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ETH Zurich – Distributed Computing Group Stephan HolzerETH Zurich – Distributed Computing – www.disco.ethz.ch

Stephan Holzer - ETH ZürichThomas Locher - ABB Switzerland

Yvonne Anne Pignolet - ABB SwitzerlandRoger Wattenhofer - ETH Zürich

Deterministic Multi-Channel

Information Exchange

ETH Zurich – Distributed Computing Group Stephan Holzer

Deterministic Multi-Channel Information Exchange

Problem:n:= # nodes



Problem:n:= # nodesk:= # information Have information

Disseminate to all!?



Problem:




Problem:




Problem:n:= # nodes

1

23

4

5

n

Unique IDs 1…n



Problem:

Disseminate to all!?Easy: O(n)

Faster?


Deterministic Multi-Channel Information ExchangeI can:

send / receive

reach each node


Deterministic Multi-Channel Information ExchangeI can:

send / receive

?reach each node



no collision detection

I can:

send / receive

reach each node



switch channels


I can:

send / receive

reach each node

101 Mhz117 Mhz132 Mhz …

synchronus



switch channels


I can:

send / receive

reach each node

complexitycomputation: freeradio: time 1

synchronus



n:= # nodesk:= # information

Time Channels[GW85]: Ω(k + log n) 1

k




kTime Channels

[GW85]: Ω(k + log n) 1[HPSW11]: O(k) n

k




kTime Channels


Optimal

k




kTime Channels


Optimal

????

k




Range of k [1, ] ( , log n ) [log n , n]Upper boundOn channels O() O() 1

[HPSW11] - Channels needed for time O(k):






Range of k [1, log n] (log n , log n loglog n)

[log n loglog n , n- log n)

[n – log n, n]

Upper boundOn channels O() O() O() 1

This paper:








[n – log n, n]


This paper:

randomized








[n – log n, n]


This paper:

randomized

deterministic








[n – log n, n]


This paper:

randomized

deterministic

Optimal?

Optimal?

Optimal?

Optimal?








[n – log n, n]

Upper boundOn channels O() O() O() 1Lower boundOn channels Ω() Ω() Ω(log ) 1

This paper:

randomized

deterministic

k



Main ingredient:

Specially taylored graphs.



Main ingredient:

Specially taylored graphs.(Inspired by use of lossless expanders in [CK08])



Main ingredient:

Specially taylored graphs.(Inspired by use of lossless expanders in [CK08])

Topology: Still single hop.Graphs used to select channel.



7

6

5

4

3

2

1V

W

Bipartite :

node IDs new names



7

6

5

4

3

2

1V

W

Bipartite :

node IDs new names



Matching Graphs:• Nodes in V have degee V

W

Δ



Matching Graphs:• Nodes in V have degee • Fixed order of edges

VW

4123




• For any of size at most k there is

at least nodes in X have a unique i-neighbor.

VW

4123






7

6

5

4

3

2

1

VW

i

ihave uniquei-neighbor






7

6

5

4

3

2

1

VW

i

ihave uniquei-neighbor






7

6

5

4

3

2

1 1

1

1

2

1

2

2

2

VW






7

6

5

4

3

2

1 1

1

1

2

1

2

2

2

VW

X






7

6

5

4

3

2

1 1

1

1

1

VW

X






7

6

5

4

3

2

1 1

1

1

1

VW

XBAD






7

6

5

4

3

2

1 1

1

1

2

1

2

2

2

VW

X






7

6

5

4

3

2

122

2

2

VW

X






7

6

5

4

3

2

122

2

2

VW

XGOOD






7

6

5

4

3

2

1 1

1

1

1

VW

X






7

6

5

4

3

2

1 1

1

1

1

VW

X




• For any of size at most K there is


exist if 7

6

5

4

3

2

1 1

1

1

2

1

2

2

2

VW



What are these graphs good for?




Renaming




Renaming

7

6

5

4

3

2

1 1

1

1

2

1

2

2

2

VW




Renaming

• To each of the k «reporters» we can assing a new unique name in |W| in time O( using |W| channels.

7

6

5

4

3

2

1 1

1

1

2

1

2

2

2

VW



What is renaming good for?



What is renaming good for?Assignment of reporters to channels!




Example: k < log n




Example: k < log n

n

Original names




Example: k < log n

n

Original names

O()

New namesof reporters



n O()

Original names





n O()

Original names





Time: O( k )

𝑆𝑖𝑧𝑒 :𝑛 /2n O()

Original names






Original names






Original names





𝑆𝑖𝑧𝑒 :𝑛 /2

Send on channel “new name” .

n O()

Original names






Send on channel “new name” .Send on channel “new name” .

n O()

Original names











𝑆𝑖𝑧𝑒 :𝑛 /2 Example: 3 channels




Channel 3

Example: 3 channels

Channel 1

{1,2}{1,3 }{2,3 }




Channel 3

Example: 3 channels

Channel 1Send 2 times

{1,2}{1,3 }{2,3 }




Channel 3

Example: 3 channels


{1,2}{1,3 }{2,3 }



{1,2}{1,3 }{2,3 }


Channel 3

Example: 3 channels





Channel 3

Example: 3 channels


{1,3 }




Unique Subset

Channel 3

Example: 3 channels


{1,3 }




Unique Subset

Channel 3

Example: 3 channels


{1,3 }




Unique Subset

Channel 3

Example: 3 channels


{1,3 }



Unique Subset

Channel 3

Example: 3 channels

Channel 1Send k times

{1,3 } O( k )




O( k )




[n – log n, n]




n O()

Original names


O() channels



in Summary … Detect / Disseminate Information!

101 Mhz117 Mhz132 Mhz …𝜣 (𝒌)

{1,3 }7

6

5

4

3

2

11

1

1

2

1

2

2

2

VW

X



[n – log n, n]





Thank You!Questions & Comments?




Thank You!Questions & Comments?

Date post:	22-Feb-2016
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