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Gabriel Kliot, Roy Friedman Technion – Israel Institute of Technology

and Chen Avin – Ben Gurion University, Israel

Probabilistic Quorum Probabilistic Quorum

Systems in Wireless Ad Systems in Wireless Ad

Hoc NetworksHoc Networks

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How Can One Find Data?

Centralized directory

Flooding lookup requests or advertisements expensive

DirectoryServer

DataOwner

DataClient

Advertise Lookup

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How Can One Find Data?

Publishing advertisements to a subset P and looking up the data in a subset L such that P and L intersect This is known as quorums

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Quorum System: A set of subsets over a universe U such that

for any Q1,Q2 in Q, Q1∩Q2≠Ф

Bi-quorum System: A couple of sets of subsets (Q1,Q2) over a

universe U such that for any Q1 in Q1 and Q2 in Q2, Q1∩Q2≠Ф

Quorums and Bi-Quorums

Majority

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Probabilistic Quorums [Malkhi, Reiter, Wool, Wright ‘01]

In probabilistic quorums, the intersection property is only ensured with some probability

The members of the probabilistic quorum are selected on each quorum access using an access strategy For example, pick nodes at random

This ensures intersection with probability Suitable for dynamic ad hoc

networks

nl2le

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Our Contributions Different accesses strategies, with varying

trade-offs Mix and Match theorem – we can mix them in

different ways, that guarantee intersection Asymmetric bi-quorum systems

Explore various combinations Theoretically, based on Random Geometric Graph

model By simulations

Along the way, some theoretical results about Random Walks in Random Geometric Graphs

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n nodes 2-dimensional unit torus [0,1]2

Uniform placement Edge between 2 nodes within Euclidian distance r No geographic knowledge We use Random Geometric Graph only for

performance analysis The correctness is ensured on any topology

Formal Network Model: 2D Random Geometric Graph

),(2 rnG

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Access Strategies in MANET

UglyUglyBadBadGoodGoodAccess CostAccess Cost(Random Geometric

Graph)

RoutingMembership / sampling service

Early halting

(if accessed serially)

RANDOMRANDOM

ln( )

nQ

n

Partial Cover Time

High

Crossing TimeCrossing Time

Revising nodes along the path

No routing

Early haltingPATH (RW)PATH (RW) , for / 2Q Q n

MAC broadcast:

•Not EE

•Low bandwidth

No Early halting

Multiple replies

No routingDepends on TTL

(no fine grained control over the cost)

FLOODINGFLOODING Q

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Mix and Match Known result [Malkhi, Reiter, Wool, Wright]:

If two quorums of size are chosen uniformly at random, then their non-intersection probability is

Our result: We show that if one of these quorums is

chosen uniformly at random, then the other quorum can be chosen in any way (including deterministically)

nl

2le

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Mix and Match Specifically, assume Qa and Qb, Qa chosen

uniformly at random and Qb chosen arbitrarily, but in a non-adversarial manner (e.g., using the PATH access strategy)

Lemma 1:

Lemma 2: In order to have intersection with probability 1-ε, the sizes of Qa and Qb must satisfy

For example, for an intersection probability of 0.9, we can pick

n

QbQa

a eQQ b

)Pr(

)/1ln( nQQ ba

2 1.15a bQ n Q n

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Optimized RANDOM strategy

Adding Cross Layer Optimization Similar to RANDOM, except that a lookup

request that passes through any intermediate node does a local lookup as well

Benefit comes from the mix&match result That is, as soon as the first lookups visit

nodes, it is likely that the object will be found Typically, after picking only a few nodes to visit

nl

125 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

50

RW TTL

Num

ber

of d

iffer

ent n

odes

vis

ited

by R

W

800 NODES - Simple RW

800 NODES - UNIQUE RW

Optimized PATH strategy - Unique PATH

By remembering the path, we can avoid revisiting nodes and speed up the walk

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Comparing the Access StrategiesComparing the Access Strategies

Q n

NoNoYesYes**

MultipleMultipleOneOne**One**# Replies

NoNoNoYesYes

Advertise Cost

YesEarly Halting

One

NoLookup Routing

Combined Cost

Lookup Cost

FLOODINGRANDOM

Advertise FLOODINGRANDOM-OPT

RANDOMLookup

nln( )

n

n

ln( )

n

n

ln( )

n

n ln( )

n

n

n n

FLOODING

nNo

Combined Cost

PATH PATH

PATH

Yes

One

No

PATH

***

***ln( )

n

n

ln( )

n

n

Yes

One

ln( )

n

n

** If accessed serially *** this is a lower bound, also validated by simulations

ln( )n n

Yes**

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Simulation setup

Simulations on JIST/SWANS http://jist.ece.cornell.edu/

Network sizes: 50, 100, 200, 400, 800 Random Waypoint mobility model

Speed between 0.5-2 m/s (walking speed) Average pause time 30 s

Transmission range ~ 220m Average number of neighbours davg=10 10 runs per data point, 1000 sec

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Simulations Scenarios 100 advertisements to a RANDOM quorum

of size nodes

1000 lookups 4 strategies:

RANDOM, RANDOM-OPT, UNIQUE-PATH, and FLOODING

On a hit, a reply was sent to the originator

Each hop is counted as one message

Hit ratio means the number of successful lookups for objects that were published

Corresponds to intersection probability

n2

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Results of RANDOM and RANDOM-OPT

Theory works… A hit ratio of 0.9 was obtained with a

quorum size of With 800 nodes, the quorum size is 33

The number of messages per lookup behaved as

RANDOM RANDOM-OPT

But, the overall communication cost was greatly affected by routing overhead, even in RANDOM-OPT

1.15 n

ln( )

nQ

nln( )Q n

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# Lookup Messages for RANDOM-OPT

2 4 6 8 10 12 140

50

100

150

200

250

300

RANDOM-OPT lookup Quorum size

Sen

t loo

kup

mes

sage

s pe

r lo

okup

50 NODES100 NODES200 NODES400 NODES800 NODES

Mobile network

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Total # Messages for RANDOM-OPT

2 4 6 8 10 12 140

500

1000

1500

2000

2500

3000

3500

4000

4500

RANDOM-OPT lookup Quorum size

Tot

al s

ent l

ooku

p m

essa

ges

per

look

up

50 NODES100 NODES200 NODES400 NODES800 NODES

This includes the cost of routing in a mobile network

195 10 15 20 25 30 35 40 45 50 55 60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

UNIQUE-PATH lookup Quorum Size

HIT

RA

TIO

50 NODES100 NODES200 NODES400 NODES800 NODES

Hit Ratio for UNIQUE-PATH

For N=400, |lookup _Q|=23~1.15*sqrt(400)

guarantees intersection of 0.9 – like in theory

Mobile network

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# Lookup Messages for UNIQUE-PATH

5 10 15 20 25 30 35 40 45 50 55 600

5

10

15

20

UNIQUE-PATH lookup Quorum Size

Sen

t loo

kup

mes

sage

s pe

r lo

okup

50 NODES100 NODES200 NODES400 NODES800 NODES

No routing overhead here!

Number of messages is smaller than quorum size!

Due to early halting.

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Hit Ratio for FLOODING

1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FLOODING lookup Quorum TTL

HIT

RA

TIO

50 NODES100 NODES200 NODES400 NODES

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1 1.5 2 2.5 3 3.5 4 4.5 50

10

20

30

40

50

60

FLOODING lookup Quorum TTL

Sen

t loo

kup

mes

sage

s pe

r lo

okup

50 NODES100 NODES200 NODES400 NODES800 NODES

# Lookup Messages for Flooding

No routing overhead here too

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Simulation summary

50 (600 with routing)

35 (140 with routing)

RANDOM_OPTRANDOM_OPT

15*14mobile

15*14staticFLOODINGFLOODINGUNIQUE-PATHUNIQUE-PATH

400 nodes #lookup msgs that guarantee 0.9 intersection Including reply

Flooding is sent by broadcast Hidden overheads

Flooding does not allow fine grained control If we want to increase the intersection probability, we must

increase TTL, which will increase the #msgs significantly

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Conclusions Examined various combinations of access

strategies for probabilistic quorums in MANETs

RANDOM RANDOM-OPT PATH UNIQUE-PATH FLOODING

Showed that it is possible to obtain efficient probabilistic quorums

In particular using asymmetric combinations Using Random walks

More about handling failures and dynamism in the paper

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Future Directions

Shared objects (with linearizable semantics)

Pub/sub

Distributed search

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Q&AQ&A

Thank You !

Contact: Gabriel Kliotgabik@cs.technion.ac.il