Broadcasting in Undirected
Ad hoc
Radio Networks
Dariusz Kowalski
University of Connecticut & Warsaw University
Andrzej Pelc
University of Quebec en Outaouais
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
Broadcasting in undirected ad hoc radio networks
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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
Broadcasting in undirected ad hoc radio networks
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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 in undirected ad hoc radio networks
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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
Broadcasting in undirected ad hoc radio networks
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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.
Broadcasting in undirected ad hoc radio networks
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Goals and results
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
O(Dlog(n/D)+log2 n)
Deterministic
this paper O(nlog n)
(Dlog(n/D)+log2 n) [ABLP,KM] (nlog n/log(n/D))
Broadcasting in undirected ad hoc radio networks
this paper
this paper
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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]
Complete-layered
network
0
L1

L2
Lj  {1,…, n}
LD-1
LD
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
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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
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Universal sequence
Remind: 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
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Idea of algorithm
Idea 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
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Algorithm
source transmits
for D:=1 to log N do
for i:=1 to aD 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.
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Complete-layered networks
QUESTION: 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)
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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

L1
0
L2
1
L*1
L3
L4
LD-3
2
LD-2
D/2-1
LD-1
LD
D/2
L*3
L*j  Lj  {D/2+1,…, n} L*D-3
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Idea of selecting worst-case network
Why 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
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Deterministic algorithm

Best known deterministic algorithm broadcasts in time
O(nlog nlog D) [CGR,KP-SIROCCO’03]
(it works also for directed networks)
 Our result: broadcasting time O(nlog 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}
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Description of algorithm
Algorithm
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)
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Conclusions
We 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
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