ETH Zurich – Distributed Computing Group Stephan Holzer 1ETH Zurich – Distributed Computing – www.disco.ethz.ch

Stephan HolzerYvonne Anne Pignolet

Jasmin SmulaRoger Wattenhofer

Time-Optimal Information Exchange

on Multiple Channels

ETH Zurich – Distributed Computing Group Stephan Holzer 2

Problem:

Time-Optimal Information Exchange on Multiple Channels

n:= # nodes

ETH Zurich – Distributed Computing Group Stephan Holzer 3

Problem:

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # information Have information

Disseminate to all!?

ETH Zurich – Distributed Computing Group Stephan Holzer 4

Problem:

Time-Optimal Information Exchange on Multiple Channels

Disseminate to all!?

ETH Zurich – Distributed Computing Group Stephan Holzer 5

Problem:

Time-Optimal Information Exchange on Multiple Channels

Disseminate to all!?Easy: O(n)

Faster?

ETH Zurich – Distributed Computing Group Stephan Holzer 6

Problem:

Time-Optimal Information Exchange on Multiple Channels

n:= # nodes

1

23

4

5

n

Unique IDs 1…n

ETH Zurich – Distributed Computing Group Stephan Holzer 7

I can:

send / receive

reach each node

Time-Optimal Information Exchange on Multiple Channels

ETH Zurich – Distributed Computing Group Stephan Holzer 8

I can:

send / receive

?reach each node

Time-Optimal Information Exchange on Multiple Channels

ETH Zurich – Distributed Computing Group Stephan Holzer 9

Time-Optimal Information Exchange on Multiple Channels

no collision detection

I can:

send / receive

reach each node

ETH Zurich – Distributed Computing Group Stephan Holzer 10

switch channels

no collision detection

I can:

send / receive

reach each node

101 Mhz117 Mhz132 Mhz …

Time-Optimal Information Exchange on Multiple Channels

synchronus

ETH Zurich – Distributed Computing Group Stephan Holzer 11

switch channels

no collision detection

I can:

send / receive

reach each node

complexitycomputation: freeradio: time 1

Time-Optimal Information Exchange on Multiple Channels

synchronus

ETH Zurich – Distributed Computing Group Stephan Holzer 12

One Information / log n bits per message

O(1)

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # information

=> Ω( k )

ETH Zurich – Distributed Computing Group Stephan Holzer 13

=> Ω( k )

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # information

+ log n[Kushilevitz, Mansour SIAM JComp 1998]

one channel ?𝜣 (𝒌)

Multi channel

ETH Zurich – Distributed Computing Group Stephan Holzer 14

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # information

TREE

What can I do?

ETH Zurich – Distributed Computing Group Stephan Holzer 15

4 7

8

3

1

2

5 6

9

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # information

10 11 12

TREE

13 14 15

16

Communicate in parallelon different channels

ETH Zurich – Distributed Computing Group Stephan Holzer 16

4

8

2

9

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # information

TREE

Communicate in parallelon different channels

ETH Zurich – Distributed Computing Group Stephan Holzer 17

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # information

TREE

ETH Zurich – Distributed Computing Group Stephan Holzer 18

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # information

TREE

ETH Zurich – Distributed Computing Group Stephan Holzer 19

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # information

TREE

ETH Zurich – Distributed Computing Group Stephan Holzer 20

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # information

TREE

ETH Zurich – Distributed Computing Group Stephan Holzer 21

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # information

TREE

ETH Zurich – Distributed Computing Group Stephan Holzer 22

TREE

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # information

O( k + log n)

ETH Zurich – Distributed Computing Group Stephan Holzer 23

Time-Optimal Information Exchange on Multiple Channels

What if k < log n?n:= # nodesk:= # information

O( k ) if k > log n

n channels TREE

ETH Zurich – Distributed Computing Group Stephan Holzer 24

Time-Optimal Information Exchange on Multiple Channels

What if k < log n?n:= # nodesk:= # information

ETH Zurich – Distributed Computing Group Stephan Holzer 25

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

ETH Zurich – Distributed Computing Group Stephan Holzer 26

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Balls into Bins

ETH Zurich – Distributed Computing Group Stephan Holzer 27

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Balls into Bins

ID=1…2k

ETH Zurich – Distributed Computing Group Stephan Holzer 28

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Balls into Bins

ID=1…2k

ETH Zurich – Distributed Computing Group Stephan Holzer 29

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Balls into Bins

Listens on channel 1Listens on channel 2Listens on channel 3Listens on channel 4

Send on random channel 1…2

ID=1…2k

k

ETH Zurich – Distributed Computing Group Stephan Holzer 30

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Balls into Bins

Listens on channel 1Listens on channel 2Listens on channel 3Listens on channel 4

ID=1…2k

Send on random channel 1…2k

ETH Zurich – Distributed Computing Group Stephan Holzer 31

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Balls into Bins

Pr no collisionID=1…2k

ETH Zurich – Distributed Computing Group Stephan Holzer 32

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

ID=1…2k

Balls into Bins

Pr no collisionTREE

ETH Zurich – Distributed Computing Group Stephan Holzer 33

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

ID=1…2k

Balls into Bins

Pr no collisionTREE

ETH Zurich – Distributed Computing Group Stephan Holzer 34

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

ID=1…2k

Balls into Bins

Pr no collisionRepeat k times

ETH Zurich – Distributed Computing Group Stephan Holzer 35

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

ID=1…2k

Balls into Bins

Pr [no collision] > 1- If

O( k ) if TREE

What if ?

Repeat k times

k2 channels

ETH Zurich – Distributed Computing Group Stephan Holzer 36

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique SubsetTime: O( k )

What if ?

𝑆𝑖𝑧𝑒 : √𝑛 log𝑛

ETH Zurich – Distributed Computing Group Stephan Holzer 37

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

𝑆𝑖𝑧𝑒 : √𝑛 log𝑛

ETH Zurich – Distributed Computing Group Stephan Holzer 38

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

𝑆𝑖𝑧𝑒 : √𝑛 log𝑛

ETH Zurich – Distributed Computing Group Stephan Holzer 39

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Send on random channel }.

𝑆𝑖𝑧𝑒 : √𝑛 log𝑛

ETH Zurich – Distributed Computing Group Stephan Holzer 40

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

𝑆𝑖𝑧𝑒 : √𝑛 log𝑛

Send on random channel }.Send on random channel .

ETH Zurich – Distributed Computing Group Stephan Holzer 41

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Not too big …Not too small …Just right!

𝑆𝑖𝑧𝑒 : √𝑛 log𝑛

Send on random channel }.Send on random channel .

Pr[at most half messages collide]1-

ETH Zurich – Distributed Computing Group Stephan Holzer 42

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Channel 3

Example: 3 channels

Channel 1

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

ETH Zurich – Distributed Computing Group Stephan Holzer 43

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Channel 3

Example: 3 channels

Channel 1Send k times

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

ETH Zurich – Distributed Computing Group Stephan Holzer 44

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Channel 3

Example: 3 channels

Channel 1Send k times

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

ETH Zurich – Distributed Computing Group Stephan Holzer 45

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

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Channel 3

Example: 3 channels

Channel 1Send k times

ETH Zurich – Distributed Computing Group Stephan Holzer 46

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Channel 3

Example: 3 channels

Channel 1

?

?

?

Send k times

{1}

{3 }

{1,3 }

ETH Zurich – Distributed Computing Group Stephan Holzer 47

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Channel 3

Example: 3 channels

Channel 1Send k times

{1,3 }

ETH Zurich – Distributed Computing Group Stephan Holzer 48

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Channel 3

Example: 3 channels

Channel 1Send k times

{1,3 }

ETH Zurich – Distributed Computing Group Stephan Holzer 49

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Channel 3

Example: 3 channels

Channel 1Send k times

{1,3 }

ETH Zurich – Distributed Computing Group Stephan Holzer 50

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Example: 3 channels

O( k )

{1,3 }

Pr[at most half messages collide]1-

ETH Zurich – Distributed Computing Group Stephan Holzer 51

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Example: 3 channels

O( k + k/2 + k/4 …

{1,3 }

Pr[at most half messages collide]1-

ETH Zurich – Distributed Computing Group Stephan Holzer 52

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique Subset

Example: 3 channels

O( k )

channelsPr[at most half messages collide]1-

Pr[this works] 1-

ETH Zurich – Distributed Computing Group Stephan Holzer 53

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known

Unique SubsetBalls into Bins

TREE

𝑘<√ log𝑛√ log𝑛≤𝑘<log𝑛log𝑛≤𝑘

ETH Zurich – Distributed Computing Group Stephan Holzer 54

Time-Optimal Information Exchange on Multiple Channels

n:= # nodesk:= # informationAssume: k known unknown

Unique SubsetBalls into Bins

TREEn channels

channels

2 channelsk

n channelsFuture directions: deterministic

less channelslower bounds

𝑘<√ log𝑛√ log𝑛≤𝑘<log𝑛log𝑛≤𝑘

ETH Zurich – Distributed Computing Group Stephan Holzer 55

in Summary … Detect / Disseminate Information!

Time-Optimal Information Exchange on Multiple Channels

101 Mhz117 Mhz132 Mhz …

𝜣 (𝒌){1,3 }

ETH Zurich – Distributed Computing Group Stephan Holzer 56ETH Zurich – Distributed Computing – www.disco.ethz.ch

Stephan HolzerYvonne Anne Pignolet

Jasmin SmulaRoger Wattenhofer

Thank You!Questions & Comments?