algorithms for radio networks winter term 2005/2006 21 dec 2005 10th lecture
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Algorithms for Radio Networks Winter Term 2005/2006 21 Dec 2005 10th Lecture. Christian Schindelhauer [email protected]. Radio Broadcasting. Broadcasting A sender distributes a message to n radio stations Radio Broadcasting Undirected Graph G=(V,E) describes possible connections - PowerPoint PPT PresentationTRANSCRIPT
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HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Algorithms for Radio NetworksWinter Term 2005/2006
21 Dec 200510th Lecture
Christian Schindelhauer
Algorithms for Radio Networks 2
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Radio Broadcasting
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 3
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Radio Broadcasting without ID
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 4
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
A simple random algorithm (I)
Every station uses the following algorithmSimple-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 endTheorem
For appropriate c>1 we have: Simple-Random informs the complete network with probability of at least 1-O(nk) within time c 2Δ/Δ (D+ log n).
Algorithms for Radio Networks 5
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Extending the Deterministic Model
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 6
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Decay (I)
Idee: randomized thinning out of the players
Decay(k,m)
begin
j ← 1
repeat
j ← j + 1
Send message to all neighborsr ← result of fair coin toss (0/1 with prob. 1/2)
until r=0 oder j > k
end
Algorithms for Radio Networks 7
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Decay (II)
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 8
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
BGI-Broadcast[Bar-Yehuda, Goldreich, Itai 1987]
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 Δt ← 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 9
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Changing the Game: New Models
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 probability1- 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 10
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian SchindelhauerDeterminism versus Probabilism
TheoremFor every distributed deterministic Radio-Broadcasting algorithm using IDs there is a graph with D=2 that cannot be completely informed within time n-2.
TheoremBGI-Broadcast informs all nodes with probability 1- in time O((D+log(n/)) log Δ) for any e>0.
TheoremFor 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 11
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Decay
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 12
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Proof of Lemma (Part I)
P(k,d): 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 13
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Proof of Lemma (Part I)
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 14
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Survey of Randomized Broadcasting Algorithms
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, 1992O((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 15
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Survey of Deterministic Algorithms
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)
16
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityChristian Schindelhauer
Thanks for your attention!End of 11th lectureNext lecture: We 18 Jan 2006, 4pm, F1.110Next exercise class: Th 19 Jan 2006, 1.15 pm, F2.211 or Tu 24 Jan 2006, 1.15 pm, F1.110Next mini exam Mo 13 Feb 2006, 2pm, FU.511