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

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

Butterfly Networkwith Two Unicast Sessions

sender s1

rate r1

sender s2

rate r2

receiver t2

receiver t1

r2

r1

Capacity Region ?

Network Coding vs Routing

y1

y2

y3

f1(y1,y2,y3)

f2(y1,y2,y3)

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

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

N

S s

N

Out s s

N

Out v In v

N

e

N

s e

Y Ys S

U Y

U U

U

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

N

( )|

*1 2 3 4 5

0,S In tt

S

Y U

Y

t T

proj con C C C C C

NR

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

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

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

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

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

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]

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]

Linear Network Coding NOTSufficient for Multiple Sessions Linear network coding is NOT generally sufficient to

achieve capacity for multiple sessions

[Dougherty, Freiling, Zeger; 2005]

P2P Networks

Upload bandwidth is bottleneck

Completely connected

overlay

; ( ), outV c v v V

P2P Network as Special Case

Networks w/edge capacities

P2P Networks

v

cout(v)

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)

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

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

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

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

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

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)

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