tries to send xi, expected throughput is:

xi

b xi

a y2dy

xi a b

xi

a xi

a b

xi a b

a

Competitive ratio is ln b

a

1 independent of xi.

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

Lower Bound contd.

Algorithm tries to send xi, expected throughput is:

xi

b xi

a y2dy

xi a b

xi

a xi

a b

xi a b

a

Competitive ratio is ln b

a

1 independent of xi.

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

b0

µb0

.

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

Our (Barely) Randomized Algorithm

This version has competitive ratio 4lgµ

8.

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

Our (Barely) Randomized Algorithm

This version has competitive ratio 4lgµ

8.

Set x0 (amount of bandwidth requested during the first step) to a uniformly random power of 2 in

1

2µ

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

Our (Barely) Randomized Algorithm

This version has competitive ratio 4lgµ

8.

Set x0 (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 2µ

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

Our (Barely) Randomized Algorithm

This version has competitive ratio 4lgµ

8.

Set x0 (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 2µ When it fails, divide by 2

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

Bandwidth Constant

µ width (bytes) x time (steps) 2x band−

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

Bandwidth Constant

µ width (bytes) x time (steps) 2xµ lg band−

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

Brief Spaces

time (steps) µ x band− width (bytes) lgµ x

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

Brief Spaces

time (steps) µ x band− width (bytes) lgµ x

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

Brief Spaces

time (steps) µ x band− width (bytes) lgµ x

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

Brief Spaces

time (steps) µ x band− width (bytes) lgµ x

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

Analysis

Consider the lgµ

2 algorithms as if they were run in parallel

2µ requests) 1 2 4 8 16 step i step i+1 (bandwidth

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

Analysis

Consider the lgµ

2 algorithms as if they were run in parallel

2µ requests) 1 2 4 8 16 step i step i+1 (bandwidth

Algorithms stay in lock-step

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

Analysis

Consider the lgµ

2 algorithms as if they were run in parallel

4µ requests) 2 4 8 16 1 2 4 8 16 2µ 2µ Bandwidth Available step i step i+1 (bandwidth

Algorithms stay in lock-step

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

Analysis

Consider the lgµ

2 algorithms as if they were run in parallel

4µ requests) 2 4 8 16 1 2 4 8 16 2µ 2µ Bandwidth Available step i step i+1 (bandwidth

Algorithms stay in lock-step Bandwidth achieved is at least 1

2 available amount

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

Analysis contd.

Bandwidth increasing or constant: Ensemble succeeds again

µ 1 2µ step i+1 (bandwidth requests) 2 4 8 16 1 4 8 16 x=2 2µ 4µ 2µ step i

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

Analysis contd.

Bandwidth increasing or constant: Ensemble succeeds again

µ 1 2µ step i+1 (bandwidth requests) 2 4 8 16 1 4 8 16 x=2 2µ 4µ 2µ step i

Bandwidth decreases to less than x and ensemble fails

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

Analysis contd.

Bandwidth increasing or constant: Ensemble succeeds again

µ 1 2µ step i+1 (bandwidth requests) 2 4 8 16 1 4 8 16 x=2 2µ 4µ 2µ step i

Bandwidth decreases to less than x and ensemble fails In the next step if ensemble misses again, bandwidth

x 2

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

Analysis contd.

Bandwidth increasing or constant: Ensemble succeeds again

µ 1 2µ step i+1 (bandwidth requests) 2 4 8 16 1 4 8 16 x=2 2µ 4µ 2µ step i

Bandwidth decreases to less than x and ensemble fails In the next step if ensemble misses again, bandwidth

x 2

Total missed bandwidth is bounded by x

x 2

x 4

2x

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

Analysis contd.

Putting it together: In step i ensemble sends x

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

Analysis contd.

Putting it together: In step i ensemble sends x Total available in step i

2x

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

Analysis contd.

Putting it together: In step i ensemble sends x Total available in step i

2x

Total missed until next success

2x

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

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

• pt

4

lgµ

2

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

Final Details

Remove assumption that µ is a power of 2: lgµ

3

algorithms, competitive ratio 4lgµ

12

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

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

8 3 lgµ

16

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

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

8 3 lgµ

16

Algorithm also works for additive adversaries: Lower bound

• n competitive ratio is ln

α

β β

1, algorithm achieves

8 3 lg

α

β β

• 16. (Use algorithm with µ

α

β β )

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

MIMD

Interesting fact: Our algorithm is Multiplicative Increase, Multiplicative Decrease

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

MIMD

Interesting fact: Our algorithm is Multiplicative Increase, Multiplicative Decrease [CJ ’89] observe that Multiplicative Increase algorithms are efficient (dynamical systems argument)

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

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

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

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

Interesting Questions

How “predictable” is real network traffic?

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

Interesting Questions

How “predictable” is real network traffic? Can we incorporate feedback effects into this model?

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

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?

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

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