On Multi-source Multi-sink Hyperedge Networks: Enumeration, Rate Region Computation, & Hierarchy

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1 On Multi-ource Multi-ink Hyperedge Network: Enumeration, Rate Region Computation, & Hierarchy Ph D Diertation Defene Congduan Li ASPITRG & MANL Drexel Univerity congduanli@gmailcom May 9, 05 C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 / 73

2 Motivation: information flow in network Coding (mix the input) outperform routing (receive-forward) in many circumtance Efficient information flow in network: admiible ource rate v channel capacitie Data Source? Network? Relay Node/ Router Uer C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 / 73

3 Motivation: information torage in data center Coding (mix ource) ave reource, compared with imple replication Efficient information torage in data center: admiible ource file ize v hard dik capacitie? Source File? Hard Drive Uer C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 3 / 73

4 Generalized model A multiource multiink hyperedge network (hyperedge MSNC) Network Path Source S Sink T In(i) i Out(i) t e E U U e R e K T Y A[R,R,] T B[H(Y ),H(Y ),] T (t) Rate/capacity region: compute Notation: (K, E U ) mean K ource, E U intermediate hyperedge We made three major contribution C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 4 / 73

5 Contribution : Revolution, ue computer for rate region calculation Conventional: (pecial) network, info ineq, manually, paper Computational: 03, 06, 0 (arbitrary) network, computer, # of paper? In(i) In(i) i U e Re t T T In(i) T T T S T i U e Re t t U e Re In(i) Out(i) i U e Re Out(i) t e e Conventional way Out(i) S S S i t e S Out(i) S In(i) Out(i) e e C Li (ASPITRG & MANL) i U e Re S T S T S T S T Computational way Ph D Diertation Defene May 9, 05 5 / 73

6 Contribution : enumeration of all network Conventional: almot impoible to lit all network due to the large number of intance Computational: enumerate 09, 0 general network Trillion of Network In(i) i t i Out(i) t U e Re e e S T S T T S T S In(i) i U e Re t In(i) Out(i) i U e Re Out(i) t e e S T S T S T S T? Conventional way C Li (ASPITRG & MANL) In(i) Out(i) U e Re Enumeration Algorithm Computational way Ph D Diertation Defene May 9, 05 6 / 73

7 Contribution 3: build hierarchy between network Conventional: what to do with o many rate region? Computational: define embedding and combination operator, build a hierarchy, analyze the rate region and ue them to olve even more network in cale Trillion of Rate Region Build hierarchy between them and olve more network in cale? Conventional way C Li (ASPITRG & MANL) Hierarchy Operator Computational way Ph D Diertation Defene May 9, 05 7 / 73

8 Outline Enumeration Network Model Minimality Network Equivalence Cla Enumeration algorithm Rate Region Computation 3 Network Operation Enumerate network Software to calculate rate region or bound Combination operator Embedding operator Embedding Operation Combination Operation Play with both New network Forbidden Minor C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 8 / 73

9 Outline Enumeration Network Model Minimality Network Equivalence Cla Enumeration algorithm Enumerate network Software to calculate rate region or bound Rate Region Computation 3 Network Operation Embedding Operation Combination Operation Play with both Combination operator New network Embedding operator Forbidden Minor C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 9 / 73

10 Network Coding: Source We aume U Out() = Y Source S Network Sink T Path In(i) i Out(i) Y t e U S e R K e T (t) Source Rate Source Independence Source Encoding H(Y ) ω H(Y S ) = S H(Y ) H(U Out() Y ) = 0 C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 0 / 73

11 Network Coding: Edge Source S Network Sink T Path In(i) i Out(i) Y t e U S e R e K T (t) Coding Rate R e H(U e ), e E C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 / 73

12 Network Coding: Node Source S Network Sink T Path In(i) i Out(i) Y t e U S e R e K T (t) Coding Contraint H(U Out(i) U In(i) )=0, i V\(S T ) C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 / 73

13 Network Coding: Sink Source S Network Sink T Path In(i) i Out(i) Y t e U S e R e K T (t) Decoding Contraint H(Y β(t) U In(t) )=0, t T C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 3 / 73

14 Special Cae: Independent Ditributed Source Coding (IDSC) Y Y U E U E U 3 E 3 D D D 3 Y Y Y Y Y g Y U U U 3 t t t3 Y Y Y Y Source available to all encoder Decoder demand variou ubet of of ource When ource are prioritized, it become MDCS C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 4 / 73

15 Special Cae: Index Coding Only one intermediate edge that tranmit all information of ource Source may be directly available at ink a ide information (K, ) hyperedge MSNC C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 5 / 73

16 Outline Enumeration Network Model Minimality Network Equivalence Cla Enumeration algorithm Enumerate network Software to calculate rate region or bound Rate Region Computation 3 Network Operation Embedding Operation Combination Operation Play with both Combination operator New network Embedding operator Forbidden Minor C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 6 / 73

17 Source minimality example Source minimality example: Source 3 redundant due to not demanded by any ink 3 Y 3 Y g U t Y Y g U t Y U t Y U t Y Y g t 3 U 3 Y Y g t 3 U 3 Y C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 7 / 73

18 Node minimality example Node minimality example: g, g can be merged due to ame input Y g U t Y Y U t Y U t g U Y t Y Y g t 3 U 3 Y Y t 3 U 3 Y C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 8 / 73

19 Edge minimality example Edge minimality example: U, U 3 are parallel and can be merged Y U U g t Y Y U g t Y Y g t U 3 Y Y g t Y U C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 9 / 73

20 Sink minimality example Sink minimality example: decoding ability of Y at t implied by t, equivalent to let t demand Y only Y g U U t t Y Y Y Y g U U t t Y Y Y g t 3 U 3 Y Y g t 3 U 3 Y C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 0 / 73

21 Outline Enumeration Network Model Minimality Network Equivalence Cla Enumeration algorithm Enumerate network Software to calculate rate region or bound Rate Region Computation 3 Network Operation Embedding Operation Combination Operation Play with both Combination operator New network Embedding operator Forbidden Minor C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 / 73

22 Equivalent Network Y g U t Y I) Y g U t Y II) III) Y Y Y $ Y & U $ U g U t Y g U t Y Y 6$ Y U Y g Y t U Y g t Y I, II are equivalent: permute ource and edge I, III are not equivalent: ource are only permuted at ource ide II, III are not equivalent: ource are only permuted at ink ide C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 / 73

23 Repreenting a Network Ordered pair (Q, W), ource K, edge K +,, K + E U ; Edge definition Q {(i, A) i {K +,, K + E U }, A {,, K + E U } \ {i}}; Sink definition W {(i, A) i {,, K}, A {,, K + E U } \ {i}}; Same i allowed to appear in W but not in Q (3, {, }) 3 (4, {, 3}) 4 Q W (, {3}) (, {3, 4}) (, {4}) C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 3 / 73

24 Equivalence Under Group Action Symmetry group G := S {,,,K} S {K+,,K+ EU }; π G, then π(q) {π((i, A)) (i, A) Q}; π((q, W)) = (π(q), π(w)); Iomorphic or Equivalent: π G uch that π((q, W )) = (Q, W ) (3, {, }) 3 (4, {, 3}) 4 :permute, Q Q 0 (3, {, }) 3 (4, {, 3}) 4 W (, {3}) (, {3, 4}) (, {4}) W 0 (, {3}) (, {3, 4}) (, {4}) C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 4 / 73

25 Canonical Network: minimal repreentative in each orbit Orbit: O (Q,W) := {(π(q), π(w)) π G} Network in an orbit are iomorphic or equivalent to each other Tranveral: one repreentative for each orbit, the canonical one Lexicographically order the pair (i, A) according to (i, A) > (j, A ) if j < i or i = j, A < A under the lexicographic ordering Canonical: apply order to (Q, W) and get the minimal one # 3 4 ` # 3 4 ` # 3 4 ` # 3 4 ` # i the canonical repreentative # Q W {(3, {, }), (4, {, 3})} {(, {3, 4}), (, {3}), (, {4})} {(3, {, }), (4, {, 3})} {(, {3}), (, {4}), (, {3, 4})} 3 {(3, {, 4}), (4, {, })} {(, {3, 4}), (, {3}), (, {4})} 4 {(3, {, 4}), (4, {, })} {(, {3}), (, {4}), (, {3, 4})} # 3 4 ` # 3 4 ` # 3 4 ` C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 5 / 73

26 Why thi repreentation Some minimality contraint eaily to take care, eg, repeated edge, redundant node, etc Smaller orbit, compared with node repreentation Orbit-tabilizer theorem: O (Q,W) = G Stab((Q,W)) # ` # ` 3 3 # ` # ` 4 4 a b c d e f # a b c d e f # ` # ` # ` # Q W {(3, {, }), (4, {, })} {(, {3, 4}), (, {3}), (, {4})} {(3, {, }), (4, {, })} {(, {3}), (, {4}), (, {3, 4})} Subgroup of S {,} S {3,4} tabilizing (G, T )=S {3,4} a b Subgroup of S {a,b} S {c,d,e,f} tabilizing (V, E) =S {d,f} a Iomorph in Node Repreentation (4) Iomorph in Edge and Sink Definition Repreentation () C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 6 / 73

27 Outline Enumeration Network Model Minimality Network Equivalence Cla Enumeration algorithm Enumerate network Software to calculate rate region or bound Rate Region Computation 3 Network Operation Embedding Operation Combination Operation Play with both Combination operator New network Embedding operator Forbidden Minor C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 7 / 73

28 Leiterpiel Algorithm [BettenBraunFripertinger, 006] Compute tranveral of orbit on ize j ubet P j (X ) of et X, incrementally in j; alo give ymmetry group (tabilizer) Lit directly the canonical repreentative atifying ome tet function f, a long a thi tet ha the inherited property, ie, if a uperet atifie, it ubet alo atify Input: et X, group G, inherited tet function f Output: tranveral of ubet of different ize until top, either fixed j or other top condition, ymmetry group (tabilizer) Tranveral: canonical repreentative Orbit on P j (X ) Extenion obeying f Orbit on P j (X ) C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 8 / 73

29 Enumeration baed on Leiterpiel Algorithm Target: (K, E U ) non-iomorphic network Recall repreentation of network: (Q, W), lit pool for Q firt Leiterpiel define edge incrementally from to E U X := {(i, A) i {K +,,K + E U }, A {,,K + E U } \ {i}} With ome minimality contraint inherited Acting group G := S {,,K} S {K+,,K+ EU } Call Leiterpiel T EU = Leiterpiel(G, P f E U (X )) Stop condition: reach number of edge C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 9 / 73

30 Enumeration baed on Leiterpiel Algorithm Now for each Q T EU, lit poible W Leiterpiel incrementally add ink from to no poible new ink Y := {(i, A) 9 a directed path in Q from i to at leat one edge in A} With ome minimality contraint inherited Acting group G := S {,,K} S {K+,,K+ EU } Call Leiterpiel T T = Leiterpiel(G, P f T (Y)) Stop condition: cannot increae j obeying f For each W {T,, T T }, tet all the other minimality condition on (Q, W), if it pae, we obtain a non-iomorphic network intance C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

31 Enumeration Reult M : number of non-iomorphic network, lited; ˆM : number of network with edge iomorphim, counted; ˆM n : number of network with node iomorphim, counted; (K, E U ) (,) (,3) (,) (,) (,3) (3,) (3,) M ˆM ˆM n 39 8, C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 3 / 73

32 All (, ) network: no direct acce btw ource & ink t Y Y g U Y U g t Y Y U g t Y Y U g t Y Y U g t YY Y U U g t Y Y g t t YY Y U U Y Y U Y U Y U Y g U g t YY g t Y g t YY g t Y Y U g t Y Y g t3 Y Intance # Intance # Intance #3 Intance #4 Intance #5 Intance #6 Intance #7 Y g Y U t YY g U Y U g t Y Y U g t Y Y U g t Y Y U g t Y Y U g t Y Y t YY U g U Y Y U U g U g t Y Y U Y Y U YY g t Y g t Y g t Y g t Y Intance #8 Intance #9 Intance #0 Intance # Intance # Intance #3 Intance #4 t Y U Y Y U g g t Y Y U g t YY Y U g t Y U Y g Y U t Y g t Y t Y U Y t Y U g U Y g Y U U t g t YY Y g t Y Y U Y g t Y Y g t Y Y g t Y U t3 Y U Y g t3 Y Intance #5 Intance #6 Intance #7 Intance #8 Intance #9 Intance #0 Intance # Y g Y U g U Y g t U YY Y g U U t Y U t YY t YY Y g Y t YY g Y U t Y g t Y U Y g U Y g U Y g t U YY Y U U U g t YY Y g t Y Y g t YY Y g t Y Intance # Intance #3 Intance #4 Intance #5 Intance #6 Intance #7 Intance #8 t Y U Y U g t Y Y U U g t YY Y g t Y Y g t Y U U U Y g t Y Y g t Y Y g t Y U U U U U Y g t Y Y U g t Y Y g t Y Y g t Y Y g t Y Y g t Y Y g t3 Y U Intance #9 Intance #30 Intance #3 Intance #3 Intance #33 Intance #34 Intance #35 t Y t Y U U Y g Y U t Y Y U t Y Y U t Y Y U Y U g t Y Y g t YY t Y g g g g t Y Y U Y U Y g t3 Y U t YY U Y t Y Y U t Y Y g t YY Y U g t Y t3 Y U Intance #4 Intance #36 Intance #37 Intance #38 Intance #39 Intance #40 Intance #4 U U U t Y Y Y g t g t Y U Y Y g t Y Y g t Y Y U Y U Y U Y g t Y g t Y g t Y g t3 Y U Intance #43 Intance #44 Intance #45 Intance #46 C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 3 / 73

33 Outline Enumeration Network Model Minimality Network Equivalence Cla Enumeration algorithm Rate Region Computation 3 Network Operation Enumerate network Software to calculate rate region or bound Combination operator Embedding operator Embedding Operation Combination Operation Play with both New network Forbidden Minor C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

34 Rate region Rate region: all poible rate and ource entropy vector atifying all network contraint Collect the N network random variable and their joint entropie Define Γ N : N -dim cone, region of valid entropy vector (reviit later) Contraint from network A: L = {h Γ N : h Y S = Σ S h Y } () L = {h Γ N : h X Out(k) Y = 0} () L 3 = {h Γ N : h X Out(i) X In(i) = 0} (3) L 4 = {(h T, R T ) T R N + E + : R e h Ue, e E} (4) L 5 = {h Γ N : h Y β(t) U In(t) = 0} (5) Rate region (cone) in term of edge rate and ource entropie (derived from [Yan, Yeung, Zhang TranIT 0]): R (A) = proj RE,H(Y S )(con(γ N L 3) L 45 ) (6) C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

35 Rate Region Example A (3, 3) network and it rate region R (A) Rate region: a cone with dimenion of all variable in the network Y Y g g g U 3 3 t Y Y R H(Y 3 ) U U R H(Y ) t Y 3 R + R H(Y )+H(Y )+H(Y 3 ) t 3 3 Y 3 Y 3 R + R 3 H(Y )+H(Y 3 ) R + R +R 3 H(Y )+H(Y )+H(Y 3 ) C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

36 Rate region Rate region: a cone in term of edge rate and ource entropie: R (A) = proj RE,H(Y S )(con(γ N L 3) L 45 ) (7) Involve Γ N : N -dim cone, region of valid entropy vector C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

37 Region of Entropic Vector Γ N : Open et in general Γ N not fully characterized for N 4: convex but contain non-polyhedral part Outer bound Inner bound N C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

38 Sandwich Bound Γ N ΓOut N : R out(a) = proj RE,H(Y S )(Γ Out N L 345) Γ N ΓIn N : R in(a) = proj RE,H(Y S )(Γ In N L 345) R (A) = R out (A) = R in (A), if R out (A) = R in (A) It become: Initial polyhedra contraint projection Our work following thi idea: Li, et al, Allerton 0, NetCod 03, ubmiion TranIT 04 Contraint P rojection Contraint N N Contraint N N In N In N C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

39 Notion of ufficiency Outer bound typically ued i Shannon outer bound R o (A); inner bound from repreentable matroid: calar and vector bound Scalar ufficiency: R (A) = R,q (A) Vector ufficiency: R (A) = R N q (A) Scalar R,q (A) Vector R N 0 q (A) Sufficient Inufficient C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

40 Rate Region Computation Reult M : number of all network intance; The other number repreent the number of intance we can cloe the gap uing variou bound, and hence exact rate region can be obtained (K, E ) M R, (A) R N+ (A) R N+ (A) R N+4 (A) (, ) (, 3) (, ) (, ) (, 3) (3, ) (3, ) C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

41 Outline Enumeration Network Model Minimality Network Equivalence Cla Enumeration algorithm Rate Region Computation 3 Network Operation Enumerate network Software to calculate rate region or bound Combination operator Embedding operator Embedding Operation Combination Operation Play with both New network Forbidden Minor C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 4 / 73

42 Outline Enumeration Network Model Minimality Network Equivalence Cla Enumeration algorithm Rate Region Computation 3 Network Operation Embedding Operation Combination Operation Enumerate network Software to calculate rate region or bound Combination operator New network Embedding operator Forbidden Minor Play with both C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 4 / 73

43 Motivated from graph and matroid theory Inheritance property regarding ufficiency of cla of code Minor-cloed graph: finite number of forbidden minor [RobertonSeymour ] Rota conjecture in matroid theory: finite number of forbidden minor for F q repreentability [Oxley0] Not Fobidden Forbidden Size N New Minimal Forbidden Minor N N C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

44 Similar characterization for network? If network have imilar characterization? Poible lit of forbidden embedded network for ufficiency of linear code over a field Network operation to obtain uch embedded network preerving inufficiency, & region relationhip Three operation Source Network Detination C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

45 Source deletion A = A \ k Source Y k deleted, ource k top ending information to the network, H(Y k ) = 0 Sink requiring Y k will no longer demand it H(Y k 0)=0 k Y k kt (t) \ k C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

46 Edge deletion A = A \ e Edge e deleted, nothing on U e, R e = H(U e ) = 0 e e R e 0 =0 C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

47 Edge contraction A = A/e Edge e contracted, input to tail of e available for head of e, R e =, H(U e ) free e e R e 0 = C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

$Source deletion: A = A \ k For each i {, q, (, q), o}, R i (A ) = Proj Y\k,R E ({R R i(a) H(Y k ) = 0}) (8) Sufficiency preerved from large to mall network a the$

48 Source deletion: A = A \ k For each i {, q, (, q), o}, R i (A ) = Proj Y\k,R E ({R R i(a) H(Y k ) = 0}) (8) Sufficiency preerved from large to mall network a the equation how Equivalently, inufficiency preerved from mall to large network R(A) H(Y k )=0 P rojection R(A 0 ) C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

49 Example: Forbidden embedded network Goal: minimal forbidden network for ufficiency Scalar binary code conidered k =,, 3; E =, 3, 4, 7360 non-iomorphic MDCS 9 ufficient, 5438 inufficient minimal forbidden minor (Li, et al ubmiion TranIT 04) 5438 / 7360 inu cient 9 / 7360 u cient forbidden embedded network C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

50 Summary for embedding operator Three embedding operator: ource deletion, edge deletion, edge contraction Rate region of maller network derivable from the aociated larger network Sufficiency of linear code preerved from larger to maller network under embedding operation Equivalently, inufficiency of linear code preerved from maller network to larger one C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

51 Outline Enumeration Network Model Minimality Network Equivalence Cla Enumeration algorithm Rate Region Computation 3 Network Operation Enumerate network Software to calculate rate region or bound Combination operator Embedding operator Embedding Operation Combination Operation Play with both New network Forbidden Minor C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 5 / 73

52 Source merge After merge: Merged ource erve a common ource to the two network C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 5 / 73

53 Sink merge After merge: Union the input and requet of ink being merged, repectively C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

54 Intermediate node merge After merge: union input and output of the two node C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

55 Edge merge Edge merge: create one extra node and four aociated edge to replace the two original edge Firt two edge are ordinary edge connecting with the extra node, the other two edge connect the extra node with all the head node of the original two edge, repectively Equivalent: create a relay node on the two edge, repectively, and then merge the two relay node C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

56 Source merge A i obtained by merging A Ŝ = A π(ŝ), then for each i {, q, (, q), o} R i (A) = Proj ((R \π(ŝ) i(a ) R i (A )) L 0 ), { } L 0 = H(Y ) = H(Y π() ), Ŝ Remark: eentially replace variable Y π() with the Y for each Ŝ R(A ) 8 Ŝ,H(Y )=H(Y () ) R(A ) + C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

57 Obtain Rate Region for Larger Network X X X3 X4 X5 X6 R R R3 3 3 R5 4 R4 5 R7 0 R4 0 R R R0 R0 0 R R R9 7 9 {,3} {,3} {5} {6} {,3} {,,4} {4} {6} {5} {6} A 4 : A : A : A 3 : ' 3' ' A 5 : X 0 X0 6' 5' 4' X3 X X X3 X4 X5 X6 0 X60 X50 X ' R9 R0 R5 R3 R8 R6 R7 R R R4 R0 0 R 8 7 R 3 ' ' {',3'} {',3'} {6} {6} {5} {,} {} {4} {4} {3} R6 H(X6) R9 H(X3 0) {5'} {6'} {4'5'} Ri H(X4),i=3, 4 R H(X) R7 + R8 H(X6) R0 H(X 0)+H(X0)+H(X30) R H(X5 R3 + R4 H(X3)+ H(X4) 0) R H(X) R H(X4 R6 + R7 H(X5)+ H(X6) R3 + R4 +R5 H(X3)+H(X4) 0)+H(X50) R + R H(X)+ H(X) R00 + R + R R6 + R7 +R8 H(X5)+H(X6) H(X40)+H(X50)+H(X60) A : 0 {4,5} Ri H(X),i=, R + R H(X)+ H(X) Ri H(X4),i=3, 4, 4 0 R3 + Ri H(X3)+H(X4),i=4, 4 0 R3 + Ri + R5 H(X3)+H(X4),i=4, 4 0 R6 H(X6) Ri + R8 H(X6),i=7, 7 0 R6 + Ri H(X5)+H(X6),i=7, 7 0 R6 + Ri +R8 H(X5)+H(X6),i=7, 7 0 R9 R0 R R R00 + R + R H(X4)+H(X5)+H(X6) H(X3) P 3 i= H(Xi) H(X5) H(X4)+H(X5) C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

58 Summary for combination operator Four combination operator: ource, ink, node, and edge merge Rate region of combined network derivable from region of maller network in the combination C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

Outline Enumeration Network Model Minimality Network Equivalence Cla Enumeration algorithm Rate Region Computation 3 Network Operation Embedding Operation Combination Operation Enumerate

59 Outline Enumeration Network Model Minimality Network Equivalence Cla Enumeration algorithm Rate Region Computation 3 Network Operation Embedding Operation Combination Operation Enumerate network Software to calculate rate region or bound Combination operator New network Embedding operator Forbidden Minor Play with both C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

60 Combination operation uffice? Anwer i NO Need cap to limit the network ize in the combination Network Cap: no beyond Large ize Moderate ize Combination reach a portion of large network Small ize Embedding enlarge the reachable portion Tiny ize C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

61 Partial cloure of network Seed lit of network Wort cae partial cloure of network: cap the predicted ize of network involved in the proce Let the pool produce new network until no new network can be generated If within network cap Combination of pair New network? Ye Add to eed lit Minor from embedding NO Stop C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 6 / 73

62 New network found from a tiny eed lit Start with the ingle (, ), ingle (, ), and the four (, ) network; :6:;75/< 939==45>75= ; ;4?9@08!"#"$!"#%$!%#%$!"#"$!"#%$!%#%$!&#"$ % % % % % %!&#%$ ' &' &' ' &' &'!(#($ " " " ) &* &+!(#"$ &" &+ &+ "' &"& &**!(#%$ ',- &'& ' *&+ +%)!"#($ ( " ( % &' &&!"#"$ (% (% (% %( "*" )""!"#%$ ' &"* &"* ' ("+& *%)&!%#($ ' ' " ' ' "!%#"$ ' ' &- ' ' %%!%#%$ ' ' (*" ' ' %%"' // %+ (,( *+) )) "%'' &&+"* C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 6 / 73

63 New network found from a tiny eed lit 6 tiny network can generate new 635 network with mall cap! :6:;75/< 939==45>75= ; ;4?9@08!"#"$!"#%$!%#%$!"#"$!"#%$!%#%$!&#"$ % % % % % %!&#%$ ' &' &' ' &' &'!(#($ " " " ) &* &+!(#"$ &" &+ &+ "' &"& &**!(#%$ ',- &'& ' *&+ +%)!"#($ ( " ( % &' &&!"#"$ (% (% (% %( "*" )""!"#%$ ' &"* &"* ' ("+& *%)&!%#($ ' ' " ' ' "!%#"$ ' ' &- ' ' %%!%#%$ ' ' (*" ' ' %%"' // %+ (,( *+) )) "%'' &&+"* C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

64 New network found from a tiny eed lit With the increae of cap ize, number of new network increae! :6:;75/< 939==45>75= ; ;4?9@08!"#"$!"#%$!%#%$!"#"$!"#%$!%#%$!&#"$ % % % % % %!&#%$ ' &' &' ' &' &'!(#($ " " " ) &* &+!(#"$ &" &+ &+ "' &"& &**!(#%$ ',- &'& ' *&+ +%)!"#($ ( " ( % &' &&!"#"$ (% (% (% %( "*" )""!"#%$ ' &"* &"* ' ("+& *%)&!%#($ ' ' " ' ' "!%#"$ ' ' &- ' ' %%!%#%$ ' ' (*" ' ' %%"' // %+ (,( *+) )) "%'' &&+"* C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

65 New network found from a tiny eed lit Embedding operation are important in the proce! :6:;75/< 939==45>75= ; ;4?9@08!"#"$!"#%$!%#%$!"#"$!"#%$!%#%$!&#"$ % % % % % %!&#%$ ' &' &' ' &' &'!(#($ " " " ) &* &+!(#"$ &" &+ &+ "' &"& &**!(#%$ ',- &'& ' *&+ +%)!"#($ ( " ( % &' &&!"#"$ (% (% (% %( "*" )""!"#%$ ' &"* &"* ' ("+& *%)&!%#($ ' ' " ' ' "!%#"$ ' ' &- ' ' %%!%#%$ ' ' (*" ' ' %%"' // %+ (,( *+) )) "%'' &&+"* C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

66 Example to ee why integrating embedding i important Start with the ingle (, ), ingle (, ), and the four (, ) network Only combination with cap (3, 4), get only 3 network with ize (, ) (,) (,) (,) ource merge = (,) ALL (,), (,), (,) MINIMAL NETWORK CODING PROBLEMS (there are 6) ALL POSSIBLE COMBINATION OPERATORS = All (,) Minimal Network Coding Problem That are direct Combination of (,),(,),and (,) Problem (There are only 3 out of 333 total (,) problem) = ource merge node merge C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

67 Example to ee why integrating embedding i important Start with the ingle (, ), ingle (, ), and the four (, ) network Conider both combination and embedding with ame cap, found (, ) network unreachable by combination only Contracting edge Contracting edge Source merge [,] with [,] Contracting edge , Contracting edge 3, Deleting edge 3 Contracting edge 4 Edge merge 3 & Merge two ink , C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

68 Summary of work thu far Enumeration, Rate Region Computation, Forbidden Minor, New Network Enumerate > non-iomorphic network Calculate their exact rate region or bound Combination operation Embedding operation New olvable network in cale Forbidden Minor for u ciency C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

69 Dream Draw or input a network Minimality & canonical converion Though we have online repoitory Want a uer-friendly interface to eaily get anwer Get the anwer Ye Ye In Databae? No Directly Computable? No Ye Reachable by contruction? No Not olvable right now C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

70 Future work When Shannon outer bound i tight? Any common tructure? I Shannon outer bound tight for all MDCS, or IDSC? I the number of forbidden minor regarding the ufficiency of a cla of linear code finite? Coverage of the operator in all problem More operation: node & edge merge, ource & ink merge A notion of forbidden minor which can harne both combination and embedding operator C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

71 Publication Lit Steven Weber, Congduan Li, and John Walh, Rate Region for a cla of delay mitigatting code and PP network, 46th Annual Conference on Information Science and Sytem, Princeton Univerity, March, 0, Invited Paper Congduan Li, John Walh, and Steven Weber, A Computational Approach for Determining Rate Region and Code uing Entropic Vector Bound, 50th Allerton Conference, UIUC, Oct 0 3 Congduan Li, Jayant Apte, John M Walh, and Steven Weber, A new computational approach for determining rate region and optimal code for coded network, The 03 IEEE International Sympoium on Network Coding (NetCod 03), Calgary, Canada, Jun 7-9, 03 4 Congduan Li, John M Walh, and Steven Weber, Matroid bound on region of entropic vector, 5th Allerton conference on Communication, Control and Computation, UIUC, IL, Oct 03 5 Jayant Apte, Congduan Li, John MacLaren Walh, and Steven Weber, Exact pepair problem with multiple ource, in The 48th annual IEEE Conference on Information Science and Sytem(CISS), Mar 04 6 Jayant Apte, Congduan Li, and John MacLaren Walh, Algorithm for Computing Network Coding Rate Region via Single Element Extenion of Matroid, IEEE International Sympoium on Information Theory (ISIT) 04, Honolulu, Hawaii 7 Congduan Li, Steven Weber, and John M Walh, Network Embedding Operation Preerving the Inufficiency of Linear Network Code, 5th Allerton conference on Communication, Control and Computation, UIUC, IL, Oct 04 8 Congduan Li, Steven Weber, and John M Walh, Multilevel Diverity Coding Sytem: Rate Region, Code, Computation, & Forbidden Minor, IEEE Tranaction on Information Theory, 04, ubmitted 9 Congduan Li, Steven Weber, and John M Walh, Computer Aided Proof for Independent Ditributed Source Coding Problem, The 05 IEEE International Sympoium on Network Coding (NetCod 05), Jun, 05 0 Congduan Li, Steven Weber, and John M Walh, Network Combination Operation Preerving the Sufficiency of Linear Network Code, The 05 IEEE Information Theory Workhop (ITW 05), ubmitted Congduan Li, Steven Weber, and John M Walh, On Multi-ource Network: Enumeration, Rate Region Computation, and Hierarchy, IEEE Tranaction on Information Theory, 05, ready to ubmit C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 7 / 73

72 Selected Reference X Yan, RW Yeung, and Z Zhang (0) An Implicit Characterization of the Achievable Rate Region for Acyclic Multiource Multiink Network Coding IEEE Tranaction on Information Theory 58(9), A Betten, M Braun, H Fripertinger (006) Error-Correcting Linear Code: Claification by Iometry and Application Springer Berlin Heidelberg N Roberton, P Seymour (004) Graph minor xx wagner conjecture Journal of Combinatorial Theory, Serie B, 9(), JG Oxley (0) Matroid Theory Oxford Univerity, 0 C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, 05 7 / 73

73 Q & A Thank you! C Li (ASPITRG & MANL) Ph D Diertation Defene May 9, / 73

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

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