Index Coding & Caching in Wireless Networks. Salim El Rouayheb. ECE IIT, Chicago

Transcription

1 Index Coding & Caching in Wireless Networks Salim El Rouayheb ECE IIT, Chicago

2 Big Data vs. Wireless Exabytes per month [Cisco]

3 Meanwhile, Storage is Getting Cheaper Storage cost per GB (USD)

4 Storage = Caching Index coding: [Birk & Kol 98] + Coded Caching: [Maddah-Ali & Niesen 13] + Femto-caching: [Golrezai et al. 12].... Content is cached (stored) on mobile devices during off-peak hours

5 Index Distributed Coding Storage Example Wants: X 1 Has: X 2 X 3 t 1 Transmission # Index code 1 Index code2 t 4 t 2 1 X 1 X 1 +X 2 2 X 2 X 3 3 X 3 X 4 Wants: X 4 Has: X 1 X 1 X 2 X 3 X 4 t 3 Wants: X 2 Has: X 1 X 3 Wants: X 3 Has: X 2 X 4 4 X 4 L=4" Min transmission rate? " Optimal schemes?" Content of the cache is given Birk & Kol, Informed-source coding-on-demand (ISCOD) over broadcast channels, INFOCOM 98 L=3" Cached data independent of a user s preferences still help

6 Talk Roadmap Graph Theory & Index Coding Rank Minimization & Index Coding Network Coding & Index Coding Privacy Problems

7 Index Coding & Coloring 1 2 X 1 +X X X 4 Side info graph G d Clique cover of G = Chromatic nbr of Ḡ X 1 +X 2 +X 3 +X G 5 X 5 Ḡ 5

8 Index Coding & Graph Coloring 1 2 X 1 +X user 1 caches X 3 3 X X 4 L* min min length of linear index code Finding L* min is NP hard by [R., Sprintson, Chaudhry ITW 07] 4 Clique cover of G= Side info graph G d Chromatic nbr of Ḡ Independ (G) apple c(g) apple L ence nbr min apple d f d (Ḡ) apple (Ḡ) Shannon capacity [Haemers 79 ] Fractional Chromatic nbr [Blasiak, Kleinberg, Lubetzky 11 ] More bounds [Dimakis et al.] [Arbobjalfoei & Kim], [Mazumdar et al.] etc

9 Index Coding on Erdős-Rényi Graphs Independence nbr (G) apple L min apple (Ḡ) When n!1, we have with prob 1 log n apple L min apple n log n G(n,p) Can improve the lower bound [Haviv & Langberg Index Coding on random graphs, ISIT 12 ] c p n apple L min apple Chromatic nbr n log n Recent results closes the gap L min = (n/ log n) [Golovnev, Regev & Weinstein, The Min Rank of Random graphs, Arxiv 16 ]

10 Talk Roadmap Graph Theory & Index Coding Rank Minimization & Index Coding Network Coding & Index Coding Privacy Problems

11 Index Coding & Rank Minimization X 1 X 2 X 3 X 4 X 1 t * 10 * 0 X 2 X 1 +X 2 X 1 +X 2 +X 3 X 3 X 3 t 2 t * 1 * * 1 10 * X 4 X 1 +X 4 X 4 t 4 10 * Linear case: L min =minrk(m) Matrix M [Bar-Yossef et al. '06] Min rank introduced by Haemers in 79 to upper bound the Shannon graph capacity Min rank can be a tighter bound on Shannon capacity then Lovász Theta function.

12 Use Matrix Completion Methods to Construct Index Codes Minimizing nuclear norm [Recht & Candes 09] does not work here because the index coding matrices have a special structure. Try other rank minimization methods [Fazel et al. 04] C D Two problems: 1) Regions not convex 2) Optimization over the reals Index coding via AP Theorem: [Alternating Projections (AP)] If C and D are convex, then an alternating projection sequence between these 2 regions converges to a point in their intersection.

13 Alternating Projection on Random Undirected Graphs [Huang & R. Index Coding via Rank minimization ITW 15] Caching prob. p Up to 13% savings over Greedy coloring. No theoretical guarantees. Recent work on min rank over finite field [Sauderson, Fazel, Hassibi ISIT 16] Index coding via LP [Blasiak et al. 10], via SDP [Chlamtac et al 14]

14 Performance with Increasing Number of Users

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16 Talk Roadmap Graph Theory & Index Coding Rank Minimization & Index Coding Network Coding & Index Coding Privacy Problems

17 Equivalence to Network Coding L Terminals: t 4 t 1 t 2 t 3 Wants: X 4 X 1 X 2 X 3 An index code of length L that satisfies all the users A network code that satisfies all the terminals

18 From Network to Index Coding Index coding is equivalent to the general network coding. If you can solve index coding efficiently you can solve any general network coding problem efficiently. How to map the codes? X 1, X 2,, X r How many? Rates? Wants:??? Has:??? Theorem: [R,Sprintson, Georghiades 08] [Effros,R,Langberg ISIT 13] For any network coding problem, one can construct an index coding problem and an integer L such that given any linear network code, one can efficiently construct a linear index code of length L, and vice versa. (same block length, same error probability).

19 Network Code è Index Code The linear case first X 1 X 2 X 1 X 2 X 1 +X 2 X 1 X 2 X X 1 +X 1 +X 2 2 Butterfly network Y e1 +X 1 Y e2 +X 1 Y e3 +X 2 Y e4 +X 2 Y e5 +X 1 +X 2 Y e6 +X 1 +X 2 Y e7 +X 1 +X 2 X 1, X 2 Y e1, Y e2,, Y e7 H(Y ei )=c(e i )=1 Equivalent index code All terminals in the index coding problem can decode Any linear network code gives an index code of length L=7

20 Implications on Index Coding Linear index codes are not optimal Vector linear codes outperform scalar linear No linear network code but a nonlinear code over alphabet of size 4 [zeger et al. 06] Only vector linear codes exist when block length is even.

21 Connections to many problems Interference management: [Jafar et al. 12] Distributed storage & caching: [Mazumdar 14], [Shanmugam et al. 14 Matroid representations: [Rouayheb et al. 09] Graph coloring: [Fragouli, Soljanin, Shokrollahi 04] [Alon et al. 08], [Shanmugam & Dimakis 13] LP bounds: [Blasiak, Kleinberg, Lubetzky 11 ] Coded caching [Maddah-Ali & Niesen 13] +

22 Variations on Index Coding 1. Data Exchange Problem [R., Sprintson, Sadeghi ITW 10] [Milosavlevijc, Pawar, R., Ramchandran, 13] [Courtade et al. 13] a No Base station (D2D). Users wants missed parts 2. Pliable index coding [Fragouli et al 15] b User 1 File: User 2 Like index coding but users want anything they don t have User 3 3. Coded Caching [Maddah-Ali & Niesen 14] Cached content is not fixed and can be designed Best paper, lots of follow up work

23 Talk Roadmap Graph Theory & Index Coding Rank Minimization & Index Coding Network Coding & Index Coding Privacy Problems

24 Caching Distributed for Private Storage Information Retrieval (PIR) PIR: user wants to hide which file it wants [chor et al 95] One server: User need to download all the data Classical PIR: data replicated on many servers Recent work: coded PIR [Jafar et al.], [Vardy et al.], [Rouayheb et al.], [Ulukus et al], [Hollanti et al.] Caching for PIR: user does not reveal cached data X 1 X 2 X M Server user Wants: X 1 Kadhe, Garcia, Heidarzadeh, R., Sprintson, PIR with Side Information, Allerton 17

25 SECURE COOPERATIVE COMPUTING IN IOT Collaboration with Hulya Seferoglu UIC Local computations on untrusted workers Homomorphic Encryption very costly New codes for security Bitar, R., Staircase Codes for Secret Sharing with Optimal Communication Overhead, Trans. on info th., R. Bitar P. Parag, R., Minimizing Latency for Secure Distributed Computing, submitted to ISIT 17

26 Acknowledgment Collaborators My students: Rawad Bitar, Razan Tajeddine, Peiwen Tian Camilla Hollanti (Aalto University, Finland) Olgica Milenkovic (UIUC) Hulya Seferoglu, (UIC) Parimal Parag (IISC, India) Funding agencies: NSF: CCF NSF: CCF ARL: W911NF

27 QUESTIONS?

28 Network Code è Index Code The linear case first X 1 X 2 X 1 X 2 X 1 +X 2 X 1 X 2 X X 1 +X 1 +X 2 2 Butterfly network Y e1 +X 1 Y e2 +X 1 Y e3 +X 2 Y e4 +X 2 Y e5 +X 1 +X 2 Y e6 +X 1 +X 2 Y e7 +X 1 +X 2 X 1, X 2 Y e1, Y e2,, Y e7 H(Y ei )=c(e i )=1 Equivalent index code All terminals in the index coding problem can decode Any linear network code gives an index code of length L=7

29 Index Code è Network Code Given a linear index code Y e1 +X 1 Y e2 +X 1 Y e3 +X 2 Y e4 +X 2 Y e5 +X 1 +X 2 Y e6 +X 1 +X 2 Y e7 +X 1 +X 2 Can always diagonalize Y e1 +Y e2 Y e2 +X 1 Y e3 +X 2 Y e4 +X 2 Y e5 +Y e4 +X 1 Y e6 +X 1 +X 2 Y e6 +Y e7 Butterfly network Any linear index code of length L=7 can be mapped to a linear network code Works for scalar linear and vector linear

30 Non-Linear Network Code è Index Code Y e1 + f e1 (X 1,X 2 ) Y e2 + f e2 (X 1,X 2 ) Y e3 + f e3 (X 1,X 2 ) Y e4 + f e4 (X 1,X 2 ) Y e5 + f e5 (X 1,X 2 ) Y e6 + f e6 (X 1,X 2 ) Y e7 + f e7 (X 1,X 2 ) Butterfly network Equivalent index code f ei (X 1,X 2 ) : message on edge e i

31 Diagonalization May Not Work for Non-Linear Butterfly network B 0 1 = g 0 1(Y e1, X) B 0 2 = g 0 2(Y e2, X) B 0 3 = g 0 3(Y e3, X) B 0 4 = g 0 4(Y e4, X) B 0 5 = g 0 5(Y e5, X) B 0 6 = g 0 6(Y e6, X) B 0 7 = g 0 7(Y e7, X) If we can we diagonalize? Given a nonlinear index code B 1 = g 1 (Ȳe, X) B 2 = g 2 (Ȳe, X) B 3 = g 3 (Ȳe, X) B 4 = g 4 (Ȳe, X) B 5 = g 5 (Ȳe, X) B 6 = g 6 (Ȳe, X) B 7 = g 7 (Ȳe, X)

32 Non-linear Index Code è Network Code Broadcast message Decoding function D Ue2 (0,X 1 ) D Ue3 (0,X 2 ) D Ue1 (0,X 1 ) D Ue4 (0,X 2 ) D Ue5 (0,Y e2,y e3 ) X1 =D Ut1 (B,Y e4,y e7 ) Y e4 = D Ue4 (B,X 2 ) Y e7 = D Ue7 (B,Y e5 ) D Ue7 (0,Y e5 ) D Ue6 (0,Y e5 ) Fix a value for B, say B=0 Destinations can decode with no errors: Recall that B=f(X 1,X 2, Y e1,,y e7 ) For a fixed B and given values of X 1 and X 2, there is a unique possible vector (Y e1,,y e7 ) Otherwise, U* cannot decode correctly

33 Dealing with Errors Consider an index code where decoding errors only happen when the broadcast message B=0 ε: Prob of error in the index code =1/2 c =1/2 7 = Prob of error in the network code =1 (bad). Claim: There exists σ, such that for B=σ, in the previous construction, the network code will have a prob of error at most ε (ε=error prob of the index code). Intuition: if for every value of B, the resulting network code will have a prob of error>ε, this implies that the prob of error in the index code >ε. A contradiction. B X=(X 1,X 2 ) Ye= Σ X =2 2 D U (B,X 1,X 2 ) : decoding error Each corresponds to a different good value of (X,Ye) Σ B =2 c Total # of <(1-ε) Σ B. Σ X But Σ B = Σ Ye è Total # of good values<(1-ε) Σ Ye. Σ X contradiction

34 Capacity Regions R X2 R B R B =7 1 P P H R N 1 R X1 :Capacity region of a network R X1 1 1 R X2 R I :Capacity region of the equivalent index code If there is a code that achieves P exactly, then P is in R I \ H, and vice versa. What if a sequence of points (not necessarily in H) converges to P. Does this mean that P is in R N? If true this will solve a long-standing open problem: Is zero-error capacity= ε-error capacity of networks? True for index coding problems [Langberg, Effros 11]

35 The Case of Co-located Sources R X2 R B R B =7 1 P P H R N 1 R X1 :Capacity region of a network 1 1 R X2 R I :Capacity region of the equivalent index code Theorem: For any network N with co-located sources one can efficiently construct an index coding problem I and an integer L such that R is in the capacity region of N iff R is in the capacity region of with broadcast length L. I