Power of Discrete Nonuniformity -- Optimizing Access to ... · presented a few days before on MSN’08 in Wuhan Cichon, Kutyłowski, Zawada´ Power of Discrete Nonuniformity. My research

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



Contents:

1 My research group

2 Old protocols, new protocolNakano-Olariu protocolCai-Lu-Wang ProtocolNew SolutionDescriptionAnalysis

3 Comparison of protocols

4 Conclusions 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




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



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 ...




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




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







Embedded systems








Embedded systems






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









money claim online










money claim online





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





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]





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↑





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 .





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 ∗)?





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





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.






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.





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




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




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





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





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

).





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.




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




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




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.




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) .




Hypothesis

HypothesisOur solution of the initialization problem is optimal in the classof algorithms which choose one station during one round in asingle-hop environment.





Power of Discrete Nonuniformity -- Optimizing Access to ... · presented a few days before on MSN’08 in Wuhan Cichon, Kutyłowski, Zawada´ Power of Discrete Nonuniformity. My research

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