Algorithms for Radio Networks Winter Term 2005/2006 21 Dec 2005 10th Lecture

1

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and ComplexityChristian Schindelhauer

Algorithms for Radio NetworksWinter Term 2005/2006

21 Dec 200510th Lecture

Christian Schindelhauer

schindel@upb.de

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