Faster Communication in Known Topology Radio Networksctag/seminars/200506/leszek-comm.pdfLeszek G!sieniec, University of Liverpool, UK!David Peleg, Weizmann Institute, Israel!Qin Xin,

11/3/05 CTAG Seminar 1

Faster Communication in Known

Topology Radio Networks

! Leszek G!sieniec, University of Liverpool, UK

! David Peleg, Weizmann Institute, Israel

! Qin Xin, University of Bergen, UK


Radio networks – communication model

! Undirected graph (of wireless connections)

! Radio communication protocol

! Entire synchronization and

! Complete knowledge of topology is assumed

transmission

conflict 1

conflict 2


Radio network parameters

! n number of nodes (processing units) in G

! ! max-degree in G

! D diameter of G

!D

n


Broadcasting versus Gossiping

! Broadcasting refers to one-to-all communication

! The goal in broadcasting is to disseminate abroadcast message from a distinguished sourcenode to all other nodes in the network

! Gossiping refers to all-to-all communication, a.k.a.total information exchange

! The goal in gossiping is to exchange messageswithin all pairs of nodes (points) in the network

! For each communication primitive the schedule oftransmissions is pre-computed based on the fullsize and knowledge about the topology of thenetwork


Broadcasting


Gossiping

! Gathering ! Broadcasting


Gossiping - known topology

! Deterministic gossiping" Arbitrary messages:

! Worst case topology " min{n,D+D1/i+2!logi+1n}

! Best case topology # log n +2 [G!sieniecPotapovXin, SIROCCO’04]

" Unit messages:

! stars & rings 2n, line 3n, trees ~3.5n,

! general graphs $(n log n) … O(n log2 n) [G!sieniecPotapov, TCS’02]


Broadcasting - known topology

! Lower bound

" Graph family of radius 2 !(log2n) [AlonBar-NoyLinialPeleg, JCSS’91]

! Upper bound

" O(D log2n) general graphs [ChlamtacWeinstein, INFOCOM’87]

" O(D logn + log2n) general graphs [KowalskiPelc, APPROX’04]

" O(D+log5n) general graphs [GaberMansour, SODA’95]

" D+O(log4n) general graphs [ElkinKortsarz, SODA’05]

" D+O(log3n) planar graphs [ElkinKortsarz, SODA’05]


Our contribution –

! Gossiping schedule

" O(D +$ log n) in general graphs! Optimal schedule in graphs with $=O(D / log n)

! Broadcast schedule" D+O(log3n) deterministic construction

" D+O(log2n) expected time randomized algorithm

! Optimal in the view of the lower bound !(D+log2n)

" 3D deterministic construction (planar graphs)

! Can beviewed as 3-approximation procedure


Tree ranking - definition

! The system of ranks in an arbitrary tree T

" Every leaf v in T has rank(v)=1

" A non-leaf node v with children v1,..,vk determines itsrank according to the rank of its childrenrank(v1),..,rank(vk), where rmax is the highest rankamong its children

" And if rmax is unique! then rank(v)= rmax! else rank(v)= rmax +1

! Lemma: in an arbitrary tree of size n the largestrank is bounded by "logn#


Tree ranking - example

1 1 1 1 1 1

1 1 1 12 2

1 1 1 122

22 21 1 1

2 2 2 111

3 2 11 1 1

3


Fast and Slow Transmissions

! Let Lk be the set of nodes placed at distance k

from the root of the tree

! Let Ri be the set of nodes with rank i in the tree

! We define two types of sets of nodes

" The fast transmission set:

! Fik={v | (v%Lk&Ri) and (parent(v) % Ri)}, and Fi='k Fik

" The slow transmission set:

! Sik={v | (v%Lk&Ri) and (parent(v) % Rj), j>i}, and Si='kSik


Gathering in Trees - in time O((D+!)log n)

! For each rank i = 1 to log n do

" Move messages in nodes in Ri to the set Si" Move messages from all nodes in Si to their parents

! The time complexity

" the time required to move messages to Si is bounded by D

" the required to move messages from Si to their parents is

bounded by $

" Altogether, the time required to gather all messages in the

root of the tree is bounded by O((D+$)log n)


Gossiping in Trees - in time O((D+!)log n)

! After the gathering is completed in the rootof the tree in time O((D+$)log n)

! The combined message (including all

individual gossip messages) is delivered to

all other nodes of the tree via naïve

broadcasting in time D-1.

! Lemma: In a known tree with parameters n,D and ! the gossiping can be completed in

time O((D+$)log n)


Gathering in General Graphs

! The gathering algorithm works in 3 stages:

" [1] Build a pre-gathering (BFS) spanning tree TPGT

" [2] Perform the pruning of the pre-gathering tree

leading to a gathering spanning tree GST

" [3] Gather messages along fast and short

transmission sets in ranked tree GST


Pruning process - checking collisions

! Function Check-Collision(i,j): pair of nodes;

" If " u,v # Fji and (u,parent(v)) # E, where u%v

! then return (u,v);

! else return (“null”);

Level i

Level i-1

u v

parent(v)


Construction of the Gathering Tree

! Procedure Gathering-Spanning-Tree(TPGT);" for i = D down to 1 do

! for j = rmax down to 1 do

" while Check-Collision(i,j) ! “null” do

# rank(parent(v)) = j+1;

# Fji = Fj

i - {u,v};

# Sji = Sj

i ' {u,v};

# EPGT = EPGT - {(u,parent(u))};

# EPGT = EPGT ' {(u,parent(v))};

# Re-rank TPGT in levels from i-1 down to 0;

# Re-compute sets in F and S in new TPGT;

! end {Gathering-Spanning-Tree}


Construction of the Gathering Tree

The pruning

process

TPGT--> GST

Nodes here have

ranks for good


Gathering in General Graphs - in timeO((D+!)log n)

! Use ranked GST to gather all messages

! For each rank i = 1 to log n do

" Move messages in nodes in Ri to the set Si

" Move messages from all nodes in Si to their parents

! The time complexity

" the time required to move messages to Si is bounded by 3D

(time multiplexing between consecutive BFS levels is required)

" the required to move messages from Si to their parents isbounded by $ (this is done with a help of minimal covering setsin bipartite graphs of degree $)

" Altogether, the time required to gather all messages in the rootof the tree is bounded by O((D+$)log n)


Gathering in General Graphs - in timeO(D+!log n)

! The gathering process can be sped up if the pattern oftransmissions of a node v at layer i with rank j in GSTis as follows" if v%F, then v transmits at time (D-i)+j·$

" Otherwise, v transmits at time (D-i)+j·$+s(v), where the value1 " s(v) " $ is determined by the use of a particular minimalcovering set

! Note that node v transmits when all its descendantsalready delivered their messages

! There is also no collision between transmissionscoming from nodes with same ranks as well asdifferent ranks


Gathering in General Graphs - in timeO(D+!log n)

O(log n) slow transmissions

O(!)

O(!)

O(!)

O(!)

O(!)

! Lemma: The gathering in arbitrary graphs canbe completed in time O(D+ $log n)


Gossiping in General Graphs - in timeO(D+!log n)

! After the gathering process is completed intime O(D+ $log n)

! The combined message is distributed byreversing direction of transmissions “along”the edges of the GST, all done with the sametime complexity

! Theorem: There exists efficient construction ingraphs with parameters n, D and ! of thegossiping schedule requiring time O(D+$log n)


Broadcasting in General Graphs

! But can the dissemination process be faster?

! I.e., with the time independent from !? Thiswould be useful in graphs with small max-degree and more consistent with previousresults in the field

! The answer is YES!

! We show efficient and explicit construction ofa broadcast schedule with time D+O(log3n)and we prove the existence of a deterministicbroadcast schedule with time D+O(log2n)


Broadcasting in General Graphs

! Note that “reversed” fast transmissions

(between nodes in F) are also collision free

" this is a simple consequence of the fact that after

the pruning process there isn’t any crossing edgebetween a node v%Fi

j and parent(u), for any u%Fij

! The slow transmissions are implemented

with a help of Chlamtac and Weinstein

procedure CW that informs second partition

of a bipartite graph of size n in time O(log2n)


Broadcasting in General Graphs - intime D+O(log3n)

O(log n) slow transmissions

O(log2n)

O(log2n)

O(log2n)

O(log2n)

O(log2n)

! Theorem: There exists efficient construction ingraphs with parameters n and D of a broadcastingschedule requiring time D+O(log3n)


Randomized Broadcasting

! In randomized algorithm we replace the mechanism

(CW procedure) of slow transmissions by a

probabilistic procedure RCW

! During execution of RCW each participating node instep 1" i " "logn# decides to transmit the messagerandomly and uniformly with probability 1/2i

! Lemma: From the moment the parent (with higher

rank) of a node v is informed the node v gets the

broadcast message (success) during each execution

of one instance of RCW with probability p>1/4e>0.


Randomized Broadcasting - in timeD+O(log2n)

! Note that on each path from the root of GST to anyleaf we need O(log n) successes during slowtransmissions

! Using RCW procedure this can be achieved with ahelp of O(log n) instances of RCW with highprobability

! Lemma: there exists a randomized algorithm thatfor any graph of size n broadcasts a message fromany node with high probability in time D+O(log2n)

! Theorem: There exists a broadcasting schedulerequiring time D+O(log2n)


Broadcasting in Planar Graphs

! Efficient construction of an asymptotically

optimal broadcast schedule with at most 3D

time steps

! In the view of the obvious lower bound D our

result can be seen as an approximation

algorithm with the approximation ratio 3

(note that the construction of an optimal

broadcast schedule is NP-hard even for

planar graphs)


Open problems

! More general time efficient radio gossiping

schedule is still unknown. We have no good

upper and lower bounds for $ >> D/logn

! Efficient construction of the O(D+log2n)-time

radio broadcasting schedule

! A broadcasting schedule in planar graphs

with better approximation ratio


Thank you

Documents

Faster Communication in Known Topology Radio Networksctag/seminars/200506/leszek-comm.pdfLeszek G!sieniec, University of Liverpool, UK!David Peleg, Weizmann Institute, Israel!Qin Xin,