Multiterminal Networks: Rate Regions, Codes, Computations, & Forbidden Minors

Transcription

1 Multiterminal etworks: Rate Regions, Codes, Computations, & Forbidden Minors Ph D Thesis Proposal Congduan Li ASPITRG & MAL Drexel University congduanli@gmailcom October 5, 204 C Li (ASPITRG & MAL) Thesis Proposal October 5, 204 / 57

2 Problems of interest A multisource multicast network: Sources etwork Destinations Paths In(i) i Out(i) Y s s U ee R e ω s t β(t) S T Rate/capacity region: compute If Shannon outer (LP) bound tight If simple linear codes sufficient S T C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

3 Outline Rate Region 2 Shannon Outer Bounds & Converse Constraints P rojection 3 Matroid Bounds and Achievability Constraints Constraints 4 Forbidden Embedding etworks Motivation Operations Results 5 Proposed Tasks and Timeline In Sources etwork Destinations Insu Insu cient cient In C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

4 Outline Rate Region 2 Shannon Outer Bounds & Converse 3 Matroid Bounds and Achievability 4 Forbidden Embedding etworks Motivation Operations Results 5 Proposed Tasks and Timeline Constraints P rojection C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

5 etwork Coding: Sources Y s Paths s In(i) Out(i) U ee R e i t S T β(t) S Source Rate Source Independence Source Encoding T H(Y s ) ω s H(Y S ) = H(Y s ) s S H(U Out(s) Y s ) = 0 C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

6 etwork Coding: Edges Y s Paths s In(i) Out(i) U ee R e i t S T β(t) S Coding Rate R e H(U e ), e E T C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

7 etwork Coding: odes Y s Paths s In(i) Out(i) U ee R e i t S T β(t) S Coding Constraints H(U Out(i) U In(i) )=0, i V\(S T ) T C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

8 etwork Coding: Sinks Y s Paths s In(i) Out(i) U ee R e i t S T β(t) S Decoding Constraints H(Y β(t) U In(t) )=0, t T T C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

9 Example: Multilevel Diversity Coding Systems (MDCS) Encoders Prioritized Sources 9 U E X X 2 >= X :K >; X K E E U E Paths Fan(D ) Fan(D D ) Decoders D D i D d D j D D Prioritized sources, no intermediate nodes X X } X :k X :K X :K } } Lev(D )= = Lev(D i )= Lev(D d )=k Lev(D j )= = Lev(D D )=K Decoders are classified to levels: level k, decode X :k otation: (k, E ) MDCS instances C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

10 Rate region Rate region: all possible rate and source entropy vectors satisfying all network constraints Collect the network random variables and their joint entropies Define Γ : 2 -dim cone, region of valid entropy vectors Constraints from network A: L = {h Γ : h Y S = Σ s S h Ys } () L 2 = {h Γ : h X Out(k) Y s = 0} (2) L 3 = {h Γ : h X Out(i) X In(i) = 0} (3) L 4 = {(h T, R T ) T R 2 + E + : R e h Ue, e E} (4) L 5 = {h Γ : h Y β(t) U In(t) = 0} (5) Rate region (cone) in terms of rates and source entropies (derived from [Yan, Yeung, Zhang TranIT 202]): R(A) = proj RE,H(Y S )(con(γ L 23) L 45 ) (6) C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

11 Region of Entropic Vectors Γ : ot all points in Euclidean space have distributions h entropic: exists a joint distribution associated with h Region of entropic vector: Γ = {h : h is entropic} Γ not fully characterized for 4: convex but contains non-polyhedral part Entropic: associated P (X,,X )exists Outer bounds Inner bounds C Li (ASPITRG & MAL) Thesis Proposal October 5, 204 / 57

12 Sandwich Bounds Γ ΓOut : R out(a) = proj RE,H(Y S )(Γ Out L 2345) Γ ΓIn : R in(a) = proj RE,H(Y S )(Γ In L 2345) It becomes: Initial polyhedra constraints projections R(A) = R out (A) = R in (A), if R out (A) = R in (A) Our work following this idea: Li, et al, Allerton 202, etcod 203, submission TransIT 204 Constraints P rojection Constraints Constraints In In C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

13 Outline Rate Region 2 Shannon Outer Bounds & Converse 3 Matroid Bounds and Achievability 4 Forbidden Embedding etworks Constraints Constraints Motivation Operations Results 5 Proposed Tasks and Timeline In In C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

14 Shannon Outer Bound Region determined by Shannon inequalities H(X i \ X i ) 0 and I (X i, X j X K ) 0, i, j, K \{i, j}, equivalent to polymatroid cone: ormalization: f ( ) = 0; 2 Monotonicity: if A B then f (A) f (B); 3 Submodularity: if A, B, f (A B) + f (A B) f (A) + f (B) Shannon ineq C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

15 Converse Proof Recall R out (A) = proj RE,H(Y S )(Γ Out L 2345) Suppose a polyhedral cone P (Γ Out L 2345): Ax 0 An inequality in the projected cone (rate region), b T x 0 Exists a vector λ 0 (weighted sum) s t A T λ = b This is a converse proof (inspired from Tian ISIT 203) 2 4 A X X X 2 X Projection 2 4 A T = 0 b b T X X X 2 X C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57 0

16 Determine Human Readable Converse Proof λ should be as sparse as possible Optimization minimize λ subject to λ 0 A T λ = b λ 0 Approximated by L -norm: λ We observed: redundant inequalities helpful, the order of inequalities determinable by computer Automatic human readable converse (Li, et al submission TransIT 204): Γ Out L b T x 0 LP solver Order determiner Converse proof C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

17 Converse Proof Example X, Y, Z R E R 2 E 2 R 3 E3 D D 2 D3 D 4 X R H(X) R 2 H(X)+H(Y ) R 3 R + R 2 X, Y X, Y, Z X, Y, Z H(X)+H(Y )+H(Z) 2H(X)+H(Y )+H(Z) R + R 2 (,2) H(U )+H(U 2) (3,4) = H(X, U )+H(X, Y, U 2) (5) (6) (7) H(X)+H(X, Y, U,U 2) H(X)+H(U,U 2) H(X)+H(X, Y, Z) (8) = 2H(X)+H(Y )+H(Z) Order Coe cients Inequality or equality R H(U ) 2 R 2 H(U 2 ) 3 H(X U )=0 4 H(X, Y U 2 )=0 5 I(U ; YU 2 X) 0 6 H(X, Y U,U 2 )=0 7 H(X, Y, Z U,U 2 )=0 8 H(X, Y, Z) =H(X)+H(Y )+H(Z) C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

18 Outline Rate Region 2 Shannon Outer Bounds & Converse 3 Matroid Bounds and Achievability 4 Forbidden Embedding etworks Constraints Constraints Motivation Operations Results 5 Proposed Tasks and Timeline In In C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

19 Matroid Definition: Rank Function Matroid: generalization of linear dependence and independence Definition (Matroid rank function) A set function r : 2 S {0,, } is a rank function of a matroid if it obeys the polymatroid axioms and two more conditions: Integrality: r(a) is integer-valued; 2 Cardinality: 0 r(a) A C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

20 Representable Matroids Definition A matroid M with ground set S of size S = and rank r M = r is representable over a field F if there exists a matrix A F r such that for each independent set I I the corresponding columns in A, viewed as vectors in F r, are linearly independent S,,S i,,s j,,s Mapping 2 i j r(s i,s j )=rank(i, j-th columns) Representable matroids usually can be characterized by forbidden minors (cannot contain such minors) Example: U 2,4 forbidden for binary (Tutte) C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

21 Minors of Matroids Definition If M is a matroid on S and T S, a matroid M on T is called a minor of M if M is obtained by any combination of deletion (\) and contraction (/) of M Illustration of contraction and deletion: e Contraction e Deletion C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

22 Inner Bounds from Representable Matroids Observation: Conic hull Γ q of representable matroids form inner bounds on region of entropic vectors Proof: It suffices to show that a rank of representable matroid is entropic Suppose the associated representation matrix is A F k q, from which we can create the random variables (X,,X )=ua, u U(F k q ) S,,S i,,s j,,s Mapping 2 i j r(s i,s j )=rank(i, j-th columns) h i,j = rank(i, j-th columns) h u = r(s i,s j ) log 2 q C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

23 Generalized Inner Bounds Γ q, : Generalize the mapping to: S,, S -partition of, > Tighter inner bounds on Γ S,,S i,,s j,,s Mapping 2 i j 0 r(s i,s j )=rank(i, j-th columns) C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

24 otion of sufficiency Recall: R in (A) = proj RE,H(Y S )(Γ In L 2345) Substitute in Γ q : each extreme ray associated with a matroid Variable to matrix: one to one mapping, scalar codes, region R s,q Substitute in Γ q, : each extreme ray associated with a projection of matroids Variable to matrix: one to multiple mapping, vector codes, region R q Scalar sufficiency: General sufficiency: R(A) = R s,q (A) R(A) = R q (A) C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

25 Representable Matroids Determine Linear Codes Basic codes: matrices associated with extreme rays A point: conic combination of extreme rays Code for the point: block diagonalize the matrices according to conic coefficients (time sharing), reshuffling columns (details in [Li, et al etcod 203]) Conic combination C, C 2, C 3 : basic codes C 2 C 3 R 2 R 3 R C R C 2 4 X S X S X S 3 R = R + R 2 + R 3 T 2 5C = 4 X S X S X S U E U E U E 3T 2 3 C O O 5 4 O C 2 O 5 O O C 3 C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

26 Matroid extremality Suffice to work on rep matroids that are extreme rays (complexity reduction, Li, et al etcod 203, further work Apte, et al ISIT 204) Extremal ranks relationship: strict containment [Li, et al Allerton 203] Extreme rays of different cones Polymatroid Matroid q-rep matroid Theorem : o matroid here! Theorem 2: o binary matroid here! Γ Γ mat Γ bin C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

27 Revolution Conventional: (special) network, info ineq, manually, paper Computational: 03, 06 (arbitrary) networks, computer, # of papers? Improvement: easy to handle small networks, further work on symmetries, inheritance In(i) In(i) s i U e Re e T T Conventional way i Out(i) t s T S T T S T t U e Re e t S In(i) Out(i) e S S Out(i) S C Li (ASPITRG & MAL) i U e Re s In(i) i In(i) Out(i) i Out(i) t s S T S T S T S T s U e Re U e Re t e e Computational way Thesis Proposal October 5, / 57

28 Outline Rate Region 2 Shannon Outer Bounds & Converse 3 Matroid Bounds and Achievability 4 Forbidden Embedding etworks Sources etwork Destinations Insu cient Motivation Operations Results 5 Proposed Tasks and Timeline Insu cient C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

29 Outline Rate Region 2 Shannon Outer Bounds & Converse 3 Matroid Bounds and Achievability 4 Forbidden Embedding etworks Sources etwork Destinations Insu cient Motivation Operations Results 5 Proposed Tasks and Timeline Insu cient C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

30 Motivated from matroid theory Inheritance property regarding insufficiency of class of codes Inspired from matroid theory: forbidden minor for linear representability, eg, U 2,4 is the forbidden minor for binary matroids Ground set size F q -representable matroids ot F q -representable matroids 2 C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

31 Similar characterization for networks? If networks have similar characterization? Possible list of forbidden embedded networks for sufficiency of linear codes over a field etwork operations to obtain such embedded networks preserving insufficiency, & region relationships Three operations [Li, et al, Allerton 204] Sources etwork Destinations Insu cient Insu cient C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

32 Outline Rate Region 2 Shannon Outer Bounds & Converse 3 Matroid Bounds and Achievability 4 Forbidden Embedding etworks Sources etwork Destinations Insu cient Motivation Operations Results 5 Proposed Tasks and Timeline Insu cient C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

33 Source deletion A = A \ k Source Y k deleted, source k stops sending information to the network, H(Y k ) = 0 Sinks requiring Y k will no longer demand it Sources etwork Destinations Paths In(i) i Out(i) Y k k U ee R e! k t S T S T (t) \ k C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

34 Example: MDCS source deletion X Y Z E E 2 E 3 D X X, Y D 2 X, Y D 3 X, Y D 4 X, Y, Z D 5 i) A(3, 3) MDCS X Y Z E E 2 E 3 X D X, Y D 2 X, Y D 3 X, Y D 4 X, Y, Z D 5 ii) Delete/contract Z Decoder D 5 no longer requires Z Redundant decoder is deleted X Y E E 2 E 3 X D X, Y D 2 X, Y D 3 X, Y D 4 iii) After deletion of Z C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

35 Source deletion: A = A \ k R(A ) = Proj Y\k,R ({R R(A) H(Y E k) = 0}), (7) R q (A ) = Proj Y\k,R ({R R E q(a) H(Y k ) = 0}), (8) R s,q (A ) = Proj Y\k,R ({R R E s,q(a) H(Y k ) = 0}) (9) Preservation: R q (A) = R(A) R q (A ) = R(A ) Scalar case: R s,q (A) = R(A) R s,q (A ) = R(A ) R(A) r R(A ) r R(A) with H(Y k ) = 0 The code to achieve r is the code to achieve r with sending nothing on source Y k All-zero is valid F q code, can reverse H(Y k )=0 P rojection R(A 0 ) C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

36 Edge deletion A = A \ e Edge U e deleted, nothing on U e, R e = H(U e ) = 0 Sources etwork Destinations Paths In(i) i Out(i) Y s s U ee R e! s t S T (t) S T C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

37 Example: MDCS encoder deletion X Y Z E E 2 E 3 E 4 D X X, Y D 2 X, Y D 3 X, Y, Z D 4 X, Y, Z D 5 i) A(3, 4) MDCS X Y Z E E 2 E 3 E 4 X D X, Y D 2 X, Y D 3 X, Y, Z D 4 X, Y, Z D 5 ii) DeleteencoderE 4 X Y Z D X E X, Y D 3 E 2 X, Y, Z D 4 E 3 X, Y, Z D 5 iii) After deletion of E 4 Decoders connected to E 4 keep same decoding ability without E 4 Redundant decoders deleted C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

38 Edge deletion: A = A \ Y k R(A ) = Proj H(YS ),R ({R R E q(a) R e = 0}), (0) R q (A ) = Proj H(YS ),R ({R R E q(a) R e = 0}), () R s,q (A ) = Proj H(YS ),R ({R R E s,q(a) R e = 0}) (2) Preservation: R q (A) = R(A) R q (A ) = R(A ) Scalar case: R s,q (A) = R(A) R s,q (A ) = R(A ) r R(A ) r R(A) with R e = H(U e ) = 0 The code to achieve r is the code to achieve r with deleting column(s) for U e, ie, sending nothing on edge e All-zero is valid F q code, can reverse R(A) R e =0 P rojection R(A 0 ) C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

39 Edge contraction A = A/e Edge e contracted, input to tail of e available for head of e, R e =, H(U e ) free Sources etwork Destinations Paths In(i) i Out(i) Y s s U ee R t U e R e! s S R e 0 = T S T (t) C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

40 Example: MDCS encoder contraction X Y Z E E 2 E 3 E 4 X D X, Y D 2 X, Y D 3 X, Y D 4 X, Y, Z D 5 X, Y, Z D 6 X Y Z E E 2 E 3 E 4 X D X, Y D 2 X, Y X D 3 Y X, Y D 4 Z X, Y, Z D 5 X, Y, Z D 6 X D X, Y D 2 X, Y D 3 X, Y, Z D 5 i) A(3, 4) MDCS ii) Contract encoder E 4 iii) After contraction of E 4 Decoders connected to E 4 directly access to all sources Requirements trivially satisfied, deleted E E 2 E 3 C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

41 Edge contraction: A = A/e R(A ) = Proj H(Yk ),k S,R E R(A), (3) R q (A ) = Proj H(Yk ),k S,R R E q(a), (4) R s,q (A ) Proj H(Yk ),k S,R E R s,q(a), (5) Preservation: R q (A) = R(A) R q (A ) = R(A ) Scalar case: R s,q (A) = R(A) R s,q (A ) = R(A ) r R(A ) r R(A) with some H(U e ) (not of interest in A ) The code to achieve r is the code to achieve r with deleting column(s) for U e, sending possibly everything available on e, possibly violating scalar code R(A 0 ) R e = R(A) Project on dimensions other than R e C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

42 Outline Rate Region 2 Shannon Outer Bounds & Converse 3 Matroid Bounds and Achievability 4 Forbidden Embedding etworks Sources etwork Destinations Insu cient Motivation Operations Results 5 Proposed Tasks and Timeline Insu cient C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

43 Example: Forbidden embedded networks Goal: minimal forbidden networks for sufficiency Scalar binary codes considered k =, 2, 3; E = 2, 3, 4, 7360 non-isomorphic MDCS 922 sufficient, 5438 insufficient 2 forbidden embedded networks (Li, et al submission TransIT 204) Open questions: complete list? Finite number for F q? 5438 / 7360 insu cient 922 / 7360 su cient 2 forbidden embedded networks C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

44 Summary of work thus far Enumerate > 0 5 networks Find > 7000 non-isomorphic ones Calculate outer bounds Calculate inner bounds Yes Obtain exact rate region &su ciency of codes Match? o Insu ciency for networks with arbitrary size? Operations preserving insu ciency Thousands of insu cient networks boiled down to 2 forbidden embedded networks C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

45 Publication List Steven Weber, Congduan Li, and John Walsh, Rate Region for a class of delay mitigatting codes and P2P networks, 46th Annual Conference on Information Sciences and Systems, Princeton University, March, 202, Invited Paper 2 Congduan Li, John Walsh, and Steven Weber, A Computational Approach for Determining Rate Regions and Codes using Entropic Vector Bounds, 50th Allerton Conference, UIUC, Oct Congduan Li, Jayant Apte, John M Walsh, and Steven Weber, A new computational approach for determining rate regions and optimal codes for coded networks, The 203 IEEE International Symposium on etwork Coding (etcod 203), Calgary, Canada, Jun 7-9, Congduan Li, John M Walsh, and Steven Weber, Matroid bounds on region of entropic vectors, 5th Allerton conference on Communication, Control and Computation, UIUC, IL, Oct Jayant Apte, Congduan Li, John MacLaren Walsh, and Steven Weber, Exact pepair problems with multiple sources, in The 48th annual IEEE Conference on Information Sciences and Systems(CISS), Mar Jayant Apte, Congduan Li, and John MacLaren Walsh, Algorithms for Computing etwork Coding Rate Regions via Single Element Extensions of Matroids, IEEE International Symposium on Information Theory (ISIT) 204, Honolulu, Hawaii 7 Congduan Li, Steven Weber, and John M Walsh, etwork Embedding Operations Preserving the Insufficiency of Linear etwork Codes, 52th Allerton conference on Communication, Control and Computation, UIUC, IL, Oct Congduan Li, Steven Weber, and John M Walsh, Multilevel Diversity Coding Systems: Rate Regions, Codes, Computation, & Forbidden Minors, IEEE Transactions on Information Theory, 204, SUBMITTED C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

46 Outline Rate Region 2 Shannon Outer Bounds & Converse 3 Matroid Bounds and Achievability 4 Forbidden Embedding etworks Motivation Operations Results 5 Proposed Tasks and Timeline C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

47 Task : network combination operations Recall: embedding operations (tools) to certify insufficiency of class of codes in arbitrary size networks but not sufficiency (no tools) Extension: certify sufficiency of class of codes ew operations: combine smaller networks into a bigger network Preserving: sufficiency of class of codes Existence: concatenation of two independent sufficient networks Sources etwork Destinations Su cient Su cient Su cient C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

48 Task : network combination operations Objectives: other operations preserving sufficiency Source Union? Sources etwork Destinations Su cient Su cient Su cient C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

49 Task : network combination operations Objectives: other operations preserving sufficiency Edge Connection? Deliverable: ISIT/etCod 205, journal with [Li, et al Allerton 204] Sources etwork Destinations Su cient Su cient Su cient C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

50 Task 2: asymmetric distributed storage exact repair Computation tools work for general networks, will try storage exact repair problems Common setup: symmetric, (n, k, d, α, β) Source n storage disks k disks reconstruct k k k d helpers repair C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

51 Task 2: asymmetric distributed storage exact repair Will investigate asymmetric setups with prioritized data Initial work with Jayant Apte [6, CISS 204]: symmetric in disks and repair, asymmetric in reconstruction Deliverable: etcod 205 n storage disks with distinct capacities asymmetric reconstruction: access not necessarily k disks; hot data, cold data R Hot Cold R m R n F m,n F,n asymmetric repair: not necessarily d helpers distinct repair bandwidth C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

52 Task 3: polymatroid extremality Recall: computation cares exclusively about matroids that are extreme rays, Connected matroids are extremal: r(a) + r(e \ A) > r(e), A E Extremal matroids are extremal polymatroids Connected polymatroid = extremal polymatroids? C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

53 Task 3: polymatroid extremality Extremal polymatroids Connected: otherwise separable, sum of two polymatroids Connected polymatroid extremal polymatroids Investigate the sufficient condition for polymatroid extremality Potential reduction in computation complexity Deliverable: ISIT 205, journal with [Li, et al Allerton 203] [H(X) H(Y ) H(XY ) H(Z) H(XZ) H(YZ) H(XY Z)] [ ] = [ ] + [ ] Connected but ot extremal ot connected Connected C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

54 Task 4: necessity of non-shannon inequalities Tighter outer bound on Γ : Shannon + non-shannon on-shannon to close the gap? Investigate necessity of non-shannon: try to construct distributed storage exact repair network associated with Vámos matroid Preserving property: if a network requires non-shannon, what about its extensions? Deliverable: ISIT/etCod 205 on-shannon ineq Shannon ineq Constraints non-shannon In C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

55 Timeline Time Task Task 2 Task 3 Task 4 Thesis Defense 0/204 Enumeration Construction /204 Start Calculation Reading Proof 2/204 Proof Writing Proof Writing 0/205 Writing Writing Writing 02/205 Writing Writing 03/205 Journal 04/205 Journal 05/205 Writing 06/205 Present Visit IC at CUHK: ov, Dec, 204 ISIT due Jan 9, 205, etcod due Feb 20, 205 C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

56 References X Yan, RW Yeung, and Z Zhang (202) An Implicit Characterization of the Achievable Rate Region for Acyclic Multisource Multisink etwork Coding IEEE Transactions on Information Theory 58(9), C Tian (203) Rate region of the (4, 3, 3) exact-repair regenerating codes IEEE International Symposium on Information Theory (ISIT), C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57

57 Q & A Thank you! C Li (ASPITRG & MAL) Thesis Proposal October 5, / 57