distributed online data aggregation in dynamic graphs · impossibility results - online adaptive...

Distributed Online Data Aggregation in Dynamic Graphs

[email protected]

Quentin Bramas, Toshimitsu Masuzawa, and Sébastien Tixeuil

NETYS 2019, Marrakech, June, 21st

Data Aggregation Problem 2

3

Data Aggregation Problem

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5


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Each node transmits at most once.

The goal: to minimize the duration

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Dynamic Data Aggregation Problem

18


We consider a dynamic network with pairwise interactions

…

s

32

15

4

t = 0 1 2 3

18


We consider a sequence of interactions


…

s

32

15

4

t = 0 1 2 3

18



A node can transmit only once


…

s

32

15

4

t = 0 1 2 3

18




Nodes may or may not have other information


…

s

32

15

4

t = 0 1 2 3

18




Nodes may or may not have other information

The goal is to aggregate all the data with

minimum duration


…

s

32

15

4

t = 0 1 2 3

Distributed Online Data Aggregation 19


…

s

32

15

4

t = 0 1 2 3



…

s

32

15

4

A Distributed Online Data Aggregation (DODA) Algorithm answers the question: Which node transmits?

t = 0 1 2 3



…

s

32

15

4

a

b


t = 0 1 2 3



…

s

32

15

4

a

b

a transmits or b transmits or no one transmits

(a node can transmit only once)


t = 0 1 2 3

has data 1 2 3 4

has transmitted

Distributed Online Data Aggregation

1

2

20

t = 0 1 2 3 4 5 6

has data 2(1) 3 4

has transmitted 1


1

2

20

t = 0 1 2 3 4 5 6

has data 2(1) 3 4

has transmitted 1


1

2

3

4

20

t = 0 1 2 3 4 5 6

has data 2(1) 3 4

has transmitted 1


1

2

1

3

3

4

20

t = 0 1 2 3 4 5 6

has data 2(1) 3 4

has transmitted 1


1

2

1

3

3

4

s

1

20

t = 0 1 2 3 4 5 6

has data 2(1) 3 4

has transmitted 1


1

2

1

3

2

3

3

4

s

1

20

t = 0 1 2 3 4 5 6

has data 2(1,3) 4

has transmitted 1 3


1

2

1

3

2

3

3

4

s

1

20

t = 0 1 2 3 4 5 6

has data 2(1,3) 4

has transmitted 1 3


1

2

1

3

2

3

2

4

3

4

s

1

20

t = 0 1 2 3 4 5 6

has data 2(1,3,4)

has transmitted 1 3 4


1

2

1

3

2

3

2

4

3

4

s

1

20

t = 0 1 2 3 4 5 6

has data 2(1,3,4)

has transmitted 1 3 4


1

2

1

3

2

3

2

4

3

4

s

1

s

2

20

t = 0 1 2 3 4 5 6

has data

has transmitted 1 2 3 4


1

2

1

3

2

3

2

4

3

4

s

1

s

2

20

t = 0 1 2 3 4 5 6

How to evaluate the performance of a DODA? 21

Is the duration of the aggregation is significant ?


Is the duration of the aggregation is significant ? No


Is the duration of the aggregation is significant ?

Even the offline optimal algorithm will struggle in the sequence:

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

…

No


Is the duration of the aggregation significant ? No

Is the ratio between the duration of the aggregation and the duration of the offline optimal algorithm significant ?




No




No

Each algorithm is either optimal or does not terminates in the sequence:

1

2

1

2

1

2

1

2

1

2

1

2

1

2

…

1

2

s

1

optimal duration = a convergecast

}

23

How to evaluate the performance of a DODA?

Definition of costI(A), for an algorithm A in a sequence I :

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}

convergecast

23



}

convergecast

}

convergecast (starting after the previous one)

23



}

convergecast

}


}convergecast

}convergecast

}

convergecast

23



}

convergecast

}


}convergecast

}convergecast

}

convergecast

execution of A

23



}

convergecast

}


}convergecast

}convergecast

}

convergecast

execution of A

costI(A) = 4

24


Definition of the costI(A) function, for an algorithm A in a sequence I :

} } }convergecast

A does not terminate

costI(A) = 4

no more convergecast convergecast convergecast

25


Definition of the costI(A) function, for an algorithm A in a sequence I :

} } }convergecast

A does not terminate

costI(A) =∞

infinitely many convergecasts

convergecast convergecast }

convergecast

26


…

s

32

15

4

t = 0 1 2 3

26


…

s

32

15

4

Who generates the sequence?

t = 0 1 2 3

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An adversary

…

s

32

15

4


t = 0 1 2 3

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An adversary

Three adversaries:

…

s

32

15

4


t = 0 1 2 3

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An adversary

Three adversaries:

…

s

32

15

4


t = 0 1 2 3

- Online Adaptive: generates the next interaction based on what happened in the past

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An adversary

Three adversaries:

…

s

32

15

4


t = 0 1 2 3


- Oblivious: generates the sequence before the execution of the algorithm

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An adversary

Three adversaries:

…

s

32

15

4


t = 0 1 2 3


- Oblivious: generates the sequence before the execution of the algorithm

- Randomized: each interaction is chosen uniformly at random.

Impossibility Results - Online Adaptive Adversary 27

- Online Adaptive: generates the next interaction based on what the algorithm decides in the current interaction

Let A be a DODA



s

1

Let A be a DODA



s

1

1 transmitsLet A be a DODA



s

1

1 transmits 1

2

s

1

Let A be a DODA



s

1

1 transmits 1

2

s

1s

2

Let A be a DODA



s

1

1 transmits 1

2

s

1s

2

2 transmits

Let A be a DODA



s

1

1 transmits 1

2

s

1s

2

2 transmits2

1

s

2

Let A be a DODA



s

1

1 transmits 1

2

s

1s

2

2 transmits2

1

s

2otherwise

Let A be a DODA



s

1

1 transmits 1

2

s

1s

2

2 transmits2

1

s

2otherwise

Let A be a DODA

A never terminates



s

1

1 transmits 1

2

s

1s

2

2 transmits2

1

s

2otherwise

Starting from any time t, the aggregation is always possible, so:

Let A be a DODA

A never terminates



s

1

1 transmits 1

2

s

1s

2

2 transmits2

1

s

2otherwise


cost(A) = ∞

Let A be a DODA

A never terminates


Theorem: If k transmissions are allowed per node, there exists an online adaptive adversary that generates for every DODA A a sequence I such that costI(A)=∞



Corollary: If k transmissions are allowed per node, there exists an oblivious adversary that generates for every deterministic DODA A a sequence I such that costI(A)=∞



Corollary: If k transmissions are allowed per node, there exists an oblivious adversary that generates for every deterministic DODA A a sequence I such that costI(A)=∞

What about randomized DODA algorithms against an oblivious adversary?

Impossibility Results - Oblivious Adversary 29

Theorem: If k transmissions are allowed per node, there exists an oblivious adversary that generates for every oblivious DODA A a sequence I such that costI(A)=∞ with high probability



Theorem: If k transmissions are allowed per node, there exists an oblivious adversary that generates for every oblivious DODA A a sequence I such that costI(A)=∞ with high probability

proof: not trivial



sketch of proof, for k=1


s

1

s

2

s

n

…



s

1

s

2

s

n

…

s

1

s

2

s

n

…



s

1

s

2

s

n

…

s

1

s

2

s

n

…


Pt = probability that no node transmits before time t


s

1

s

2

s

n

…

s

1

s

2

s

n

…


Pt = probability that no node transmits before time tlemma: if Pt > 1/n then there exists a node u that has not transmitted w.h.p.


s

1

s

2

s

n

…

s

1

s

2

s

n

…



if Pt > 1/n for all t, then it’s over


s

1

s

2

s

n

…

s

1

s

2

s

n

…



if Pt > 1/n for all t, then it’s over , otherwise there is a time t0 such that:


s

1

s

2

s

n

…

s

1

s

2

s

n

…



if Pt > 1/n for all t, then it’s over , otherwise there is a time t0 such that:• Pt0 -1 > 1/n • Pt0 ≤1/n


s

1

s

2

s

n

…

s

1

s

2

s

n

…




t0

s

1

s

2

s

n

s

1

s

2

s

n

… …

, otherwise there is a time t0 such that:• Pt0 -1 > 1/n • Pt0 ≤1/n


s

1

s

2

s

n

…

s

1

s

2

s

n

…




t0

s

1

s

2

s

n

s

1

s

2

s

n

… …

, otherwise there is a time t0 such that: : one node u has not transmitted w.h.p.• Pt0 -1 > 1/n

• Pt0 ≤1/n


s

1

s

2

s

n

…

s

1

s

2

s

n

…




t0

s

1

s

2

s

n

s

1

s

2

s

n

… …


• Pt0 ≤1/n : one node has transmitted w.h.p.


s

1

s

2

s

n

…

s

1

s

2

s

n

…




t0

s

1

s

2

s

n

s

1

s

2

s

n

… …

n

u

n-1

n

s

1

…




s

1

s

2

s

n

…

s

1

s

2

s

n

…




t0

s

1

s

2

s

n

s

1

s

2

s

n

… …

n

u

n-1

n

s

1

…

n

u

n-1

n

s

1

…




sketch of proof, for k>1

require nk nodes

Ik-1

Ik-1

Ik-1

…

Each group of nodes acts like a node that can transmit only once


What about randomized DODA algorithms against oblivious adversary?

Theorem: If k transmissions are allowed per node, there exists an oblivious adversary that generate for every oblivious DODA A a sequence I such that costI(A)=∞ with high probability




What about non-oblivious randomized DODA algorithms against oblivious adversary?




What about non-oblivious randomized DODA algorithms against oblivious adversary?

open question

DODA against randomized adversary

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If you have no information about the future, always transmit is optimal. It takes O(n2) interactions in average.

If you know everything about the future, it takes O(nlog(n)) interactions in average.

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35


The meetTime information: each node knows when will be its next interaction with the sink

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The meetTime information: each node knows when will be its next interaction with the sink

1

meetTime = 3

2

meetTime = 6

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Greedy Algorithm: if I meet the sink after the other node, then I transmit my data

36


Greedy Algorithm: if I meet the sink after the other node, then I transmit my data

1

meetTime = 3

2

meetTime = 5

37


-Waiting Greedy Algorithm: if at least one of the nodes has meetTime > , then we apply Greedy Algorithm

37



1

meetTime = 3

2

meetTime = 5=10

37



1

meetTime = 3

2

meetTime = 5

1

meetTime = 3

2

meetTime = 15

=10

37



1

meetTime = 3

2

meetTime = 5

1

meetTime = 3

2

meetTime = 15

=10

1

meetTime = 13

2

meetTime = 15

37



1

meetTime = 3

2

meetTime = 5

1

meetTime = 3

2

meetTime = 15

=10

1

meetTime = 13

2

meetTime = 15

1

meetTime = 7

s

38


n√n log(n)-Waiting Greedy Algorithm is optimal


38



Without knowledge: 𝞗(n2) interactions in average


38




With full knowledge: 𝞗(n log(n)) interactions w.h.p.


38




With full knowledge: 𝞗(n log(n)) interactions w.h.p.

meetTime information: 𝞗(n√n log(n)) interactions w.h.p.


Conclusion 39

Conclusion 39

Online Data Aggregation: - hard in general - optimal algorithms exist in randomized networks

Conclusion 39

Perspective


Conclusion 39

Perspective


More realistic networks?

Conclusion 40

Perspective


More realistic networks?

Thank you for your attention!

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What if nodes are allowed to transmit more than once?

s

1

1

2

1 transmits s

1s

2

2 transmits2

1

s

2otherwise


cost(A) = ∞

Let A be a DODA



s

1

1

2

s

1

s

2

1

2

s

1

…



s’

1

1

2

s’

1

s’

2

1

2

s’

1

…



s’

1

1

2

s’

1

s’

2

1

2

s’

1

…

s’

s



s’

1

1

2

s’

1

s’

2

1

2

s’

1

…

s’

s

s’

s



s’

1

1

2

s’

1

s’

2

1

2

s’

1

…

s’

s

s’

s

s’

s



s’

1

1

2

s’

1

s’

2

1

2

s’

1

…

s’

s

s’

s

s’

s

s’

s



s’

1

1

2

s’

1

s’

2

1

2

s’

1

…

s’

s

s’

s

s’

s

s’

s

s’

s



s’

1

1

2

s’

1

s’

2

1

2

s’

1

…

s’

s

s’

s

s’

s

s’

s

s’

s

s’

s



s’

1

1

2

s’

1

s’

2

1

2

s’

1

…

s’

s

s’

s

s’

s

s’

s

s’

s

s’

s

if s’ owns all the data:



s’

1

1

2

s’

1

s’

2

1

2

s’

1

…

s’

s

s’

s

s’

s

s’

s

s’

s

s’

s


then, a node has transmitted twice



s’

1

1

2

s’

1

s’

2

1

2

s’

1

…

s’

s

s’

s

s’

s

s’

s

s’

s

s’

s



and s does not own all the data



s’

1

1

2

s’

1

s’

2

s’

s

s’

s

s’

s






s’

1

1

2

s’

1

s’

2

s’

s

s’

s

s’

s



1

2

1

s

1

s’




s’

1

1

2

s’

1

s’

2

s’

s

s’

s

s’

s



1

2

1

s

1

s’

and s will never own all the data



s’

1

1

2

s’

1

s’

2

s’

s

s’

s

s’

s



1

2

1

s

1

s’

and s will never own all the data

yet a convergecast is always possible

distributed online data aggregation in dynamic graphs · impossibility results - online adaptive...

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