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My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Power of Discrete Nonuniformity – OptimizingAccess to Shared Radio Channel in Ad Hoc
Networks
Jacek Cichon Mirosław Kutyłowski Marcin Zawada
Institute of Mathematics and Computer ScienceWrocław University of Technology
Poland
presented a few days before on MSN’08 in Wuhan
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Contents:
1 My research group
2 Old protocols, new protocolNakano-Olariu protocolCai-Lu-Wang ProtocolNew SolutionDescriptionAnalysis
3 Comparison of protocols
4 Conclusions and Hypothesis
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer ScienceWrocław University of Technology
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer ScienceWrocław University of Technology
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer ScienceWrocław University of Technology
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer Sciencesecurity and algorithms
Teaching• engineer and master degrees in computer science,
bachelor and master in mathematics• PhD studies
Some research directions in my group• security• distributed algorithms• fuzzy optimization• bioinformatics
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer Sciencesecurity, privacy and cryptography
Prof. Kutyłowski Prof. Cichon Dr Klonowski Mr. Zagorski
security
• privacy protection,
• e-voting,
• key distribution,
• digital signatures,
• lightweight devices,
• side channel attacksCichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer Sciencealgorithms
Prof. Cichon Prof. Kutyłowski Dr Korzeniowski Dr Zawada
distributed algorithms
• P2P technologies,
• self-organization of ad hoc networks,
• access to radio channel,
• communication in case of failures and malicious adversaries
• sensor networks ...
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer Scienceoptimization, bioinformatics
Dr Zielinski Dr Bogdan
optimizationdiscrete optimization for algorithms with inputs given as (fuzzy)intervals and not exact values
bioinformaticsstatistical data mining in genetic data
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer Sciencesome projects
FRONTSEU project on pervasive systems of tiny artefacts
sensor networksR&D, large sensor network for environment monitoring
Embedded systems
protection against kleptography in high speed networks
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer Sciencesome projects
FRONTSEU project on pervasive systems of tiny artefacts
sensor networksR&D, large sensor network for environment monitoring
Embedded systems
protection against kleptography in high speed networks
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer Sciencesome projects
FRONTSEU project on pervasive systems of tiny artefacts
sensor networksR&D, large sensor network for environment monitoring
Embedded systems
protection against kleptography in high speed networks
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer Sciencesome projects
Internet voting in Poland
fully resistant to malicious machines, focused on political large scaleelections, also in IACR competition
electronic signatures in public administration
new kind of PKI for public sector in Poland
money claim online
electronic court - automatic processing of small claims
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer Sciencesome projects
Internet voting in Poland
fully resistant to malicious machines, focused on political large scaleelections, also in IACR competition
electronic signatures in public administration
new kind of PKI for public sector in Poland
money claim online
electronic court - automatic processing of small claims
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Institute of Mathematics and Computer Sciencesome projects
Internet voting in Poland
fully resistant to malicious machines, focused on political large scaleelections, also in IACR competition
electronic signatures in public administration
new kind of PKI for public sector in Poland
money claim online
electronic court - automatic processing of small claims
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
Description of problem
Problem• consider single–hop, ad hoc network with n-stations• there is one additional node called a coordinator• we have to choose a unique station• a station may transmit and listen using common
radio-channel• a station can recognize the following states of the radio
channel: IDLE, SINGLE, COLLISION
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
Common parameters
Parameters• λ - the maximal transmission delay• δ - the length of the shortest message
Some possible values:
• distance d = 3 [km], light speed c = 3 · 105 [km/sec]:λ ≈ 1
105 [sec]
• transmission speed 1[Mb/sec], length 128 bits: δ ≈ 1104
[sec]
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
Nakano - Olariu leader election protocol
There are n stations. Time is divided into small slots.
1 each station generates ξ = random();2 if ξ < 1
n , then the station transmits a message of length δ;3 if only one station transmits, then the coordinator sends the
message OK,else the coordinator sends the message CONTINUE.
r r r r r r↑
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
Nakano-Olariu protocol: Analysis
Analysis
1 the length of a slot: (δ + λ) + (δ + λ)
2 probability of success in one slot p =(n
1
)1n (1− 1
n )n−1 ∼ 1e
3 Fact: the expected value of a random variable withgeometric distribution with parameter p equals 1
p .
TheoremLet NOn be the time complexity of the Nakano-Olariu leaderelection protocol. Then
E [NOn] ≈ 2 · e(λ + δ) ≈ 5.43656 · δ + 5.43656 · λ .
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
Cai-Lu-Wang Protocol
Fix probability p and a time-slot [0, T ].
Basic idea1 each station generates ξ = random();2 if ξ < p, then
1 a station chooses a random time t ∈ [0, T ]2 if at time t channel is idle, then the station starts a
transmission to the end of the slot [0, T ]
3 if in the interval [0, T ] there was no collision, then thecoordinator sends the message OK,else it sends the message CONTINUE.
What is the optimal pair (p∗, T ∗)?
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
Cai-Lu-Wang: Analysis 1
Collision1 Let X1:n ≤ X2:n ≤ . . . ≤ Xn:n be the moments chosen by n
participating stations.2 There is no collision, if X2:n − X1:n > λ.3 Pr[X2:n − X1:n > λ] = (1− λ
T )n.
r r< λ
r r r rCichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
Cai-Lu-Wang: Analysis 2
Let CLWp,T = (random variable) the time necessary to choosea leader in this algorithm. Then E [CLWp,T ] =
T + 2(δ + λ)
Np(1− p)N−1 + 1λ≤T (∑N
k=2(N
k
)((1− λ/T )p)k (1− p)n−k )
1 The behavior of this algorithm depends on a proper settingof parameters p and T for given λ, δ and n.
2 No analytical formula for the optimal choice of parametersis known.
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
Cai-Lu-Wang: Analysis 3
clw(δ, λ) = min{E [CLWp,δ,λ,T ] : 0 ≤ p ≤ 1 ∧ T ≥ 1} .
1 If δ < 3.1 · λ, then Nakano-Olariu is better thanCai-Lu-Wang.
2 If δ > 3.1 · λ, then Cai-Lu-Wang better than Nakano-Olariu.3 If δ ∈ [λ, 200 · λ], then
clw(δ, λ) ≈ 3.89456 · λ + 2.25012 · δ + 8.28525 · λ · ln δ
λ.
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
New solution: CKZ Protocol
Description
1 Fix k and time-points 0, λ, 2λ, . . . (k − 1)λ.2 Fix probabilities p0, p1, p2, . . . , pk−1, such that
p0 + . . . + pk−1 ≤ 1.3 Each station chooses one of these time-points, the i th
point chosen with probability pi .4 A station starts transmission at a chosen point, if the
channel is idle (from its point of view).
r r r r r rCichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
CKZ: Analysis 1
Problem: given N, λ, δ, k , find optimal probabilitiesp0, p1, p2, . . . , pk−1.
1 Probability of the success is
k∑i=1
(N1
)pi (1− (p1 + . . . + pi))
N−1
2 Good approximation (pi = ai/N)
fk (a1, . . . , ak ) =k∑
i=1
aie−(a1+...+ai ) .
3 So we need to find the maximum of the function fkCichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
CKZ: Analysis 2
Recurrence definition{M0 = 0Mi+1 = e−1+Mi for all i
Theorem1 The following point is an extremal point of fk :
(1−Mk−1, 1−Mk−2, . . . , 1−M1, 1−M0)
2 The maximal value of fk is Mk
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
CKZ: Analysis 3
Recurrence {M0 = 0Mi+1 = e−1+Mi
Properties of sequence
First five values of the sequence (Mi)i≥0:
0, 1/e, e−1+1/e, e−1+e−1+1/e, e−1+e−1+e−1+1/e
which are approximately equal to
0, 0.367879, 0.531464, 0.625918, 0.68792
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
CKZ: Analysis 4
Recurrence {M0 = 0Mi+1 = e−1+Mi
TheoremThe sequence (Mi) is monotonically convergent to 1. Moreover
Mk = 1− 2k
+(2/3) ln k
k2 + o(
ln kk2
).
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Nakano-Olariu protocolCai-Lu-Wang ProtocolDescriptionAnalysis
CKZ: Analysis 5
Let CKZk = our protocol with probabilities
(p1, . . . , pk ) = (1−Mk−1
N, . . . ,
1−M1
N,
1N
) .
Abusing notation: CKZk = (random variable) the timenecessary to choose a leader in this protocol.
TheoremFor each k ≥ 1 we have
E [CKZk ] ≈ 2δ + (k + 2)λ
Mk.
Moreover, for large k we have E [CKZk ] ≈ 2δ + (k + 3)λ.
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
All together
r1nNakano-Olariu
r r r r r r r runiform distributionCai-Lu-Wang
r r r r r r1n
0.632n
0.468n
0.374n
0.312nCKZ
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Comparison with CLW protocol
Last problem: given λ, δ find optimal k . For δ ≥ λ we have
kopt ≈ 2√
δ/λ .
δ/λ CLW CKZopt1 13.5234 · λ 9.4079 · λ10 44.0127 · λ 35.301 · λ50 148.6400 · λ 130.610 · λ100 268.0110 · λ 242.196 · λ
Table: Expected run-times for N = 100
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Summary
Conclusions1 There are combinations of the Nakano-Olariu protocol and
Cai-Lu-Wang protocol which improve the run-time overboth of them.
2 There are precise analytical formulas for the optimal choiceof parameters controlling behavior of our protocols.
3 Our protocol can be easily transformed into initializationalgorithms.
4 The strategies can be adapted to the case of an unknownnumber of stations.
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Summary
TheoremThe task of initializing an n-station with known n terminates,with probability exceeding 1− 1
n , can be accomplished in1
Mkn + O(
√n log n) time slots.
Therefore the task of initializing an n-station with the known nterminates, with probability exceeding 1− 1
n , in
(1 +2k
)n + O(√
n log n)
time slots. Let us recall that for the original Nakano-Olariuprotocol the bound is
e · n + O(√
n log n) .
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Hypothesis
HypothesisOur solution of the initialization problem is optimal in the classof algorithms which choose one station during one round in asingle-hop environment.
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity
My research groupOld protocols, new protocol
Comparison of protocolsConclusions and Hypothesis
Cichon, Kutyłowski, Zawada Power of Discrete Nonuniformity