Institute of Computer ScienceUniversity of Wroclaw

Geometric Aspects of Online Packet BufferingAn Optimal Randomized Algorithm for Two Buffers

Marcin BienkowskiInstitute of Computer Science,

University of Wroclaw, Poland

Aleksander MądryCSAIL, MIT, US

Institute of Computer ScienceUniversity of Wroclaw

Marcin Bieńkowski: Online Packet Buffering 2

Network switch (1)

Discrete time divided into rounds In one round any number of packets arrive We may transmit one of them

network network

switchoutput m input queues (buffers)

Round 1 Round 3Round 2

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Marcin Bieńkowski: Online Packet Buffering 3

Network switch (2)

Each buffer has size B No place in the buffer packets get lost

Goal: maximize throughput, i.e. the number of sent packets

switchoutput m input buffers

Round 4

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Marcin Bieńkowski: Online Packet Buffering 4

Online problem, algorithm does not know the future Adversary: adds packets to buffers = creates input Algorithm: decides from which buffer to transmit

Performance ratio on :

Competitive ratio:

Competitive analysis

throughput of the optimal offline algorithm

throughput of online algorithm

Institute of Computer ScienceUniversity of Wroclaw

Marcin Bieńkowski: Online Packet Buffering 5

Previous results

Competitive ratio :

Deterministic algorithms Any reasonable algorithm: [Azar, Richter ’03] B = 1: [Azar, Richter ’03]

Any B, large m: [Albers, Schmidt ’04]

Semi-Greedy alg., any m: [Albers, Schmidt ’04]

Randomized algorithms Random permutation algorithm, [Schmidt

’05]

any B,m:

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Marcin Bieńkowski: Online Packet Buffering 6

, m = 2, : [Schmidt ’05]

Most related results

For any randomized algorithm, for any B: [Albers, Schmidt ’04]

M 2 3 4 …

h(m) …

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Marcin Bieńkowski: Online Packet Buffering 7

Our results

We consider the two-buffer case (m = 2) and any B

Algorithmdeterministic 16/13-competitive algorithm for fractional model (dividing packets possible)

Algorithm randomized 16/13-competitive algorithm for standard model

Two-dimensional

randomized rounding

Optimal

competitiveness

THIS TALK

Institute of Computer ScienceUniversity of Wroclaw

Marcin Bieńkowski: Online Packet Buffering 8

Input description

Round in which adversary adds packets to buffer 0 and packets to buffer 1:

Round in which adversary does not add packets:

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Marcin Bieńkowski: Online Packet Buffering 9

Bad input for GREEDY

The competitive ratio of GREEDY is in the fractional model

Input sequence incurring competitive ratio 9/7:

Greedy:

[Schmidt ’05]

OPT:

Buffer 0 Buffe

r 1

Greedy policy

go to the anti-diagonal

loss = 1/3 Bloss = 2/3 B

In total: packets added

Institute of Computer ScienceUniversity of Wroclaw

Marcin Bieńkowski: Online Packet Buffering 10

How can we improve GREEDY?

Input sequence:

Buffer 0 Buffe

r 1

GREEDY state

OPT state

set of possible OPT states (computed by the algorithm)

Institute of Computer ScienceUniversity of Wroclaw

Marcin Bieńkowski: Online Packet Buffering 11

How can we improve GREEDY?

Input sequence:

Buffer 0 Buffe

r 1

OPT does not lose packets as long a part of is within the square

Algorithm PB: stay as close to the Perpendicular Bisector of L as possible

GREEDY state

OPT state

set of possible OPT states (computed by the algorithm)

PB state loss = 1/6 B

loss = 1/3 B

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Marcin Bieńkowski: Online Packet Buffering 12

Main Theorem

The competitive ratio of PB is at most i.e. the performance ratio on any sequence is at most

Main idea: find hardest (in terms of competitive ratio), but regular sequences and prove the bound on the performance ratio for them.

all sequences

proper sequences

Proper sequences:

(i) start with full buffers,

(ii) L is always above the main diagonal

i.e. the total number of OPT packets >= B

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Marcin Bieńkowski: Online Packet Buffering 13

Proof outline

1. Proper sequences are hardest ones:

2. On proper sequences

3. For any proper sequence ,On a proper sequence, always looks like this:

perpendicular bisector = main anti-diagonal

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Marcin Bieńkowski: Online Packet Buffering 14

Proper sequences are hardest

Idea: step-wise transformationnon-proper proper preserving A) spatial relations between L and state of PBB) length of L

throughput on is the same as throughput on

Institute of Computer ScienceUniversity of Wroclaw

Marcin Bieńkowski: Online Packet Buffering 15

“Properisation” preserving spatial relations

Step of step of

Case 1: , = const do nothing

Case 2: , decreases in

Case 3: , = const do nothing

Case 4: , increases in

Case 5: , decreases

in

Assume that already starts with

Institute of Computer ScienceUniversity of Wroclaw

Marcin Bieńkowski: Online Packet Buffering 16

Proof outline

1. Proper sequences are hardest ones:

2. On proper sequences

3. For any proper sequence ,

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Marcin Bieńkowski: Online Packet Buffering 17

Nemesis proper sequence for GREEDY?

Buffer 0 Buffe

r 1

This is the worst possible behavior of the adversary (potential-like proof)

k packets added

GREEDY loss

Institute of Computer ScienceUniversity of Wroclaw

Marcin Bieńkowski: Online Packet Buffering 18

Outlook

We show an optimal randomized algorithm for two buffers

and any buffer size B For we can get the same ratio for deterministic

variant of PB Geometry is neat, actual technical details are gory

Open questions:

Is it possible to extend this approach to m > 2? How well the deterministic version performs for small

B?

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Thank you for your attention!