Index Coding Part II of tutorial NetCod 2013

1. Index CodingPart II of tutorialNetCod 2013 Michael Langberg Open University of Israel Caltech (sabbatical)

2. Outline This 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.

3. General theme Network communication challenging: combines topology withinformation. Reduction separates information from topology. Significantly simplifies the study of Network Comm. Index Coding is a simple but representative instance of general network communication. s1 s2 s1 s2 s3 s4 s5 s6 t3 t2 t1 NC IC t1 t2 t3 t4 t5 t6 Obtain solution to NC Solve IC 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.

4. General Network Coding s3 s1 • 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. s2 t3 t2 t1

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. S1 S2 t2 t1 S1 T1 N S2 T2 S3 T3 S4 T4 5

6. NC preliminaries Communication at rate R = (R1,…,Rk) is achievable over instance NCwith 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 Sireqired. R=(R1,…Rk) is ”n-feasible” if  code with block length n. Alternatively we say that NC is (R,n)-feasible. X1 fe s1 t1 X2 Xe s2 t2 X3 s3 t3 s4 t4 6

7. 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 cB t1 t2 t3 t4

8. 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 cB IC is a simple instance of the NC problem: only a single encoding node. t1 t2 t3 t4

9. Connecting NC to IC s1 s2 NC IC s1 s2 s3 s4 s5 s6 Obtain solution to NC Solve IC t2 t1 t1 t2 t3 t4 t5 t6 Step 1: Need to define reduction from NC to IC. Step 2: Need to prove NC is (R,n)-feasible iffIC is (R’,n)-feasible. Would like: Reduction/code const. to be very efficient.

10. Outline s1 s2 NC IC s1 s2 s3 s4 s5 s6 Theorem: For any NC, R one can construct IC,R’ such that for any n: NC is (R,n)-feasible iffIC is (R’,n)-feasible. t2 t1 t1 t2 t3 t4 t5 t6 Step 1: Present reduction from NC to IC. Step 2: Equivalence for linear and general encoding/decoding. [ElRouayhebSprintsonGeorghiades], [EffrosElRouayhebLangberg].

11. The reduction NC sources NC edges NC sources Network: edges NC IC NC terminals NC edges NC term. X1 • 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! X2 Xe X3

12. The reduction in more detail NC sources NC edges NC sources Network: edges NC IC NC terminals NC edges NC term. X1 • 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 ain In(e). • One special terminal tall: wants {Se’}and has{Si’}. X2 ti X3

13. The reduction in more detail NC edges NC sources NC sources Network: 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 ain 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’=Riand Re’=ce. NC edges NC terminals NC term. X1 X2 ti X3 Theorem: For any NC, R one can construct IC,R’ such that for any n: NC is (R,n)-feasible iffIC is (R’,n)-feasible.

14. Theorem NC sources NC edges NC sources NC edges s1 NC term. s2 NC terminals NC edges s1 s2 s3 s4 s5 s6 t2 t1 t1 t2 t3 t4 t5 t6 Theorem: NC is (R,n)-feasible iffIC is (R’,n)-feasible.

15. Geometric view NC sources NC edges NC sources NC edges s1 NC term. s2 NC terminals NC edges s1 s2 s3 s4 s5 s6 t2 t1 Theorem: NC is (R,n)-feasible iffIC is (R’,n)-feasible. t1 t2 t3 t4 t5 t6 e: Re’=ce

16. What now? Theorem: For any NC, R one can construct IC,R’ such that for any n: NC is (R,n)-feasible iffIC is (R’,n)-feasible. 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).

17. NC IC 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 ewe have fe= function from incoming R.V.’sXe1,…,Xe,in(e)toXe(i.e., Xe=fe(Xe1,…,Xe,in(e))). Decoding: for each terminal tiwe define a decoding function yielding sourcesSirequired. Theorem: For any NC, R one can construct IC,R’ such that for any n: NC is (R,n)-feasible iffIC 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 …).

18. NC IC Basic idea:simulate the NC solution! fe(Xa1,Xa2,Xa3)=Xe Xa1 Xa2 Xe Xa3 te’will simulate the NC solution on edge e. 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 feto 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.

19. 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).

20. Linear: IC NC tall only has {Si’} and wants all {Se’} Rate:Sources S’ = ({Si’},{Se’}) of support [2Ri’n], [2Re’n]. Bottleneck: XB = fB({Si’},{Se’}) of support [2cBn]. AS AE + 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!

21. Linear: IC NC AS AE + sE’ sI’ Value = 0 fB(sI’,sE’) Will define NC by ‘projecting’ fBonto the white curve! fB({Si’}, {Se’})= AE is square and full rank. Crucial property: For all sI’ exists sE’s.t.fB(sI’, sE’)=0.

22. Linear: IC NC Xa1 sE’ Xa2 Xe Xa3 sI’ Value = 0 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). feis a valid local encoding function. NC decoding defined similarly.

23. Linear: IC NC Xa1 sE’ Xa2 Si Xa3 sI’ 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). Bothfe and giare valid encoding/decoding functions. Need to prove correct decoding!

24. 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’ Decoding: IC NC Decoderi=gi(xa1,xa2,xa3) = gi’(xa1,xa2,xa3, 0) = gi’(sa1’,sa2’,sa3’, fB(sI’,sE’)) = si’ = si We get a valid NC! Xa1 sE’ Xa2 Si/Sa Xa3 sI’ Value = 0 Basic idea:NC is simulating the IC solution! 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 sIcorresponds 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

25. 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).

26. General: IC NC Number of different XBvalues is exactly equal to number of different SE’ values. sE’ sI’ Value = 0 differ 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!

27. 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].

28. Multicast vs Unicast 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?

29. Multicast vs Unicast 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.

30. Some open problems e: Re’=ce 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].

31. 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 …

32. Conclusions NC sources NC edges NC sources NC edges • Network communication challenging: combines topology with information. • Discussed equivalencebetween 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. NC term. NC terminals NC edges Thanks! 37