Network Routing Capacitycode.ucsd.edu/~jcannons/05CaDoFrZe-DIMACS.pdf1 Network Routing Capacity Jillian Cannons (University of California, San Diego) Randy Dougherty (Center for Communications

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Network Routing Capacity

Jillian Cannons(Universityof California,SanDiego)

Randy Dougherty(Centerfor CommunicationsResearch,La Jolla)

Chris Freiling(CaliforniaStateUniversity, SanBernardino)

KenZeger(Universityof California,SanDiego)

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�Detailed resultsfound in:

� R. Dougherty, C. Freiling,andK. Zeger

“Linearity andSolvability in MulticastNetworks”

IEEE Transactions on Information Theory

vol. 50, no. 10, pp.2243-2256,October2004.

� R. Dougherty, C. Freiling,andK. Zeger

“Insufficiency of LinearCodingin Network InformationFlow”

IEEE Transactions on Information Theory

(submittedFebruary27,2004,revisedJanuary6, 2005).

� J.Cannons,R. Dougherty, C. Freiling,andK. Zeger

“Network RoutingCapacity”

IEEE/ACM Transactions on Networking

(submittedOctober16,2004).

Manuscriptson-lineat: code.ucsd.edu/zeger

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�Definitions

� An alphabet is a finite set.

� A network is afinite d.a.g.with sourcemessagesfrom a fixedalphabetand

messagedemandsatsinknodes.

� A network is degenerate if somesourcemessagecannotreachsomesink

demandingit.

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Definitions - scalar coding

� Eachedgein a network carriesanalphabetsymbol.

� An edge function mapsin-edgesymbolsto anout-edgesymbol.

� A decoding function mapsin-edgesymbolsata sink to a message.

� A solution for a givenalphabetis anassignmentof edgefunctionsanddecoding

functionssuchthatall sinkdemandsaresatisfied.

� A network is solvable if it hasa solutionfor somealphabet.

� A solutionis a routing solution if theoutputof every edgefunctionequalsa

particularoneof its inputs.

� A solutionis a linear solution if theoutputof every edgefunctionis a linear

combinationof its inputs(typically, finite-field alphabetsareassumed).

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Definitions - vector coding

� Eachedgein a network carriesa vectorof alphabetsymbols.

� An edge function mapsin-edgevectorsto anout-edgevector.

� A decoding function mapsin-edgevectorsata sink to amessage.

� A network is vector solvable if it hasasolutionfor somealphabetandsomevector

dimension.

� A solutionis a vector routing solution if every edgefunction’s outputcomponents

arecopiedfrom (fixed)input components.

� A vector linear solution hasedgefunctionswhich arelinearcombinationsof

vectorscarriedon in-edgesto a node,wherethecoefficientsarematrices.

� A vectorroutingsolutionis reducible if it hasat leastonecomponentof an edge

functionwhich,whenremoved,still yieldsa vectorroutingsolution.

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Definitions - � � ��� � fractional coding

� Messagesarevectorsof dimension� .

Eachedgein a network carriesa vectorof atmost � alphabetsymbols.

� A � � � fractional linear solution hasedgefunctionswhich arelinear

combinationsof vectorscarriedon in-edgesto anode,wherethecoefficientsare

rectangularmatrices.

� A � � � fractionalsolutionis a fractional routing solution if every edgefunction’s

outputcomponentsarecopiedfrom (fixed)input components.

� A � � � fractionalroutingsolutionis minimal if it is not reducibleandif no

� � ��� fractionalroutingsolutionexistsfor any � � � .

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Definitions - capacity

� Theratio � � � in a � � � fractionalroutingsolutionis calledan

achievable routing rate of thenetwork.

� Therouting capacity of a network is thequantity

� � �� � � all achievableroutingrates��

� Notethatif a network hasa routingsolution,thentheroutingcapacityof the

network is at least .

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Someprior work

� Somesolvablenetworksdo not have routingsolutions(AhCaLiYe2000).

� Every solvablemulticastnetwork hasa scalarlinearsolutionover somesufficiently large

finite field alphabet(LiYeCa2003).

� If anetwork hasavectorroutingsolution,thenit doesnotnecessarilyhave ascalarlinear

solution(MeEfHoKa2003).

� For multicastnetworks,solvability over a particularalphabetdoesnot imply scalarlinear

solvability over thesamealphabet(RaLe,MeEfHoKa,Ri 2003,DoFrZe2004).

� For non-multicastnetworks,solvability doesnot imply vectorlinearsolvability

(DoFrZe2004).

� For somenetworks,thesizeof thealphabetneededfor a solutioncanbesignificantly

reducedusingfractionalcoding(RaLe2004).

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�Our results

� Routingcapacitydefinition.

� Routingcapacityof examplenetworks.

� Routingcapacityis alwaysachievable.

� Routingcapacityis alwaysrational.

� Every positive rationalnumberis theroutingcapacityof somesolvablenetwork.

� An algorithmfor determiningtheroutingcapacity.

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�Somefacts

� Solvablenetworksmayor maynot have routingsolutions.

� Every non-degeneratenetwork hasa � � � fractionalroutingsolutionfor some�and � (e.g.take � � and � equalto thenumberof messagesin thenetwork).

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Example of routing capacity

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4

5

2 3

6 7

x, y

x, y x, y

This network has a linear coding solution but no

routingsolution.

Each of the � � messagecomponents must be

carried on at least two of the edges � � ��� � � ��� , � � �� .

Hence,� � � � � � , andso � � � � .

Now, we will exhibit a � � � fractional routing

solution...

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Exampleof routing capacity continued...

y3

y2x3

x2

x3

x2 y3

y2

x1x2x3y

1

y1

y2y3x1

x1x2x3y

1

y1

y2y3x1

1

4

5

2 3

6 7

x, y

x, y x, y

Let � � � and � � � .

This is a fractionalroutingsolution.

Thus, � � � is anachievableroutingrate,so � � � � � .

Therefore,theroutingcapacityis � � � � � .

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Example of routing capacity

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3

4

65x, yx, y x, yx, y

x y

Theonly way to get � to � � is � � � � � � � � � � � .

Theonly way to get � to � is � � � � � � � � � � .

� � � � musthave enoughcapacityfor bothmessages.

Hence,� � � , so � � � .

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Exampleof routing capacity continued...

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3

4

65x, yx, y x, yx, y

x y

xyx

x y

y

y x

Let � � and � � � .

This is a fractionalroutingsolution.

Thus, � � is anachievableroutingrate,so � � � � .

Therefore,theroutingcapacityis � � � � .

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Example of routing capacity

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3 54

96 7 8

, ba

, db, cb, da, ca

, dc

This network is dueto R. Koetter.

Eachsourcemustemit at least � � componentsandthe

total capacityof eachsource’s two out-edgesis � � .

Thus, � � � � , yielding � .

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Exampleof routing capacity continued...

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3 54

96 7 8b1b2c1c2

b2b1 c2

a2

a1

b1b2

c1c2d1d2

d1d2

c1a2a1

d2a2

d2

b1

b1

c1

c1a2

a1a2d1d2

b1b2d1d2

a1a2c1c2

Let � � � and � � � .

This is a fractionalroutingsolution

(asgivenin MeEfHoKa,2003).

Thus, � � � is anachievableroutingrate,so � � .

Therefore,theroutingcapacityis � � .

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Example of routing capacity

(1),x x m( )... ,

N+2(1),x x m( )... ,

N+1N32

1

(1),x x m( )

II

...

...

...

...

... ,IN

+1+N

Eachnodein the3rd layerreceivesauniquesetof � edgesfrom the2ndlayer.

Every subsetof � nodesin layer2 mustreceive all � � messagecomponentsfrom the

source.Thus,eachof the � � messagecomponentsmustappearat least � � � � � timeson the � out-edgesof thesource.Sincethetotal numberof symbolson the �

sourceout-edgesis � � , we musthave � � � � � � � � � � or equivalently

� � � � � � � � � � � � . Hence, � � � � � � � � � � .

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Exampleof routing capacity continued...

(1),x x m( )... ,

N+2(1),x x m( )... ,

N+1N32

1

(1),x x m( )

II

...

...

...

...

... ,IN

+1+N

Let � � � and � � � � � � � � Thereis a fractionalroutingsolutionwith theseparameters(theproof is somewhatinvolvedandwill beskippedhere).

Therefore,� � � � � � � � � is anachievableroutingrate,so

� � � � � � � � � � � .

Therefore,theroutingcapacityis � � � � � � � � � � � .

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(1),x x m( )... ,

N+2(1),x x m( )... ,

N+1N32

1

(1),x x m( )

II

...

...

...

...

... ,IN

+1+N

Somespecialcasesof thenetwork:

� � � � ��� � �� , � (AhRi 2004)

No binaryscalarlinearsolutionexist. It hasa non-linearbinaryscalarsolutionusinga � � � �� � � �Nordstrom-Robinsonerrorcorrectingcode.We computethattheroutingcapacityis � � � � � � .

� � � � ��� � � , � � (RaLe2003)

Thenetwork is solvable,if thealphabetsizeis at leastequalto thesquareroot of thenumberof sinks.

We computethattheroutingcapacityis � � � �� � ��� � � � .

� � � � ,� � � �Illustratesthatthenetwork’s routingcapacitycanbegreaterthan1. We obtain � � � � .

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21

3

4

65x, yx, y x, yx, y

x y For eachmessage� , a directedsubgraphof � is an

� -tree if it has exactly one directedpath from the

sourceemitting � to each destinationnode which

demands� , and the subgraphis minimal with re-

spectto thisproperty(similar to directedSteinertrees).

Let � � � � � � � beall such � -treesof a network.

e.g.,thisnetwork hastwo � -treesandtwo � -trees:

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4

65x, yx, y x, yx, y

1

x

3

4

65x, yx, y x, yx, y

1

x

3

4

65x, yx, y x, yx, y

2

y

3

4

65x, yx, y x, yx, y

2

y

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Definethefollowing index sets:

� � � � ��� � � � is an � -tree�� � � � ��� � � � containsedge� ��

Denotethetotal numberof trees� � by � .For agivennetwork, wecall thefollowing 4conditionsthenetwork inequalities:

� �� � � � � ��� � � �

� �� �� � � � �� � ��

� �

� � �where � � � � � � arerealvariables.If asolution � � � � � � � to thenetwork

inequalitieshasall rationalcomponents,thenit is saidto bea rational solution.

( � � representsthenumberof messagecomponentscarriedby � � .)

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Lemma: If a non-degeneratenetwork hasaminimal fractionalroutingsolutionwith

achievableroutingrate � � � , thenthenetwork inequalitieshave arationalsolution

with � � � � .

Lemma: If thenetwork inequalitiescorrespondingto a non-degeneratenetwork have a

rationalsolutionwith � � � , thenthereexistsa fractionalroutingsolutionwith

achievableroutingrate � � .

By formulatinga linearprogrammingproblem,we obtain:

Theorem: Theroutingcapacityof every non-degeneratenetwork is achievable.

Theorem: Theroutingcapacityof every network is rational.

Theorem: Thereexistsanalgorithmfor determiningthenetwork routingcapacity.

Theorem: For eachrational � � � thereexistsasolvablenetwork whoserouting

capacityis � .

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Network Coding Capacity

� Thecoding capacity is

� � � � � � � � � �� � � � fractionalcodingsolution� �

� routingcapacity linearcodingcapacity codingcapacity

� Routingcapacityis independentof alphabetsize.

Linearcodingcapacityis not independentof alphabetsize.

� Theorem: Thecodingcapacityof a network is independentof thealphabetused.

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TheEnd.