Adaptiveness vs. obliviousness and randomization vs. determinism Dariusz Kowalski University of Connecticut & Warsaw University Andrzej Pelc University

adaptiveness vs. obliviousness and

randomization vs. determinism

Dariusz Kowalski

University of Connecticut & Warsaw University

Andrzej PelcUniversity of Quebec en Outaouais

Time of Radio Broadcasting

Time of Radio Broadcasting 2

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

– every neighbor z w of node v in network G is receiving in step i

Otherwise node v receives nothing


Broadcasting problem

Broadcasting problem: some node, called source, has the message, and transmits it in step 0Goal: all nodes must know the source message

Spontaneous transmission allowed:every node knows global steps, starting step and can transmit since this time

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


Obliviousness and randomization

Algorithms may be adaptive (deterministic or randomized) oblivious

– deterministic - the sequence of transmissions of every node is fixed prior the broadcasting

– randomized - the sequence of probabilities of transmission is fixed, for every node, prior the broadcasting


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

radio broadcast. J. of Computer and System Sciences, 1991.[CGLP] B. Chlebus, L. Gasieniec, A. Lingas, A. Pagourtzis:

Oblivious gossiping in ad-hoc radio networks. DIALM, 2001.[CMS-ft] A. Clementi, A. Monti, R. Silvestri: Round robin is optimal

for fault-tolerant broadcasting on wireless networks. ESA, 2001.

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

[CGGPR] B. Chlebus, L. Gasieniec, A. Gibbons, A. Pelc, W. Rytter: Deterministic broadcasting in unknown radio networks. Distributed Computing, 2002.

[KP] D. Kowalski, A. Pelc: Broadcasting in undirected ad hoc radio networks. PODC, 2003, to appear.


Randomized Deterministic

Adaptive O(Dlog(n/D)+log2 n) [KP]

(D+ log2 n) [ABLP]

O(n) [CGGPR]

(n) this paper

Oblivious O(n min{D,log n}) this paper

(n) this paper

O(n min{D,n1/2}) this paper

(n min{D,n1/2}) this paper

Goals and results

GOAL: understand impact of obliviousness and/or randomization for broadcasting time


Adaptive deterministic algorithms Algorithm broadcasting in time O(n) in [CGGPR] Lower bound (n) for n-node networks with constant

diameter - correct proof unknown

How to choose S,R to get linear broadcasting time of algorithm A on GS,R

0

1 nS

R

Network GS,R

R {n+1,…,2n}

S {1,…, n}

layer 0

layer 1

layer 2


Lower boundTheorem 1: For every broadcasting algorithm A and every n,

there is a network GS,R on (n) nodes such that broadcasting time of algorithm A on GS,R is (n).

Proof :

We construct sets S,R starting from sets S0 = {1,…,n} and R0

= {n+1,…,2n}. We proceed construction until step n/2 of algorithm A, to obtain sets S = Sn/2 and R = Rn/2 .

Problem : network G is not defined

Solution : – introducing abstract object corresponding to the real ones: history

and transmitters, and preserving theirs required properties

– for constructed network, real and abstract objects are equal


Proof of Theorem 1 - objects

For every step k n/2 define (abstract) objects :

Hk(v) : the history of received messages by the end of step k, for every node v {0,…,2n}

Tk : set of nodes v transmitting in step k under given history Hk(v)

Sk Sk-1 : a subset of {1,…,n} being the output of function MODIFY(Sk-1,T), where T = {T1,…,Tk-1}; initially S0 = {1,…,n}

Rk Rk-1 : a subset of {n+1,…,2n} being the output of function MODIFY(Rk-1,T), where T = {T1,…,Tk-1}; initially R0 = {n+1,…,2n}


Proof of Theorem 1 - construction

Procedure MODIFY(S,T) set stop:=0 while stop = 0 do

– stop:=1

– if there is a set TlT such that |Tl S | = 1 then

• choose such a set with smallest index, say Tk, such that Tk S = {i}

• remove node i from S

• set stop:=0


Proof of Theorem 1 - invariantThe following invariant is preserved after step k of the construction, according to sets Sk and Rk and objects:

– No single transmitter : for every set Tl , l k, |Tl Sk| 1 and |Tl Rk| 1.

– Removed nodes correspond to disjoint transmitters’ sets : At least n-|Sk| sets Tl are disjoint with Sk , and

at least n-|Rk| sets Tl are disjoint with Rk , for l k.

– Large size : |Sk| n-k and |Rk| n-k .

– No message in second layer : if v Rk then Hk(v) is the empty history.


Oblivious randomized algorithms Algorithm broadcasting in time O(n log2 n) in [CGLP] Lower bound (D+ log2 n) follows from [ABLP]

Algorithm Randomized-Oblivious count := 1 repeat N2/log N times

for l := 1 to log N do => iteration of stages

(a.) each node transmits independently with probability 2-l

(b.) node with label count transmits, count := count+1 mod N


Analysis of randomized algorithmTheorem 2: Algorithm Randomized-Oblivious broadcasts in

time O(n min{D,log n}) on any n-node network with diameter D.

Proof : D < log n : broadcasting completed during first nD

executions of instruction (b.), by round-robin property D log n : consider a shortest path v0,…,vk=v from the

source to a node v ; let di+1 be degree of node vi+1 .

Claim: vi receives a message from vi+1 during di+1 consecutive stages with (positive) constant probability.

Since i k di 2n we get expected number O(nlog n) of steps


Lower bound for randomized algorithmsTheorem 3: For every oblivious randomized broadcasting algorithm A and every sufficiently large

n, there exists an n-node network GA of diameter 3, such that the algorithm A requires time (n),

with probability at least 1/2, to complete broadcasting on GA.

Idea of the proof :

Select network GA,v with uniform

probability, among v = 1,…,n-2 .

With probability at least 1/2 node

vn-1 receives a source message

in algorithm A in time (n).

Network GA,v

n-2

1

0 v n-1


Oblivious deterministic algorithms Oblivious algorithm in [CGLP] broadcasts in time O(n3/2) Lower bound (n log D) in [CMS]

Observation : Interleaving algorithm from [CGLP] with round-robin algorithm we obtain algorithm broadcasting in time O(n min{D,n1/2}).

Theorem 4: For all parameters n,D such that 1 < D < n, and for any deterministic oblivious broadcasting scheme A, there exists an n-node network GA of radius D, such that scheme A requires time (n min{D,n1/2}) to broadcast on GA.


Strongly-selective familiesStrongly-selective family [CMS] :

A family F of subsets of R is called (|R|,k)-strongly-selective, for k |R|, if

for every subset Z of R such that |Z| k, and

for every element z Z,

there is a set F F such that Z F = {z}.

Lemma [CMS]: Let F be an (|R|,k)-strongly-selective family (ssf in short). Then (a) if 3 k < (2|R|)1/2 then |F| (k2 log |R|)/(48 log k) ,

(b) if k (2|R|)1/2 then |F| |R| .


Proof of lower bound : case D (n/8)1/2 Suppose sets X0, X1,…, Xi constructed

by step t of scheme A, i < D/2

Let Ri+1 contain

remaining nodes, |Ri+1| > n/2

Consider family Tt+1,…,Tt+n/2 of transmitters in steps t+1,…,t+n/2;it is not (|Ri+1|,n/(2D))-ssf

Define Xi+1 Ri+1 and vi+1 Xi+1 s.t.

Xi+1 Tt+j {vi+1}

|Xi+1| n/(2D)

layer 0

layer 1

layer 2

layer 3

layer D/2+3

layer D/2+4

layer D-1

layer DRemaining nodes

Path

v0

v1

v2

v3X1


Proof of lower bound : case D > (n/8)1/2 Suppose sets X0, X1,…, Xi constructed

by step t of scheme A, i < D’ = n1/2/4

Let Ri+1 contain

remaining nodes, |Ri+1| > n/2

Consider family Tt+1,…,Tt’ of transmitters in steps t+1,…,t’= t+n/2;it is not (|Ri+1|,n/(2D’))-ssf

Define Xi+1 Ri+1 and vi+1 Xi+1 s.t.

Xi+1 Tt+j {vi+1}

|Xi+1| n/(2D’)

layer 0

layer 1

layer 2

layer 3

layer D’/2+3

layer D’/2+4

layer D-1

layer DRemaining nodes

Path

v0

v1

v2

v3X1


Concluding remarksWe analyzed impact of obliviousness and randomization

for radio broadcasting randomization is better than determinism for D such that

log n << D << n adaptive algorithms are faster than oblivious for D s.t.

1 << D << n

Conjecture : randomization is better than determinism for D iff

1 << D << n adaptive algorithms are faster than oblivious for D iff

1 << D

Documents

Adaptiveness vs. obliviousness and randomization vs. determinism Dariusz Kowalski University of Connecticut & Warsaw University Andrzej Pelc University