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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.
