packing multicast trees philip a. chou sudipta sengupta minghua chen jin li kobayashi workshop on...
TRANSCRIPT
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Packing Multicast Trees
Philip A. Chou
Sudipta Sengupta
Minghua Chen
Jin Li
Kobayashi Workshop on Modeling and Analysis of Computer and Communication Systems, Princeton University, May 9, 2008
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Problem
What is the capacity region What is a transmission scheme that achieves a given
transmission rate vector
Given Directed graph (V,E) with
edge capacities Multiple multicast sessions
{(si,Ti)}, each with Sender si
Receiver set Ti
Transmission rate ri
i
i
2
2
1
1
1 1
1
( ), c e e E
1 | |{achievable ( ,..., )}?Sr rR
1 | |( ,..., ) ?Sr r R
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Butterfly Networkwith Two Unicast Sessions
sender s1
rate r1
sender s2
rate r2
receiver t2
receiver t1
r2
r1
Capacity Region ?
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Network Coding vs Routing
y1
y2
y3
f1(y1,y2,y3)
f2(y1,y2,y3)
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Butterfly Networkwith Two Unicast Sessions
sender s1
rate r1
sender s2
rate r2
receiver t2
receiver t1
a
b
b
a
XOR
XOR
XOR
a+b
r2
r1
Capacity RegionR
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Characterization of Capacity for Acyclic Graphs
[Yan, Yeung, Zhang; ISIT 2007]
( )
( ) ( )
* 2 1
2 11
2 12 |
2 13 |
2 14
2 15
{ : } { : } -- random variables
: is entropic
:
: 0,
: 0, \ ( )
: ( ),
:
N
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S s
N
Out s s
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Out v In v
N
e
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s e
Y Ys S
U Y
U U
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Y s S U e E N
C h h
C h s S
C h v V S T
C h c e e E
C h
h h
h
h
h
h
h
N
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( )|
*1 2 3 4 5
0,S In tt
S
Y U
Y
t T
proj con C C C C C
NR
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Single Unicast Session
r ≤ MinCut(s,t) MinCut(s,t) is achievable, i.e., MaxFlow(s,t) = MinCut(s,t),
by packing edge-disjoint directed paths
s
t
Value ofs-t cut
[Menger; 1927]
Given Directed graph (V,E) with
edge capacities Single unicast session with
Sender s Receiver t Transmission rate r
( ), c e e E
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Single Broadcast Session
r ≤ minvЄV MinCut(s,v)
minvЄV MinCut(s,v) is achievable (“broadcast capacity”)by packing edge-disjoint directed spanning trees
[Edmonds; 1972]
Given Directed graph (V,E) with
edge capacities Single broadcast session with
Sender s Receiver set T=V Transmission rate r
( ), c e e E
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Single Broadcast Session
r ≤ minvЄV MinCut(s,v)
minvЄV MinCut(s,v) is achievable (“broadcast capacity”)by packing edge-disjoint directed spanning trees
[Edmonds; 1972]
Given Directed graph (V,E) with
edge capacities Single broadcast session with
Sender s Receiver set T=V Transmission rate r
( ), c e e E
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Single Multicast Session
r ≤ mintЄT MinCut(s,t)
mintЄT MinCut(s,t) is NOT always achievableby packing edge-disjoint multicast (Steiner) trees
Given Directed graph (V,E) with
edge capacities Single multicast session with
Sender s Receiver set Transmission rate r
T V
( ), c e e E
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Packing Multicast Trees Insufficient to achieve MinCut
optimal routingthroughput = 1
network codingthroughput = 2
a,b
a
a
a
a
bb
b
b
a+b a+b
a+b
,a
,b
[Alswede, Cai, Li, Yeung ; 2000]
mintЄT MinCut(s,t) is always achievableby network coding
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Linear Network CodingSufficient to achieve MinCut Linear network coding is sufficient to achieve
mintЄT MinCut(s,t)
Polynomial time algorithm for finding coefficients
y1
y2
y3
α1y1+ α2y2+ α3y3
β1y1+ β2y2+ β3y3
[Jaggi, Chou, Jain, Effros; Sanders, et al.; 2003][Erez, Feder; 2005]
[Cai, Li, Yeung; 2003] [Koetter and Médard; 2003]
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Linear Network CodingSufficient to achieve MinCut Packing a maximum-rate set of multicast trees is NP hard
Rate gap to mintЄT MinCut(s,t) can be a factor of log n
[Jaggi, Chou, Jain, Effros; Sanders, et al.; 2003]
[Jain, Mahdian, Salavatipour; 2003]
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Linear Network Coding NOTSufficient for Multiple Sessions Linear network coding is NOT generally sufficient to
achieve capacity for multiple sessions
[Dougherty, Freiling, Zeger; 2005]
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P2P Networks
Upload bandwidth is bottleneck
Completely connected
overlay
; ( ), outV c v v V
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P2P Network as Special Case
Networks w/edge capacities
P2P Networks
v
cout(v)
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P2P Network as Set of Networks
Corollary to Edmonds’ Theorem: Given a P2P network with a single broadcast session (i.e., a single
sender and all other nodes as receivers), the maximum throughput is achievable by routing over a set of directed spanning trees
( )P2P network network w/edge capacities
( ) (; , ( , ); , : ) ,( ) ( )e Ou
out outt v
V v V V E e E v Vc v c e c e c v
cout(v) v
c(e)
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Throughput:
Mutualcast
[Li, Chou, Zhang; 2005]
1( )
1outc t
N 2( )
1outc t
N ( )
1out nc t
N ( )
1out Nc t
N 1
( )
1
Nout n
n
c t
N 1
( )( ) /
1
Nout n
outn
c tc s N
N
1
1
( )( ) /
1
( )
1
Nout n
outn
Nout n
n
c tc s N
N
c t
N
Cutset analysis: Throughput achieves mincut bound
t1
s
tN
…t2
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Throughput:
Mutualcast with Helpers
[Li, Chou, Zhang; 2005]
1( )
1outc t
N ( )
1out nc t
N ( )
1out Nc t
N 1
( )
1
Nout n
n
c t
N 1 1
( ) ( )( ) /
1
N Mout n out n
outn n
c t c hc s N
N N
1 1
1 1
( ) ( )( ) /
1
( ) ( )
1
N Mout n out n
outn n
N Mout n out n
n n
c t c hc s N
N N
c t c h
N N
( )out nc h
N1
( )Mout n
n
c h
N
1 1
( ) ( )
1
N Mout n out n
n n
c t c h
N N
t1
s
tN
Cutset analysis: Throughput achieves mincut bound
…
hn
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Given P2P network Source s, receiver set T, helper set H = V T s
Then any rate r C can be achieved by routingover at most |V| depth-1 and depth-2 multicast trees
Capacity is given by
Remarks: Despite existence of helpers, no network coding needed! |V| trees instead of O(|V||V|) Simple, short trees
Mutualcast Theorem
1 1 1 1
( ) ( ) ( ) ( )min ( ), ( ) /
1 1
N M N Mout n out n out n out n
out outn n n n
c t c h c t c hC c s c s N
N N N N
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Given P2P network Multiple sources si and receiver sets Ti, i=1,…,|S|,
s.t. sets (siUTi) are identical or disjoint; no connections between the disjoint receiver sets; H = V U(siUTi)
Then any rate vector can be achievedby routing over at most |V||S| depth-1 and depth-2 multicast trees
Multi-sessionMutualcast Theorem
1 | |( ,..., )Sr r R
[Sengupta, Chen, Chou, Li; ISIT 2008]
H
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Capacity region is given by set of linear inequalities:
Remark: Despite multiple sessions (and helpers), still no inter-
session or intra-session network coding is needed! Still |V| trees instead of O(|V||V|); still simple, short
Multi-sessionMutualcast Theorem
[Sengupta, Chen, Chou, Li; ISIT 2008]
1 | | , 1, ,| |i i V ix x r i S
deg ( ), ij vij outij
x c v v V
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Applicationto Video Conferencing
Maximizing over allcan be found by maximizingover all tree rates xij, i=1,…,|S|, j=1,…,|V|, s.t.
, i=1,…,|S|
Network Utility Maximization problem
( )i iU r 1 | |( ,..., )Sr r R
1 | |i i V ix x r deg ( ), ij ij out
ij
x c v v V
( )i iU r
U1(r1)
U2(r2)
U4(r4)
U5(r5)
U3(r3)
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For general multi-session multicast over directed graphs with edge capacities
Capacity region is stated in terms of * Hard to know if a rate vector r is in capacity region
For restricted case of multi-session multicast with identical or disjoint receiver sets over P2P networks
Capacity region is given by |V| inequalities over at most |V||S| variables (rates on each tree)
Any point in this region is achievable by routing over at most |V||S| depth-1 and depth-2 trees
Summary