HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Algorithms for Radio Networks
Winter Term 2005/2006
21 Dec 2005
10th Lecture
Christian Schindelhauer
[email protected]
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HEINZ NIXDORF INSTITUTE
Radio Broadcasting
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
 Broadcasting
– A sender distributes a message to n radio stations
 Radio Broadcasting
– Undirected Graph G=(V,E) describes possible connections
• If edge {u,v} exists, u can transmit to v and vice versa
• If no edge exists, then there is no reception and no interference
– One frequency, stations communicate in a round model
– If more than one neighbored station send at the same time, no signal
is received (not even an interference signal)
 Main problem:
– Graph G=(V,E) is unknown to the participants
– Distributed algorithm avoiding conflicts
Algorithms for Radio Networks
2
Radio Broadcasting
without ID
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
 Theorem
There is no deterministic broadcasting algorithm for the radio
broadcasting problem (without id)
 Proof:
Consider the following graph:
1. Blue node sends (at any time)
a message to the neighbors
2. As soon they are informed, they
behave completely synchronously
– because they use the same algorithm
– so, they send (or do not send) always at
the same time
3. Red node does not receive any message.
Algorithms for Radio Networks
3
HEINZ NIXDORF INSTITUTE
A simple random algorithm (I)
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Every station uses the following algorithm
Simple-Random(t)
begin
if message m is available then
for i ← 1 to t do
r ← result of a fair coin toss (0/1 with prob. 1/2)
if r = 1 then
send m to all neighbors
fi
od
fi
end
Theorem
For appropriate c>1 we have: Simple-Random informs the complete
network with probability of at least 1-O(nk) within time c 2Δ/Δ (D+ log n).
Algorithms for Radio Networks
4
HEINZ NIXDORF INSTITUTE
Extending the Deterministic Model
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
 Model too restrictiv
 New deterministic model:
– Every of the n players knows his unique id number from the
set {1,..,n}
 Probabilistic model:
– Die number n of players is known
– The maximal degree Δ is known
– But no ID is available
Algorithms for Radio Networks
5
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Decay (I)
 Idee:
randomized thinning out of the players
Decay(k,m)
begin
j ← 1
repeat
j ← j + 1
Send message to all neighbors
r ← result of fair coin toss (0/1 with prob. 1/2)
until r=0 oder j > k
end
Algorithms for Radio Networks
6
HEINZ NIXDORF INSTITUTE
Decay (II)
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
 d neighbors are informed
 All d neighbors start simultanously (k,m)
 P(k,d):
Prob. that message is received by d neighbors within
at most k rounds:
Lemma
For d≥2 :
Algorithms for Radio Networks
7
HEINZ NIXDORF INSTITUTE
BGI-Broadcast
[Bar-Yehuda, Goldreich, Itai 1987]

University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
All informed players have synchronized round counters, i.e.
– Time is attached to each message
– and incremented in each round
BGI-Broadcast(Δ,)
begin
k ← 2 log Δ
t ← 2 log (N/)
wait until message arrives
for i ← 1 to t do
wait until (Time mod k) = 0
Decay(k,m)
od
end
Theorem
BGI-Broadcast informs all nodes with probability 1- in time O((D+log(n/)) log Δ)
Algorithms for Radio Networks
8
HEINZ NIXDORF INSTITUTE
Changing the Game: New Models
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
 Probabilistic mode:
– Number n of players is known
– The maximal degree Δ is known
– But no ID
 Restriction: What if the maximal degree is not known?
– Corollary
• BGI-Broadcast informs all nodes with probability
1- in time O((D+log(n/)) log n)
 Determinististic model:
– Each of the n players knows a unique identifier (id) of the
set {1,..,n} and knows n
Algorithms for Radio Networks
9
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Determinism versus Probabilism
 Theorem
For every distributed deterministic Radio-Broadcasting algorithm using IDs
there is a graph with D=2 that cannot be completely informed within time n-2.
 Theorem
BGI-Broadcast informs all nodes with probability 1- in time
O((D+log(n/)) log Δ) for any e>0.
 Theorem
For any constant >0 BGI-Broadcast informs all nodes of a graph with D=2
with probability 1- in time O((log n)2).
Algorithms for Radio Networks
10
HEINZ NIXDORF INSTITUTE
Decay
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
 d neighbors are informed
 All d neighbors start simultanously (k,m)
 P(k,d):
Prob. that message is received from d neighbors
within at most k rounds:
Lemma
For d≥2 :
Algorithms for Radio Networks
11
HEINZ NIXDORF INSTITUTE
Proof of Lemma (Part I)
 P(k,d):
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Prob. that the message is received from d neighbors
within at most k rounds
 0 neighbored players are informed:
– P(1,0)= 0
Chance of being informed in the first round by nobody
– P(2,0)= 0
– P(3,0)= 0
– ...
 1 neighbored player is informed:
– P(1,1)= 1
One player cannon cause any conflict
– P(2,1)= 1
stays informed in the next roundd
– P(3,1)= 1
etc.
– ...
Algorithms for Radio Networks
12
HEINZ NIXDORF INSTITUTE
Proof of Lemma (Part I)
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
P(k,d):
–Prob. that the message is received from d
neighbors within at most k rounds
2 neighbored players are informed:
–P(2,1)= 0
•Two nodes send in the first round.
•No chance
–P(2,2)= P(no player continues) P(1,0) +
P(one player continues) P(1,1) +
P(two players continue) P(1,1)
= 1/4 P(1,0) + 1/2 P(1,1) + 1/4 P(2,1)
= 0 + 1/2 + 0 = 1/2
Algorithms for Radio Networks
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Survey of Randomized Broadcasting
Algorithms
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
 Lower bounds for random algorithms concerning expected round time:
– Alon, Bar-Noy, Linial, Peleg, 1991
(log2n) for diameter D=1
– Kushiletz, Mansour, 1998
(D log (n/D))
 Expected round time of random algorithms
– Gaber, Mansour, 2003
O(D+ log5 n) if the network is known
– Bar-Yehuda, Goldreich, Itai, 1992
O((D+log n) log n)
(presented here)
– Czumaj, Rytter, 2003:
O(D log (n/D) + log2 n)
– Bar-Yehuda, Goldreich, Itai, 1992
O(n log n)
if D is unknown
Algorithms for Radio Networks
14
HEINZ NIXDORF INSTITUTE
Survey of Deterministic Algorithms
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
 Lower bounds for deterministic algorithms concerning expected round
time:
– Bar-Yehuda, Goldreich, Itai, 1992
(n)
(presented here)
 Worst case time of deterministic algorithms
– Chlebus, Gasieniec, Gibbons, Pelc, Rytter, 1999
O(n11/6)
– Chlebus, Gasieniec, Östlin, Robson, 2000
O(n3/2)
– Chrobak, Gasieniec, Rytter, 2001,
O(n log2 n)
– Kowalski, Pelc, 2002
O(n log n log D)
Algorithms for Radio Networks
15
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Thanks for your attention!
End of 11th lecture
Next lecture:
Next exercise class:
Next mini exam
We 18 Jan 2006, 4pm, F1.110
Th 19 Jan 2006, 1.15 pm, F2.211 or
Tu 24 Jan 2006, 1.15 pm, F1.110
Mo 13 Feb 2006, 2pm, FU.511
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