faster communication in known topology radio networksctag/seminars/200506/leszek-comm.pdfleszek...
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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
11/3/05 CTAG Seminar 2
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
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11/3/05 CTAG Seminar 3
Radio network parameters
! n number of nodes (processing units) in G
! ! max-degree in G
! D diameter of G
!D
n
11/3/05 CTAG Seminar 4
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
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11/3/05 CTAG Seminar 5
Broadcasting
11/3/05 CTAG Seminar 6
Gossiping
! Gathering ! Broadcasting
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11/3/05 CTAG Seminar 7
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 $(n log n) … O(n log2 n) [G!sieniecPotapov, TCS’02]
11/3/05 CTAG Seminar 8
Broadcasting - known topology
! Lower bound
" Graph family of radius 2 !(log2n) [AlonBar-NoyLinialPeleg, JCSS’91]
! Upper bound
" O(D log2n) general graphs [ChlamtacWeinstein, INFOCOM’87]
" O(D logn + log2n) general graphs [KowalskiPelc, APPROX’04]
" O(D+log5n) general graphs [GaberMansour, SODA’95]
" D+O(log4n) general graphs [ElkinKortsarz, SODA’05]
" D+O(log3n) planar graphs [ElkinKortsarz, SODA’05]
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11/3/05 CTAG Seminar 9
Our contribution –
! Gossiping schedule
" O(D +$ 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
11/3/05 CTAG Seminar 10
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#
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11/3/05 CTAG Seminar 11
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
11/3/05 CTAG Seminar 12
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
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11/3/05 CTAG Seminar 13
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+$)log n)
11/3/05 CTAG Seminar 14
Gossiping in Trees - in time O((D+!)log n)
! After the gathering is completed in the rootof the tree in time O((D+$)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)
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11/3/05 CTAG Seminar 15
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
11/3/05 CTAG Seminar 16
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)
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11/3/05 CTAG Seminar 17
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}
11/3/05 CTAG Seminar 18
Construction of the Gathering Tree
The pruning
process
TPGT--> GST
Nodes here have
ranks for good
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11/3/05 CTAG Seminar 19
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)
11/3/05 CTAG Seminar 20
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
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11/3/05 CTAG Seminar 21
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)
11/3/05 CTAG Seminar 22
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)
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11/3/05 CTAG Seminar 23
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)
11/3/05 CTAG Seminar 24
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)
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11/3/05 CTAG Seminar 25
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)
11/3/05 CTAG Seminar 26
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.
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11/3/05 CTAG Seminar 27
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)
11/3/05 CTAG Seminar 28
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)
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11/3/05 CTAG Seminar 29
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
11/3/05 CTAG Seminar 30
Thank you