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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:n:= # nodes

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

kTime Channels

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

kTime Channels

Optimal

kTime Channels

Optimal

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?

[n – log n, n]

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

This paper:

randomized

deterministic

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.

Bipartite :

node IDs new names

Bipartite :

node IDs new names

Matching Graphs:• Nodes in V have degee V

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

• For any of size at most k there is

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

ihave uniquei-neighbor

• For any of size at most K there is

exist if 7

What are these graphs good for?

Renaming

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

What is renaming good for?

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

Example: k < log n

Original names

Example: k < log n

Original names

New namesof reporters

Original names

Time: O( k )

𝑆𝑖𝑧𝑒 :𝑛 /2n O()

Original names

𝑆𝑖𝑧𝑒 :𝑛 /2

Send on channel “new name” .

Original names

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

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 }

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]

Original names

O() channels

in Summary … Detect / Disseminate Information!

101 Mhz117 Mhz132 Mhz …𝜣 (𝒌)

{1,3 }7

[n – log n, n]

Thank You!Questions & Comments?

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