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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?