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1
Index Coding
Part II of tutorialNetCod 2013
Michael LangbergOpen University of Israel
Caltech (sabbatical)
OutlineThis part of tutorial:
•Will show an equivalence between the network coding and index coding problems.
• Outline:• Preliminary: Network Coding model.
• Preliminary: Index Coding model.
• Equivalence for linear encoding/decoding [ElRouayhebSprintsonGeorghiades].
• Equivalence for general encoding/decoding [EffrosElRouayhebLangberg].
•Multicast vs. Unicast Index Coding [MalekiCadambeJafar].
•Open Questions.
2
General theme
Will show an equivalence between the network coding and index coding problems.
• An efficient reduction that allows to solve NC using any scheme to solve IC.
3
s1
t2t1
t3
s2
s1 s2 s3 s4 s5 s6
t1 t2 t3 t4 t5 t6
Solve ICObtain solution to NC
NC IC
• Network communication challenging: combines topology with information.
• Reduction separates information from topology.
• Significantly simplifies the study of Network Comm.
• Index Coding is a simple but representative instance of general network communication.
• Directed acyclic network N.• Edge e has capacity ce.• Source vertices S.• Terminal vertices T.• Requirement matrix:
• Transfer information from S to T.
• Objective: • Information flow using Network Coding that satisfies
terminals.
4
General Network Codings1
t2t1
t3
s2
s3
5
Assumptions
• Sources Si hold independent information.• Zero error in communication.
• We consider the multiple unicast communication requirement (w.l.o.g. [DoughertyZeger]):• k source/terminal pairs (Si,Ti) that wish to communicate over
N.
NS2
S1
S4
S3
T2
T1
T4
T3
S1
t2t1
S2
6
NC preliminaries
Communication at rate R = (R1,…,Rk) is achievable
over instance NC with block length n if random variables {Si},{Xe}:
• Rate: Source Si = R.V. independent and uniform over [2Rin].
• Edge capacity: For each edge e: Xe = R.V. with support [2cen].
• Functionality: for each edge e we have fe = function from
incoming R.V.’s Xe1,…,Xe,in(e) to Xe (i.e., Xe=fe(Xe1,…,Xe,in(e))).
• Decoding: for each terminal ti we define
a decoding function yielding sources Si reqired.
• R=(R1,…Rk) is ”n-feasible” if code with block length
n.
• Alternatively we say that NC is (R,n)-feasible.
s2
s1
s4
s3
t2
t1
t4
t3
X1
X2
X3
Xe
fe
Index Coding [Birk,Bar-Yossef et al.]
• IC is a special case of NC
• A set S of sources.
• A set T of terminals.
• Each terminal has some subset of sources (as side info.) and wants some subset of sources.
• Broadcast link has capacity cB.
•Other links have unlimited cap.
• Objective: To satisfy all terminals. using broadcast rate cB.
s1 s2 s3 s4
t1 t2 t3 t4
cB
Index Coding Communication at rate R = (R1,…,Rk) is achievable
with block length n if random variables {Si},XB:
• Rate: Source Si = R.V. independent and uniform over [2Rin].
• Encoding: XB = fB(S1,…,Sk) is R.V. with support [2cBn].
• Decoding: for each terminal ti we define a decoding function gi taking as input the broadcasted message XB and the side information of ti; and returning the sources Si wanted by ti.
• R=(R1,…Rk) is ”n-feasible” if
code with block length n.
• Will use notation: IC is (R,n)-feasible.
s1 s2 s3 s4
t1 t2 t3 t4
IC is a simple instance of the NC problem: only a single encoding node.
cB
Connecting NC to IC
• Step 1: Need to define reduction from NC to IC.
• Step 2: Need to proveNC is (R,n)-feasible iff IC is (R’,n)-feasible.
• Would like: Reduction/code const. to be very efficient. 9
s1
t2t1
s2
s1 s2 s3 s4 s5 s6
t1 t2 t3 t4 t5 t6
Solve ICObtain solution to NC
NC IC
10
Outline
• Step 1: Present reduction from NC to IC.
• Step 2: Equivalence for linear and general encoding/decoding. [ElRouayhebSprintsonGeorghiades],
[EffrosElRouayhebLangberg].
s1
t2t1
s2
s1 s2 s3 s4 s5 s6
t1 t2 t3 t4 t5 t6
NC IC
Theorem: For any NC, R one can construct IC, R’ such that for any n: NC is (R,n)-feasible iff IC is (R’,n)-feasible.
The reductionNC sources
Network: edges
NC terminals
NC sources NC edges
NC term. NC edges
NC IC
•Index Coding instance:•Sources corresponding to NC sources, and NC edges.
•Terminals corresponding to NC term., NC edges, special terminal.
•For edge e: terminal te in IC wants IC source Xe and has as side information all IC sources incoming to e in NC.
IC encodes topology of NC in its terminals!
X1
X2
X3
Xe
The reduction in more detail
NC sources
Network: edges
NC terminals
NC sources NC edges
NC term. NC edges
NC IC
•Sources: |S|+|E| sources, one for each source of NC and one for each edge of NC: {Si’} and {Se’}.
•Terminals: |T|+|E|+1 terminals:•One terminal ti’ for each ti: wants Si’ and has {Se’} for e in In(ti).
• te’ for each edge e: wants Se’ and has {Sa’} for edge a in In(e).
•One special terminal tall: wants {Se’} and has {Si’}.
X1
X2
X3
ti
The reduction in more detail
NC sources
Network: edges
NC terminals
NC sources NC edges
NC term. NC edges
•Sources: |S|+|E| sources, one for each source of NC and one for each edge of NC: {Si’} and {Se’}.
•Terminals: |T|+|E|+1 terminals:
•One for each terminal ti: wants Si’ and has {Se’} for e in In(ti).
•One for each edge e: wants Se’ and has {Sa’} for edges a in In(e).
•One special terminal tall: wants {Se’} and has {Si’}.
•Bottle neck edge of capacity cB=ce.
•Given rate vector R=(R1,…,Rk) we construct rate vector R’=({Ri’};{Re’}):
•Ri’=Ri and Re’=ce.
Has Wants
ti’ {Se’} for e in In(ti).
Si’
te’ {Sa’} for a in In(e).
Se’
tall {Si’} {Se’} Theorem: For any NC, R one can construct IC, R’ such that for any n: NC is (R,n)-feasible iff IC is (R’,n)-feasible.
Reduction
NC IC
Sources S1,…,Sk {Si’}, {Se’}
Terminals t1,…,tk {ti’},{te’},tall
Capacities ce cB=ce
Rate R1,…,Rk {Ri’}, {Re’}Ri’=Ri
Re’=ce
X1
X2
X3
ti
Theorem
•Theorem: NC is (R,n)-feasible iff IC is (R’,n)-feasible.
NC sources
NC edges
NC terminals
NC sources NC edges
NC term. NC edges
Reduction
NC IC
Sources S1,…,Sk {Si’}, {Se’}
Terminals t1,…,tk {ti’},{te’},tall
Capacities
ce cB=ce
Rate R1,…,Rk
{Ri’}, {Re’}Ri’=Ri
Re’=ce
s1 s2 s3 s4 s5 s6
t1 t2 t3 t4 t5 t6
s1
t2t1
s2Has Wants
ti’ {Se’} for e in In(ti). Si’
te’ {Sa’} for a in In(e). Se’
tall {Si’} {Se’}
Geometric view
•Theorem: NC is (R,n)-feasible iff IC is (R’,n)-feasible.
NC sources
NC edges
NC terminals
NC sources NC edges
NC term. NC edges
Reduction
NC IC
Sources S1,…,Sk {Si’}, {Se’}
Terminals t1,…,tk {ti’},{te’},tall
Capacities
ce cB=ce
Rate R1,…,Rk
{Ri’}, {Re’}Ri’=Ri
Re’=ce
s1 s2 s3 s4 s5 s6
t1 t2 t3 t4 t5 t6
s1
t2t1
s2
e: Re’=ce
What now?
Outline:
• NC feasible implies IC feasible (works for both linear and non-linear).
• IC feasible implies NC feasible (will show new proof for linear that modifies to non linear).
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Theorem: For any NC, R one can construct IC, R’ such that for any n: NC is (R,n)-feasible iff IC is (R’,n)-feasible.
• Use global NC encoding functions.
• Seen that Xe = fe(XIn(e)).
• Edge e also has a function Fe:
• Xe = Fe(S1,…,Sk).
• IC: We need to define XB of rate cB.
• Recall that cB=Σce.
• Recall that Xe of rate ce.
• For all e let XB(e)=Se’+Fe(S1’,…,Sk’).
• There is a separation between {Si’} and {Se’}.
• Lets see that this works (decoding …).17
NC ICReduction
NC IC
Sources S1,…,Sk {Si’}, {Se’}
Terminals t1,…,tk {ti’},{te’},tall
Capacities
ce cB=ce
Rate R1,…,Rk {Ri’}, {Re’}Ri’=Ri
Re’=ceHas Wants
ti’ {Se’} for e in In(ti).
Si’
te’ {Sa’} for a in In(e).
Se’
tall{Si’} {Se’}
Theorem: For any NC, R one can construct IC, R’ such that for any n: NC is (R,n)-feasible iff IC is (R’,n)-feasible.
•Rate: Source Si = R.V. independent and uniform over
[2Rin].
•Edge capacity: For each edge e: Xe = R.V. with support
[2cen].
•Functionality: for each edge e we have fe = function
from incoming R.V.’s Xe1,…,Xe,in(e) to Xe (i.e., Xe=fe(Xe1,
…,Xe,in(e))).
•Decoding: for each terminal ti we define a decoding function yielding sources Si required.
• Use global encoding functions of NC.
• Each edge e has a function Fe such that Fe(S1,…,Sk)=Xe.
• Recall Xe of support [2cen].
• Recall cB=Σce.
• We need to define XB of total support [2cBn].
• Let XB(e)=Se’+Fe(S1’,…,Sk’).
• Decoding:• Consider terminal te’: wants Se’ and has {Sa’} for edges a in
In(e).
• te’ also receives the broadcast XB.
• For each a compute XB(a)-Sa’ = Sa’+Fa(S1’,…,Sk’)-Sa’ = Fa(S1’,…,Sk’).
• Use local encoding function fe to compute:
fe(Fa1(S1’,…,Sk’),…, Fa3(S1’,…,Sk’)) = Fe(S1’,…,Sk’)
• Compute XB(e)-Fe(S1’,…,Sk’) = Se’+Fe(S1’,…,Sk’)-Fe(S1’,…,Sk’) = Se’.
• Same process for other terminals.
18
NC ICReduction
NC IC
Sources S1,…,Sk {Si’}, {Se’}
Terminals t1,…,tk {ti’},{te’},tall
Capacities
ce cB=ce
Rate R1,…,Rk {Ri’}, {Re’}Ri’=Ri
Re’=ce
Xa1
Xa2
Xa3
Xe
Has Wants
ti’ {Se’} for e in In(ti).
Si’
te’ {Sa’} for a in In(e).
Se’
tall{Si’} {Se’}
fe(Xa1,Xa2,Xa3)=Xe
te’ will simulate the NC solution on edge e.
Basic idea: simulate the NC solution!
What now?
Outline:
• NC feasible implies IC feasible (works for both linear and non-linear).
• IC feasible implies NC feasible (will show new proof for linear that modifies to non linear).
19
20
• Given a linear code for IC, how do we build one for NC?
• Encoding for IC includes a linear encoding function fB
fB({Si’}, {Se’}) =
• Can prove that AE is square and full rank.
• Crucial property:
• Fix any value sI’ for SI’=S1’,…,Sk’
• There exists unique value sE’ for SE’=Se1’,…,Sem’ such that fB(sI’, sE’) =0.
• This will allow the construction of a NC!
AS AE+
Linear: IC NCReduction
NC IC
Sources S1,…,Sk {Si’}, {Se’}
Terminals T1,…,Tk {ti’},{te’},tall
Capacities
ce cB=ce
Rate R1,…,Rk {Ri’}, {Re’}Ri’=Ri
Re’=ce
tall only has {Si’} and wants all {Se’}
Has Wants
ti’ {Se’} for e in In(ti).
Si’
te’ {Sa’} for a in In(e).
Se’
tall{Si’} {Se’}
•Rate: Sources S’ = ({Si’},{Se’}) of support [2Ri’n], [2Re’n].
•Bottleneck: XB = fB({Si’},{Se’}) of support [2cBn].
21
fB({Si’}, {Se’}) =
• AE is square and full rank.
• Crucial property: For all sI’ exists sE’ s.t. fB(sI’, sE’) =0.
AS AE+
Linear: IC NCReduction
NC IC
Sources S1,…,Sk {Si’}, {Se’}
Terminals T1,…,Tk {ti’},{te’},tall
Capacities
ce cB=ce
Rate R1,…,Rk {Ri’}, {Re’}Ri’=Ri
Re’=ce
sI’
sE’
fB(sI’, sE’)
Value = 0Will define NC by ‘projecting’ fB onto the white curve!
Has Wants
ti’ {Se’} for e in In(ti).
Si’
te’ {Sa’} for a in In(e).
Se’
tall{Si’} {Se’}
22
• Consider edge e in NC.
• We will define local encoding function fe(Xa1,Xa2,Xa3)=Xe.
• Will define fe based on decoding ge’ function of IC for terminal te’.
ge’(Sa1’,Sa2’,Sa2’, fB({Si’}, {Se’}))=Se’.
• fe(Xa1,Xa2,Xa3)=ge‘(Xa1,Xa2,Xa3, 0).
• fe is a valid local encoding function.
• NC decoding defined similarly.
Xa1
Xa2
Xa3
Xe
Linear: IC NCReduction
NC IC
Sources S1,…,Sk {Si’}, {Se’}
Terminals T1,…,Tk {ti’},{te’},tall
Capacities
ce cB=ce
Rate R1,…,Rk {Ri’}, {Re’}Ri’=Ri
Re’=ce
sI’
sE’
Value = 0
• Consider terminal i in NC.
• We need to define local decoding function gi(Xa1,Xa2,Xa3)=Si.
• Will define gi based on decoding gi’ function of IC for terminal ti’.
gi‘(Sa1’,Sa2’,Sa2’, fB({Si’}, {Se’}))=Si’.
• gi(Xa1,Xa2,Xa3)=gi‘(Xa1,Xa2,Xa3, 0).
• Recall: fe(Xa1,Xa2,Xa3)=ge‘(Xa1,Xa2,Xa3, 0).
• Both fe and gi are valid encoding/decoding functions.
• Need to prove correct decoding!23
Xa1
Xa2
Xa3
Si
Linear: IC NCReduction
NC IC
Sources S1,…,Sk {Si’}, {Se’}
Terminals T1,…,Tk {ti’},{te’},tall
Capacities
ce cb=ce
Rate R1,…,Rk {Ri’}, {Re’}Ri’=Ri
Re’=ce
sI’
sE’
Value = 0
Has Wants
ti’ {Se’} for e in In(ti).
Si’
te’ {Sa’} for a in In(e).
Se’
tall{Si’} {Se’}
24
• fe(Xa1,Xa2,Xa3)=ge’(Xa1,Xa2,Xa3, 0).
• gi(Xa1,Xa2,Xa3)=gi’(Xa1,Xa2,Xa3, 0).
• Consider source info sI=sI’=s1,…,sk
• Let sE’ be corresponding value on curve.
• Will show by induction that running NC on input sI corresponds to running IC on input (sI’, sE’).
• Inductive claim: information xe in NC is exactly se’.
• Now for decoding si at terminal i use gi
Xa1
Xa2
Xa3
Si/Sa
Decoding: IC NC
Reduction
NC IC
Sources S1,…,Sk {Si’}, {Se’}
Terminals T1,…,Tk {Ti’},{Te’},Tall
Capacities
ce cE=ce
Rate R1,…,Rk {Ri’}, {Re’}Ri’=Ri
Re’=ce
sI’
sE’
Value = 0
xe=fe(xa1,xa2,xa3) =
ge’(xa1,xa2,xa3, 0) =
ge’(xa1,xa2,xa3, fB(sI’,sE’)) =
ge’(sa1’,sa2’,sa3’, fB(sI’,sE’)) = se’Decoderi=gi(xa1,xa2,xa3) =
gi’(xa1,xa2,xa3, 0) =
gi’(sa1’,sa2’,sa3’, fB(sI’,sE’)) = si’ = si
Basic idea: NC is simulating the IC solution!
We get a valid NC!
What now?
Outline:
• NC feasible implies IC feasible (works for both linear and non-linear).
• IC feasible implies NC feasible (will show new proof for linear that modifies to non linear).
25
• We use exact same proof!
• Where did we use linearity?Crucial property: For all sI’ exists sE’ s.t. fB(sI’, sE’) =0.
• Need to prove property for general encoding functions.
• Property follows from terminal tall.
• Given SI’ and XB=fB(SI’, SE’) we must be able to decode SE’
• Thus fixing sI’, fB is 1-1 as a function of SE’
• Support of XB equals support of SE’.
• Each row is a permutation.
• Thus property holds!26
General: IC NC
Reduction
NC IC
Sources S1,…,Sk {Si’}, {Se’}
Terminals T1,…,Tk {ti’},{te’},tall
Capacities
ce cB=ce
Rate R1,…,Rk {Ri’}, {Re’}Ri’=Ri
Re’=ce
sI’
sE’
Value = 0
Has Wants
ti’ {Se’} for e in In(ti).
Si’
te’ {Sa’} for a in In(e).
Se’
tall{Si’} {Se’}
differ
Number of different XB values is exactly equal to number of different SE’ values.
What now?
Outline:
• NC feasible implies IC feasible (works for both linear and non-linear).
• IC feasible implies NC feasible (will show new proof for linear that modifies to non linear).
• Multicast IC can be represented by Unicast IC (linear only) [MalekiCadambeJafar].
27
• In previous reduction we use an IC instance which “multicasts” information to different terminals.• Same information is wanted by more than one terminal.
• In NC, any (multiple) multicast can be reduced to (multiple) unicast [DoughertyZeger].
• Does the same phenomena hold for IC?28
Multicast vs Unicast
Reduction
NC IC
Sources S1,…,Sk {Si’}, {Se’}
Terminals T1,…,Tk {ti’},{te’},tall
Capacities
ce cB=ce
Rate R1,…,Rk {Ri’}, {Re’}Ri’=Ri
Re’=ce
Has Wants
ti’ {Se’} for e in In(ti).
Si’
te’ {Sa’} for a in In(e).
Se’
tall{Si’} {Se’}
• In previous reduction we use an IC instance which “multicasts” information to different terminals.
• Same information is wanted by more than one terminal.
• In NC, any (multiple) multicast can be reduced to (multiple) unicast.
• Does the same phenomena hold for IC?
• Recent work by [MalekiCadambeJafar] show that unicast suffices in case (if restricted to linear encoding/decoding).
• Implies that for linear encoding: NC reduces to (multiple) unicast IC! • Each terminal wants different message.
• Same number of sources and terminals.
• IC can be characterized by side information graph, rather than side information hypergraph.
29
Multicast vs Unicast
Some open problems
• Multicast vs. Unicast for general encoding.• Would be surprising: problems known to be more
difficult in the multiple-multicast setting (e.g., IC via cycle packing [ChaudhryAsadSprintsonLangberg]).
• Capacity: can one determine if rate R is in capacity region of NC via knowledge of capacity region of IC?• Reduction is not robust enough to withhold the closure
operation in the definition of capacity.
• Answer is yes for linear case. Also for co-located NC sources [WongLangbergEffros].
35
e: Re’=ce
Some open problems
• vs zero error in communication:• Does allowing some error increase rate in NC/IC?
• IC there is no advantage [LangbergEffros] to allowing small error in communication … can this extend to NC?
• NC – not known! In NC “no advantage” known for co-located [ChanGrant] [LangbergEffros] and other cases.
• Can we use equivalence between NC and IC?
• Intriguing connections to other problems such as the edge removal problem [HoEffrosJalali].
• Algorithms!: • Wide open … both in NC setting and IC setting …
36
Conclusions
• Network communication challenging: combines topology with information.
• Discussed equivalence between the network coding and index coding problems.
• Reduction separates information from topology.
• Significantly simplifies the study of Network Comm.
• Index Coding is a simple but representative instance of general network communication.
37
Thanks!
NC sources
NC edges
NC terminals
NC sources NC edges
NC term. NC edges