distributed online data aggregation in dynamic graphs · impossibility results - online adaptive...
TRANSCRIPT
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Distributed Online Data Aggregation in Dynamic Graphs
Quentin Bramas, Toshimitsu Masuzawa, and Sébastien Tixeuil
NETYS 2019, Marrakech, June, 21st
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Data Aggregation Problem 2
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Data Aggregation Problem
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Data Aggregation Problem
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Data Aggregation Problem
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Data Aggregation Problem
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Data Aggregation Problem
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Data Aggregation Problem
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Data Aggregation Problem
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Data Aggregation Problem
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Data Aggregation Problem
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Data Aggregation Problem
Each node transmits at most once.
The goal: to minimize the duration
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Data Aggregation Problem
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Data Aggregation Problem
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Data Aggregation Problem
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Data Aggregation Problem
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Data Aggregation Problem
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Dynamic Data Aggregation Problem
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Dynamic Data Aggregation Problem
We consider a dynamic network with pairwise interactions
…
s
32
15
4
t = 0 1 2 3
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18
Dynamic Data Aggregation Problem
We consider a sequence of interactions
We consider a dynamic network with pairwise interactions
…
s
32
15
4
t = 0 1 2 3
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18
Dynamic Data Aggregation Problem
We consider a sequence of interactions
A node can transmit only once
We consider a dynamic network with pairwise interactions
…
s
32
15
4
t = 0 1 2 3
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18
Dynamic Data Aggregation Problem
We consider a sequence of interactions
A node can transmit only once
Nodes may or may not have other information
We consider a dynamic network with pairwise interactions
…
s
32
15
4
t = 0 1 2 3
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18
Dynamic Data Aggregation Problem
We consider a sequence of interactions
A node can transmit only once
Nodes may or may not have other information
The goal is to aggregate all the data with
minimum duration
We consider a dynamic network with pairwise interactions
…
s
32
15
4
t = 0 1 2 3
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Distributed Online Data Aggregation 19
We consider a dynamic network with pairwise interactions
…
s
32
15
4
t = 0 1 2 3
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Distributed Online Data Aggregation 19
We consider a dynamic network with pairwise interactions
…
s
32
15
4
A Distributed Online Data Aggregation (DODA) Algorithm answers the question: Which node transmits?
t = 0 1 2 3
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Distributed Online Data Aggregation 19
We consider a dynamic network with pairwise interactions
…
s
32
15
4
a
b
A Distributed Online Data Aggregation (DODA) Algorithm answers the question: Which node transmits?
t = 0 1 2 3
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Distributed Online Data Aggregation 19
We consider a dynamic network with pairwise interactions
…
s
32
15
4
a
b
a transmits or b transmits or no one transmits
(a node can transmit only once)
A Distributed Online Data Aggregation (DODA) Algorithm answers the question: Which node transmits?
t = 0 1 2 3
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has data 1 2 3 4
has transmitted
Distributed Online Data Aggregation
1
2
20
t = 0 1 2 3 4 5 6
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has data 2(1) 3 4
has transmitted 1
Distributed Online Data Aggregation
1
2
20
t = 0 1 2 3 4 5 6
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has data 2(1) 3 4
has transmitted 1
Distributed Online Data Aggregation
1
2
3
4
20
t = 0 1 2 3 4 5 6
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has data 2(1) 3 4
has transmitted 1
Distributed Online Data Aggregation
1
2
1
3
3
4
20
t = 0 1 2 3 4 5 6
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has data 2(1) 3 4
has transmitted 1
Distributed Online Data Aggregation
1
2
1
3
3
4
s
1
20
t = 0 1 2 3 4 5 6
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has data 2(1) 3 4
has transmitted 1
Distributed Online Data Aggregation
1
2
1
3
2
3
3
4
s
1
20
t = 0 1 2 3 4 5 6
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has data 2(1,3) 4
has transmitted 1 3
Distributed Online Data Aggregation
1
2
1
3
2
3
3
4
s
1
20
t = 0 1 2 3 4 5 6
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has data 2(1,3) 4
has transmitted 1 3
Distributed Online Data Aggregation
1
2
1
3
2
3
2
4
3
4
s
1
20
t = 0 1 2 3 4 5 6
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has data 2(1,3,4)
has transmitted 1 3 4
Distributed Online Data Aggregation
1
2
1
3
2
3
2
4
3
4
s
1
20
t = 0 1 2 3 4 5 6
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has data 2(1,3,4)
has transmitted 1 3 4
Distributed Online Data Aggregation
1
2
1
3
2
3
2
4
3
4
s
1
s
2
20
t = 0 1 2 3 4 5 6
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has data
has transmitted 1 2 3 4
Distributed Online Data Aggregation
1
2
1
3
2
3
2
4
3
4
s
1
s
2
20
t = 0 1 2 3 4 5 6
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How to evaluate the performance of a DODA? 21
Is the duration of the aggregation is significant ?
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How to evaluate the performance of a DODA? 21
Is the duration of the aggregation is significant ? No
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How to evaluate the performance of a DODA? 21
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
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How to evaluate the performance of a DODA? 22
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 ?
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How to evaluate the performance of a DODA? 22
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
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How to evaluate the performance of a DODA? 22
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
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
}
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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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23
How to evaluate the performance of a DODA?
Definition of costI(A), for an algorithm A in a sequence I :
}
convergecast
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23
How to evaluate the performance of a DODA?
Definition of costI(A), for an algorithm A in a sequence I :
}
convergecast
}
convergecast (starting after the previous one)
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23
How to evaluate the performance of a DODA?
Definition of costI(A), for an algorithm A in a sequence I :
}
convergecast
}
convergecast (starting after the previous one)
}convergecast
}convergecast
}
convergecast
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23
How to evaluate the performance of a DODA?
Definition of costI(A), for an algorithm A in a sequence I :
}
convergecast
}
convergecast (starting after the previous one)
}convergecast
}convergecast
}
convergecast
execution of A
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23
How to evaluate the performance of a DODA?
Definition of costI(A), for an algorithm A in a sequence I :
}
convergecast
}
convergecast (starting after the previous one)
}convergecast
}convergecast
}
convergecast
execution of A
costI(A) = 4
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24
How to evaluate the performance of a DODA?
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
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25
How to evaluate the performance of a DODA?
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
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26
Distributed Online Data Aggregation
…
s
32
15
4
t = 0 1 2 3
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26
Distributed Online Data Aggregation
…
s
32
15
4
Who generates the sequence?
t = 0 1 2 3
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26
Distributed Online Data Aggregation
An adversary
…
s
32
15
4
Who generates the sequence?
t = 0 1 2 3
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26
Distributed Online Data Aggregation
An adversary
Three adversaries:
…
s
32
15
4
Who generates the sequence?
t = 0 1 2 3
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26
Distributed Online Data Aggregation
An adversary
Three adversaries:
…
s
32
15
4
Who generates the sequence?
t = 0 1 2 3
- Online Adaptive: generates the next interaction based on what happened in the past
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26
Distributed Online Data Aggregation
An adversary
Three adversaries:
…
s
32
15
4
Who generates the sequence?
t = 0 1 2 3
- Online Adaptive: generates the next interaction based on what happened in the past
- Oblivious: generates the sequence before the execution of the algorithm
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26
Distributed Online Data Aggregation
An adversary
Three adversaries:
…
s
32
15
4
Who generates the sequence?
t = 0 1 2 3
- Online Adaptive: generates the next interaction based on what happened in the past
- Oblivious: generates the sequence before the execution of the algorithm
- Randomized: each interaction is chosen uniformly at random.
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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
![Page 61: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/61.jpg)
Impossibility Results - Online Adaptive Adversary 27
- Online Adaptive: generates the next interaction based on what the algorithm decides in the current interaction
s
1
Let A be a DODA
![Page 62: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/62.jpg)
Impossibility Results - Online Adaptive Adversary 27
- Online Adaptive: generates the next interaction based on what the algorithm decides in the current interaction
s
1
1 transmitsLet A be a DODA
![Page 63: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/63.jpg)
Impossibility Results - Online Adaptive Adversary 27
- Online Adaptive: generates the next interaction based on what the algorithm decides in the current interaction
s
1
1 transmits 1
2
s
1
Let A be a DODA
![Page 64: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/64.jpg)
Impossibility Results - Online Adaptive Adversary 27
- Online Adaptive: generates the next interaction based on what the algorithm decides in the current interaction
s
1
1 transmits 1
2
s
1s
2
Let A be a DODA
![Page 65: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/65.jpg)
Impossibility Results - Online Adaptive Adversary 27
- Online Adaptive: generates the next interaction based on what the algorithm decides in the current interaction
s
1
1 transmits 1
2
s
1s
2
2 transmits
Let A be a DODA
![Page 66: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/66.jpg)
Impossibility Results - Online Adaptive Adversary 27
- Online Adaptive: generates the next interaction based on what the algorithm decides in the current interaction
s
1
1 transmits 1
2
s
1s
2
2 transmits2
1
s
2
Let A be a DODA
![Page 67: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/67.jpg)
Impossibility Results - Online Adaptive Adversary 27
- Online Adaptive: generates the next interaction based on what the algorithm decides in the current interaction
s
1
1 transmits 1
2
s
1s
2
2 transmits2
1
s
2otherwise
Let A be a DODA
![Page 68: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/68.jpg)
Impossibility Results - Online Adaptive Adversary 27
- Online Adaptive: generates the next interaction based on what the algorithm decides in the current interaction
s
1
1 transmits 1
2
s
1s
2
2 transmits2
1
s
2otherwise
Let A be a DODA
A never terminates
![Page 69: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/69.jpg)
Impossibility Results - Online Adaptive Adversary 27
- Online Adaptive: generates the next interaction based on what the algorithm decides in the current interaction
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
![Page 70: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/70.jpg)
Impossibility Results - Online Adaptive Adversary 27
- Online Adaptive: generates the next interaction based on what the algorithm decides in the current interaction
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:
cost(A) = ∞
Let A be a DODA
A never terminates
![Page 71: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/71.jpg)
Impossibility Results - Online Adaptive Adversary 28
![Page 72: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/72.jpg)
Impossibility Results - Online Adaptive Adversary 28
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)=∞
![Page 73: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/73.jpg)
Impossibility Results - Online Adaptive Adversary 28
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)=∞
![Page 74: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/74.jpg)
Impossibility Results - Online Adaptive Adversary 28
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)=∞
What about randomized DODA algorithms against an oblivious adversary?
![Page 75: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/75.jpg)
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
What about randomized DODA algorithms against an oblivious adversary?
![Page 76: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/76.jpg)
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
proof: not trivial
What about randomized DODA algorithms against an oblivious adversary?
![Page 77: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/77.jpg)
Impossibility Results - Oblivious Adversary 30
sketch of proof, for k=1
![Page 78: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/78.jpg)
Impossibility Results - Oblivious Adversary 30
s
1
s
2
s
n
…
sketch of proof, for k=1
![Page 79: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/79.jpg)
Impossibility Results - Oblivious Adversary 30
s
1
s
2
s
n
…
s
1
s
2
s
n
…
sketch of proof, for k=1
![Page 80: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/80.jpg)
Impossibility Results - Oblivious Adversary 30
s
1
s
2
s
n
…
s
1
s
2
s
n
…
sketch of proof, for k=1
Pt = probability that no node transmits before time t
![Page 81: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/81.jpg)
Impossibility Results - Oblivious Adversary 30
s
1
s
2
s
n
…
s
1
s
2
s
n
…
sketch of proof, for k=1
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.
![Page 82: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/82.jpg)
Impossibility Results - Oblivious Adversary 30
s
1
s
2
s
n
…
s
1
s
2
s
n
…
sketch of proof, for k=1
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.
if Pt > 1/n for all t, then it’s over
![Page 83: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/83.jpg)
Impossibility Results - Oblivious Adversary 30
s
1
s
2
s
n
…
s
1
s
2
s
n
…
sketch of proof, for k=1
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.
if Pt > 1/n for all t, then it’s over , otherwise there is a time t0 such that:
![Page 84: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/84.jpg)
Impossibility Results - Oblivious Adversary 30
s
1
s
2
s
n
…
s
1
s
2
s
n
…
sketch of proof, for k=1
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.
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
![Page 85: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/85.jpg)
Impossibility Results - Oblivious Adversary 30
s
1
s
2
s
n
…
s
1
s
2
s
n
…
sketch of proof, for k=1
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.
if Pt > 1/n for all t, then it’s over
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
![Page 86: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/86.jpg)
Impossibility Results - Oblivious Adversary 30
s
1
s
2
s
n
…
s
1
s
2
s
n
…
sketch of proof, for k=1
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.
if Pt > 1/n for all t, then it’s over
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
![Page 87: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/87.jpg)
Impossibility Results - Oblivious Adversary 30
s
1
s
2
s
n
…
s
1
s
2
s
n
…
sketch of proof, for k=1
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.
if Pt > 1/n for all t, then it’s over
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 : one node has transmitted w.h.p.
![Page 88: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/88.jpg)
Impossibility Results - Oblivious Adversary 30
s
1
s
2
s
n
…
s
1
s
2
s
n
…
sketch of proof, for k=1
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.
if Pt > 1/n for all t, then it’s over
t0
s
1
s
2
s
n
s
1
s
2
s
n
… …
n
u
n-1
n
s
1
…
, otherwise there is a time t0 such that: : one node u has not transmitted w.h.p.• Pt0 -1 > 1/n
• Pt0 ≤1/n : one node has transmitted w.h.p.
![Page 89: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/89.jpg)
Impossibility Results - Oblivious Adversary 30
s
1
s
2
s
n
…
s
1
s
2
s
n
…
sketch of proof, for k=1
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.
if Pt > 1/n for all t, then it’s over
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
…
, otherwise there is a time t0 such that: : one node u has not transmitted w.h.p.• Pt0 -1 > 1/n
• Pt0 ≤1/n : one node has transmitted w.h.p.
![Page 90: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/90.jpg)
Impossibility Results - Oblivious Adversary 31
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
![Page 91: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/91.jpg)
Impossibility Results - Oblivious Adversary 32
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
![Page 92: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/92.jpg)
Impossibility Results - Oblivious Adversary 32
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?
![Page 93: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/93.jpg)
Impossibility Results - Oblivious Adversary 32
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?
open question
![Page 94: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/94.jpg)
DODA against randomized adversary
![Page 95: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/95.jpg)
34
DODA against randomized adversary
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.
![Page 96: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/96.jpg)
35
DODA against randomized adversary
![Page 97: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/97.jpg)
35
DODA against randomized adversary
The meetTime information: each node knows when will be its next interaction with the sink
![Page 98: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/98.jpg)
35
DODA against randomized adversary
The meetTime information: each node knows when will be its next interaction with the sink
1
meetTime = 3
2
meetTime = 6
![Page 99: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/99.jpg)
36
DODA against randomized adversary
Greedy Algorithm: if I meet the sink after the other node, then I transmit my data
![Page 100: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/100.jpg)
36
DODA against randomized adversary
Greedy Algorithm: if I meet the sink after the other node, then I transmit my data
1
meetTime = 3
2
meetTime = 5
![Page 101: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/101.jpg)
36
DODA against randomized adversary
Greedy Algorithm: if I meet the sink after the other node, then I transmit my data
1
meetTime = 3
2
meetTime = 5
![Page 102: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/102.jpg)
37
DODA against randomized adversary
-Waiting Greedy Algorithm: if at least one of the nodes has meetTime > , then we apply Greedy Algorithm
![Page 103: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/103.jpg)
37
DODA against randomized adversary
-Waiting Greedy Algorithm: if at least one of the nodes has meetTime > , then we apply Greedy Algorithm
1
meetTime = 3
2
meetTime = 5=10
![Page 104: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/104.jpg)
37
DODA against randomized adversary
-Waiting Greedy Algorithm: if at least one of the nodes has meetTime > , then we apply Greedy Algorithm
1
meetTime = 3
2
meetTime = 5
1
meetTime = 3
2
meetTime = 15
=10
![Page 105: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/105.jpg)
37
DODA against randomized adversary
-Waiting Greedy Algorithm: if at least one of the nodes has meetTime > , then we apply Greedy Algorithm
1
meetTime = 3
2
meetTime = 5
1
meetTime = 3
2
meetTime = 15
=10
![Page 106: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/106.jpg)
37
DODA against randomized adversary
-Waiting Greedy Algorithm: if at least one of the nodes has meetTime > , then we apply Greedy Algorithm
1
meetTime = 3
2
meetTime = 5
1
meetTime = 3
2
meetTime = 15
=10
1
meetTime = 13
2
meetTime = 15
![Page 107: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/107.jpg)
37
DODA against randomized adversary
-Waiting Greedy Algorithm: if at least one of the nodes has meetTime > , then we apply Greedy Algorithm
1
meetTime = 3
2
meetTime = 5
1
meetTime = 3
2
meetTime = 15
=10
1
meetTime = 13
2
meetTime = 15
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37
DODA against randomized adversary
-Waiting Greedy Algorithm: if at least one of the nodes has meetTime > , then we apply Greedy Algorithm
1
meetTime = 3
2
meetTime = 5
1
meetTime = 3
2
meetTime = 15
=10
1
meetTime = 13
2
meetTime = 15
1
meetTime = 7
s
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38
DODA against randomized adversary
n√n log(n)-Waiting Greedy Algorithm is optimal
-Waiting Greedy Algorithm: if at least one of the nodes has meetTime > , then we apply Greedy Algorithm
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38
DODA against randomized adversary
n√n log(n)-Waiting Greedy Algorithm is optimal
Without knowledge: 𝞗(n2) interactions in average
-Waiting Greedy Algorithm: if at least one of the nodes has meetTime > , then we apply Greedy Algorithm
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38
DODA against randomized adversary
n√n log(n)-Waiting Greedy Algorithm is optimal
Without knowledge: 𝞗(n2) interactions in average
With full knowledge: 𝞗(n log(n)) interactions w.h.p.
-Waiting Greedy Algorithm: if at least one of the nodes has meetTime > , then we apply Greedy Algorithm
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38
DODA against randomized adversary
n√n log(n)-Waiting Greedy Algorithm is optimal
Without knowledge: 𝞗(n2) interactions in average
With full knowledge: 𝞗(n log(n)) interactions w.h.p.
meetTime information: 𝞗(n√n log(n)) interactions w.h.p.
-Waiting Greedy Algorithm: if at least one of the nodes has meetTime > , then we apply Greedy Algorithm
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Conclusion 39
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Conclusion 39
Online Data Aggregation: - hard in general - optimal algorithms exist in randomized networks
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Conclusion 39
Perspective
Online Data Aggregation: - hard in general - optimal algorithms exist in randomized networks
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Conclusion 39
Perspective
Online Data Aggregation: - hard in general - optimal algorithms exist in randomized networks
More realistic networks?
![Page 117: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/117.jpg)
Conclusion 40
Perspective
Online Data Aggregation: - hard in general - optimal algorithms exist in randomized networks
More realistic networks?
Thank you for your attention!
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Impossibility Results - Online Adaptive Adversary 41
What if nodes are allowed to transmit more than once?
s
1
1
2
1 transmits s
1s
2
2 transmits2
1
s
2otherwise
Starting from any time t, the aggregation is always possible, so:
cost(A) = ∞
Let A be a DODA
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Impossibility Results - Online Adaptive Adversary 42
What if nodes are allowed to transmit more than once?
s
1
1
2
s
1
s
2
1
2
s
1
…
![Page 120: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/120.jpg)
Impossibility Results - Online Adaptive Adversary 43
What if nodes are allowed to transmit more than once?
s’
1
1
2
s’
1
s’
2
1
2
s’
1
…
![Page 121: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/121.jpg)
Impossibility Results - Online Adaptive Adversary 43
What if nodes are allowed to transmit more than once?
s’
1
1
2
s’
1
s’
2
1
2
s’
1
…
s’
s
![Page 122: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/122.jpg)
Impossibility Results - Online Adaptive Adversary 43
What if nodes are allowed to transmit more than once?
s’
1
1
2
s’
1
s’
2
1
2
s’
1
…
s’
s
s’
s
![Page 123: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/123.jpg)
Impossibility Results - Online Adaptive Adversary 43
What if nodes are allowed to transmit more than once?
s’
1
1
2
s’
1
s’
2
1
2
s’
1
…
s’
s
s’
s
s’
s
![Page 124: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/124.jpg)
Impossibility Results - Online Adaptive Adversary 43
What if nodes are allowed to transmit more than once?
s’
1
1
2
s’
1
s’
2
1
2
s’
1
…
s’
s
s’
s
s’
s
s’
s
![Page 125: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/125.jpg)
Impossibility Results - Online Adaptive Adversary 43
What if nodes are allowed to transmit more than once?
s’
1
1
2
s’
1
s’
2
1
2
s’
1
…
s’
s
s’
s
s’
s
s’
s
s’
s
![Page 126: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/126.jpg)
Impossibility Results - Online Adaptive Adversary 43
What if nodes are allowed to transmit more than once?
s’
1
1
2
s’
1
s’
2
1
2
s’
1
…
s’
s
s’
s
s’
s
s’
s
s’
s
s’
s
![Page 127: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/127.jpg)
Impossibility Results - Online Adaptive Adversary 43
What if nodes are allowed to transmit more than once?
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:
![Page 128: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/128.jpg)
Impossibility Results - Online Adaptive Adversary 43
What if nodes are allowed to transmit more than once?
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:
then, a node has transmitted twice
![Page 129: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/129.jpg)
Impossibility Results - Online Adaptive Adversary 43
What if nodes are allowed to transmit more than once?
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:
then, a node has transmitted twice
and s does not own all the data
![Page 130: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/130.jpg)
Impossibility Results - Online Adaptive Adversary 44
What if nodes are allowed to transmit more than once?
s’
1
1
2
s’
1
s’
2
s’
s
s’
s
s’
s
if s’ owns all the data:
then, a node has transmitted twice
and s does not own all the data
![Page 131: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/131.jpg)
Impossibility Results - Online Adaptive Adversary 44
What if nodes are allowed to transmit more than once?
s’
1
1
2
s’
1
s’
2
s’
s
s’
s
s’
s
if s’ owns all the data:
then, a node has transmitted twice
1
2
1
s
1
s’
and s does not own all the data
![Page 132: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/132.jpg)
Impossibility Results - Online Adaptive Adversary 44
What if nodes are allowed to transmit more than once?
s’
1
1
2
s’
1
s’
2
s’
s
s’
s
s’
s
if s’ owns all the data:
then, a node has transmitted twice
1
2
1
s
1
s’
and s will never own all the data
![Page 133: Distributed Online Data Aggregation in Dynamic Graphs · Impossibility Results - Online Adaptive Adversary 28 Theorem: If k transmissions are allowed per node, there exists an online](https://reader033.vdocuments.us/reader033/viewer/2022050521/5fa4456791bed029eb68a061/html5/thumbnails/133.jpg)
Impossibility Results - Online Adaptive Adversary 44
What if nodes are allowed to transmit more than once?
s’
1
1
2
s’
1
s’
2
s’
s
s’
s
s’
s
if s’ owns all the data:
then, a node has transmitted twice
1
2
1
s
1
s’
and s will never own all the data
yet a convergecast is always possible