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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
Institute of Computer ScienceUniversity of Wroclaw
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
Institute of Computer ScienceUniversity of Wroclaw
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:
Institute of Computer ScienceUniversity of Wroclaw
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) …
Institute of Computer ScienceUniversity of Wroclaw
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:
Institute of Computer ScienceUniversity of Wroclaw
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
Institute of Computer ScienceUniversity of Wroclaw
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
Institute of Computer ScienceUniversity of Wroclaw
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
Institute of Computer ScienceUniversity of Wroclaw
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 ,
Institute of Computer ScienceUniversity of Wroclaw
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?
Institute of Computer ScienceUniversity of Wroclaw
Thank you for your attention!