Dariusz Kowalski University of Connecticut & Warsaw University Andrzej Pelc

Dariusz Kowalski

University of Connecticut & Warsaw University

Andrzej PelcUniversity of Quebec en Outaouais

Broadcasting in Undirected

Ad hoc

Radio Networks

Broadcasting in undirected ad hoc radio networks 2

Structure of the presentation Preliminaries

– Model of ad-hoc radio network

– Broadcasting problem - definition and prior work

– Goals and results

Efficient randomized algorithm matching lower bound for randomized algorithms

Complete-layered networks Lower bound for deterministic algorithms Efficient deterministic algorithm based on technique of

solving collision Conclusions


Radio network n nodes with different labels 1,…,N (N=(n))

communicate via radio network modeled by symmetric graph G

node v knows only it own label and parameter N communication is in synchronous steps in every step, node v is either

– transmitting, or– receiving


Message delivery Node v receives a message from node w in step i if

– node v : • is receiving in step i

– node w : • is a neighbor of node v in network G, and

• is transmitting in step i

– node z w :• if z is a neighbor of node v in network G then z is receiving

in step i

Otherwise node v receives nothing


Broadcasting problem

Broadcasting problem: some node, called source, has the message, called

the source message, and transmits it in step 0 every node different than source is receiving until

it receives the source message (no-spontaneous)Goal: all nodes must know the source message

Measure of performance: time by the first step when all nodes have the source message


Bibliography[ABLP] N. Alon, A. Bar-Noy, N. Linial, D. Peleg: A lower bound for

radio broadcast. J. of Computer and System Sciences, 1991.

[BGI] R. Bar-Yehuda, O. Goldreich, A. Itai: On the time complexity of broadcast in radio networks: an exponential gap between determinism and randomization. JCSS, 1992.

[CMS] A. Clementi, A. Monti, R. Silvestri: Selective families, superimposed codes, and broadcasting on unknown radio networks. SODA, 2001.

[CGR] M. Chrobak, L. Gasieniec, W. Rytter: Fast broadcasting and gossiping in radio networks. FOCS, 2000.

[KP] D. Kowalski, A. Pelc: Deterministic broadcasting time in radio networks of unknown topology, FOCS, 2002.

[KM] E. Kushilevitz, Y. Mansour: An (Dlog(n/D)) lower bound for broadcast in radio networks. SIAM J. Comp. 1998.


Goals and results

Randomized Deterministic

O(Dlog(n/D)+log2 n) this paper

(Dlog(n/D)+log2 n) [ABLP,KM]

O(nlog n) this paper

(nlog n/log(n/D)) this paper

GOAL: understand better what are the properties of graphs on which deterministic/randomized broadcasting is time-consuming

RESULT: more advanced property of graphs, which are hard to broadcast by deterministic algorithms, yields

randomization is better


Randomized algorithms - lower bounds Lower bound (Dlog(n/D)) for expected broadcasting

time for n-node networks (complete-layered) with diameter D - proved by Kushilevitz and Mansour [KM]

Lower bound (log2 n) for broadcasting time for n-node networks with constant diameter

proved by Alon et al. [ABLP] even for known network and deterministic algorithms

0

L1

Lj {1,…, n}L2 LD-1 LD

Complete--layerednetwork


Randomized algorithms Randomized algorithm with O(Dlog n + log2 n) expected

broadcasting time introduced by Bar-Yehuda, Goldreich, Itai [BGI]

Our result: algorithm broadcasting in expected time

O(Dlog(n/D) + log2 n)

matching lower bound.

Presentation:– Combinatorial tools : universal sequence

– Idea of construction

– Algorithm and remarks


Universal sequenceRemind: N,D are fixed.

Definition: An infinite sequence (pi)i=1,…, of reals from the interval [0,1] is called universal sequence if the following conditions hold:

for every j = log(N/D)+1, … , log(N/(4 log N)) , the sequence pi+1, pi+2, … , pi+3Dx/N contains at least one value 1/x, where x=2j ;

for every j = log(N/(4 log N))+1, … , log N , the sequence pi+1, pi+2,…, pi+3Dx/(Nlog N) contains at least one value 1/x, where x=2j.

Lemma: There exists universal sequence.

Proof: Idea of construction of universal sequence: – put values 2-j to nodes of the complete binary tree of N leaves according to

some rule

– traverse this tree, writing values of visiting nodes


Idea of algorithmIdea of algorithm (assuming known D): partition into stages, each taking log(N/D) + 2 steps in steps j of stage, for j = 0,1,…,log(N/D) , we want to

assure fast transmission to the node having informed neighbor and of degree close to 2j -

- hence we transmit with probability 2-j

in step j = log(N/D) + 1 of stage i we want to assure fast transmission to the node having informed neighbor and of degree greater than N/D -

- hence we transmit with probability pi according to the universal sequence


Algorithmsource transmits

for D:=1 to log N do

for i:=1 to aD do -- executing stage(D,i)

if node v received the source message before stage(D,i) then

for k=0 to log(N/D) do transmit with probability 2-k transmit with probability pi

Expected broadcasting time: O(Dlog(n/D) + log2 n)

Remark: Complete-layered graphs are among most difficult to

broadcast by randomized algorithms.


Complete-layered networksQUESTION: are complete-layered networks among most

difficult graphs to broadcast by deterministic algorithms?

Clementi, Monti, Silvestri in [CMS] claimed that

every deterministic algorithm needs time (nlog D) to broadcast on some complete-layered graph of n nodes

and diameter D

Claim is wrong, and answer for the QUESTION is NOT (unlike for randomized algorithms)

We showed [KP-STACS’03] deterministic algorithm broadcasting on complete-layered networks in time

O(Dlog(n/D) + log2 n)


Deterministic lower bound For D n1/2 : lower bound (n) claimed in [BGI] and

proved by us is [KP-SIROCCO’03]

In this case Dlog(n/D) + log2 n = o(n) For D > n1/2 we prove lower bound (nlog n / log(n/D))

on star-layered graphs

0

L*1 L*

j Lj {D/2+1,…, n}

L*3

L*D-3

LD-2

1 2 D/2-1 D/2

L1 L2 L3 L4 LDLD-1LD-3


Idea of selecting worst-case networkWhy are complete-layered networks bad? Fast broadcasting using selective-family (see also [CMS]) Fast broadcasting using leader election in every front layer

To construct layer L2j-1 we need in the same time:

Keep size |L2j-1| = O(n/D)

Select set L*2j-1 to assure that node 2j will not receive a

message from set L*2j-1 during (n/D)log D steps after

activation of nodes in L*2j-1

Not allow nodes in layer L2j-1 to receive a message from node 2(j-1) during (n/D)log D steps after activation of nodes in L2j-1


Deterministic algorithm Best known deterministic algorithm broadcasts in time

O(nlog nlog D) [CGR,KP-SIROCCO’03] (it works also for directed networks)

Our result: broadcasting time O(nlog n)

Procedure SELECT(p,o,s) [KP] Using node p and procedure ECHO, node o “asks” if

there exists unvisited neighbor in range {1,…,N/2} O(1) If YES then node o recursively restricts the range of

SELECT from {1,…,N} to {1,…,N/2} If NO then node o recursively restricts the range of

SELECT from {1,…,N} to {N/2+1,…,N}


Description of algorithmAlgorithm

Traverse a DFS tree on network G by a token (source starts): owner of a token transmits O(1) owner selects a successor using SELECT O(log

n) owner sends a token to successor O(1)

Until token in source and no successor selected in SELECT

Length of a DFS-traverse: O(n)

Broadcasting time: O(nlog n)


ConclusionsWe considered problem of broadcasting on radio networks: Randomization is better than determinism Complete-layered networks are among most hard

networks to broadcast by randomized algorithms, but not by deterministic algorithms

Remaining open problem Closing gap between lower and upper bounds on

broadcasting time for deterministic algorithms

Dariusz Kowalski University of Connecticut & Warsaw University Andrzej Pelc

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