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Tighter Cut-Based Bounds for k-pairs Communication Problems

Tighter Cut-Based Bounds for k-pairs Communication Problems. Nick Harvey Robert Kleinberg. Overview. Definitions Sparsity and Meagerness Bounds Show these bounds very loose Define Informational Meagerness Based on Informational Dominance Show that it can be slightly loose. M 2. M 1.

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Tighter Cut-Based Bounds for k-pairs Communication Problems

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  1. Tighter Cut-BasedBounds for k-pairs Communication Problems Nick Harvey Robert Kleinberg

  2. Overview • Definitions • Sparsity and Meagerness Bounds • Show these bounds very loose • Define Informational Meagerness • Based on Informational Dominance • Show that it can be slightly loose

  3. M2 M1 M1⊕M2 k-pairs Communication Problem S(1) S(2) T(2) T(1)

  4. Concurrent Rate • Source i desires communication rate di. • Rate r is achievable if rate vector[ rd1, rd2, …, rdk ] is achievable • Rate region interval of R+ Def: “Network coding rate” (or NCR) := sup { r : r is achievable }

  5. M2 M1 M1⊕M2 k-pairs Communication Problem S(1) S(2) d1 = d2 = 1ce = 1 eE Rate 1 achievable T(2) T(1)

  6. Upper bounds on rate [Classical]: Sparsity bound for multicommodity flows [CT91]: General bound for multi-commodity information networks [B02]: Application of CT91 to directed network coding instances; equivalent to sparsity. [KS03]: Bound for undirected networks with arbitrarytwo-way channels [HKL04]: Meagerness [SYC03], [HKL05]: LP bound [KS05]: Bound based on iterative d-separation

  7. Vertex-Sparsity Def: For U V, • VS(G) := minUVVS(U) Claim: NCR VS(G) Capacity of edges crossing between U and U Demand of commodities separated by U VS(U) :=

  8. Edge-Sparsity • Def: For A E, • ES(G) = minAEES(A) Claim: Max-Flow  ES(G) But:Sometimes NCR > ES(G) Capacity of edges in A Demand of commodities separated in G\A ES(A) :=

  9. NCR > Edge-Sparsity • Cut {e} separates S(1) and S(2) ES({e}) = 1/2 • But rate 1 achievable! S(1) S(2) e T(2) T(1)

  10. Meagerness • Def: For A E and P  [k],A isolates P if for all i,j P,S(i) and T(j) disconnected in G\A. • M(G) := minAEM(A) Claim: NCR M(G) Capacity of edges in A Demand of commodities in P M(A) := minP isolated by A

  11. Meagerness & Vtx-Sparsity are weak • Thm: M(Gn) = VS(Gn) = (1),but NCR 1/n. fn-1 f3 f2 f1 S(n) S(n-1) S(3) S(2) S(1) gn gn-1 g3 g2 g1 Gn:= hn-1 h3 h2 h1 T(n) T(n-1) T(3) T(2) T(1)

  12. A Proof Tool Def: Let A,BE. B is downstream of Aif B disconnected from sources in G\A.Notation: A B. Claim: If A B then H(A)  H(A,B). Pf: Because S  A  B form Markov chain.

  13. Lemma: NCR 1/n fn-1 f3 f2 f1 S(n) S(n-1) S(3) S(2) S(1) Proof: {gn}  {gn,T(1),h1} gn gn-1 g3 g2 g1 Gn:= hn-1 h3 h2 h1 T(n) T(n-1) T(3) T(2) T(1)

  14. Lemma: NCR 1/n fn-1 f3 f2 f1 S(n) S(n-1) S(3) S(2) S(1) Proof: {gn}  {gn,T(1),h1}  {S(1),f1,g1,h1} gn gn-1 g3 g2 g1 Gn:= hn-1 h3 h2 h1 T(n) T(n-1) T(3) T(2) T(1)

  15. Lemma: NCR 1/n fn-1 f3 f2 f1 S(n) S(n-1) S(3) S(2) S(1) Proof: {gn}  {gn,T(1),h1}  {S(1),f1,g1,h1}  {S(1),f1,T(2),h2} gn gn-1 g3 g2 g1 Gn:= hn-1 h3 h2 h1 T(n) T(n-1) T(3) T(2) T(1)

  16. Lemma: NCR 1/n fn-1 f3 f2 f1 S(n) S(n-1) S(3) S(2) S(1) Proof: {gn}  {gn,T(1),h1}  {S(1),f1,g1,h1}  {S(1),f1,T(2),h2}  {S(1),S(2),f2,g2,h2} gn gn-1 g3 g2 g1 Gn:= hn-1 h3 h2 h1 T(n) T(n-1) T(3) T(2) T(1)

  17. Lemma: NCR 1/n fn-1 f3 f2 f1 S(n) S(n-1) S(3) S(2) S(1) Proof: {gn}  {gn,T(1),h1}  {S(1),f1,g1,h1}  {S(1),f1,T(2),h2}  {S(1),S(2),f2,g2,h2}  {S(1),S(2),f2,T(3),h3} gn gn-1 g3 g2 g1 Gn:= hn-1 h3 h2 h1 T(n) T(n-1) T(3) T(2) T(1)

  18. Lemma: NCR 1/n fn-1 f3 f2 f1 S(n) S(n-1) S(3) S(2) S(1) Proof: {gn}  …  {S(1),S(2),…,S(n)} Thus 1 H(gn)  H(S(1),…,S(n)) = n∙r So 1/n  r gn gn-1 g3 g2 g1 Gn:= hn-1 h3 h2 h1 T(n) T(n-1) T(3) T(2) T(1)

  19. Towards a stronger bound • Our focus: cut-based bounds • Given A E, we want to infer thatH(A) H(A,P) where P{S(1),…,S(k)} • Meagerness uses Markovicity:(sources in P)  A  (sinks in P) • Markovicity sometimes not enough…

  20. i  Informational Dominance Def: A dominates B if information in A determines information in Bin every network coding solution.Denoted AB. • Trivially implies H(A) H(A,B) • How to determine if A dominates B? • [HKL05] give combinatorial characterization and efficient algorithm to test if A dominates B.

  21. Capacity of edges in A Demand of commodities in P i  Informational Meagerness • Def: For A  E and P  {S(1),…,S(k)},A informationally isolates P ifAP P. • iM(A) = minPfor P informationally isolated by A • iM(G) = minA  EiM(A) Claim: NCR iM(G).

  22. iMeagerness Example s1 s2 • “Obviously” NCR = 1. • But no two edges disconnect t1 and t2 from both sources! t1 t2

  23. iMeagerness Example s1 s2 • After removing A, still a path from s2 to t1! Cut A t1 t2

  24. s1 s2 t1 t2 i  Informational Dominance Example • Our characterization shows A {t1,t2} • H(A) H(t1,t2) and iM(G) = 1 Cut A

  25. s(ε) s(0) s(1) s(10) s(11) s(00) s(01) H2 q(10) q(01) q(11) q(00) r(00) r(01) r(10) r(11) Capacity 2-n t(00) t(01) t(10) t(11) t(0) t(1) t(ε) A bad example: Hn Thm: iMeagerness gap of Hn is (log |V|)

  26. s(ε) s(0) s(1) s(00) s(01) s(10) s(11) Tn= Binary tree of depth n Source S(i) iTn

  27. s(ε) s(0) s(1) s(00) s(01) s(10) s(11) t(00) t(01) t(10) t(11) t(0) t(1) t(ε) Tn= Binary tree of depth n Source S(i) iTn Sink T(i) iTn

  28. s(ε) s(0) s(1) s(00) s(01) s(10) s(11) q(10) q(01) q(11) q(00) r(00) r(01) r(10) r(11) t(00) t(01) t(10) t(11) t(0) t(1) t(ε) Nodes q(i) and r(i) for every leaf i of Tn

  29. s(ε) s(0) s(1) s(00) s(01) s(10) s(11) q(10) q(01) q(11) q(00) r(00) r(01) r(10) r(11) t(00) t(01) t(10) t(11) t(0) t(1) t(ε) Complete bip. graph between sources and q’s

  30. s(ε) s(0) s(1) s(00) s(01) s(10) s(11) q(10) q(01) q(11) q(00) r(00) r(01) r(10) r(11) t(00) t(01) t(10) t(11) t(0) t(1) t(ε) (r(a),t(b)) if b ancestor of a in Tn

  31. s(ε) s(0) s(1) s(00) s(01) s(10) s(11) q(10) q(01) q(11) q(00) r(00) r(01) r(10) r(11) t(00) t(01) t(10) t(11) t(0) t(1) t(ε) (s(a),t(b)) if a and b cousins in Tn

  32. s(ε) s(0) s(1) s(00) s(01) s(10) s(11) q(10) q(01) q(11) q(00) r(00) r(01) r(10) r(11) t(00) t(01) t(10) t(11) t(0) t(1) t(ε) All edges have capacity  except (q(i),r(i)) Capacity 2-n

  33. s(ε) s(0) s(1) s(00) s(01) s(10) s(11) q(10) q(01) q(11) q(00) r(00) r(01) r(10) r(11) Demand of source at depth i is 2-i Capacity 2-n t(00) t(01) t(10) t(11) t(0) t(1) t(ε)

  34. Properties of Hn Lemma: iM(Hn) = (1) Lemma: NCR < 1/n Corollary: iMeagerness gap is n=(log |V|)

  35. Properties of Hn Lemma: iM(Hn) = (1) Lemma: NCR < 1/n Corollary: iMeagerness gap is n=O(log |V|) We will prove this

  36. Proof Ingredients • Entropy moneybags • i.e., sets of RVs • Entropy investments • Buying sources and edges, putting into moneybag • Loans may be necessary • Profit • Via Downstreamness or Info. Dominance • Earn new sources or edges for moneybag • Corporate mergers • Via Submodularity • New Investment Opportunities and Debt Consolidation • Debt repayment

  37. Submodularity of Entropy Claim: Let A and B be sets of RVs.Then H(A)+H(B)  H(AB)+H(AB) Pf: Equivalent to I( X; Y | Z )  0.

  38. Lemma: NCR < 1/n Proof: • Two entropy moneybags: • F(a) = { S(b) : b not an ancestor of a } • E(a) = F(a)  { (q(b),r(b)) : b is descendant of a }

  39. r(00) r(01) r(10) r(11) Entropy Investment s(ε) s(0) s(1) • Let a be a leaf of Tn • Take a loan and buy E(a). s(00) s(01) s(10) s(11) a q(10) q(01) q(11) q(00)

  40. r(00) r(01) r(10) r(11) Earning Profit s(ε) s(0) s(1) Claim: E(a) T(a) Pf: Cousin-edges not from ancestors.Vertex r(00) blocked by E(a). s(00) s(01) s(10) s(11) a q(10) q(01) q(11) q(00) t(00)

  41. r(00) r(01) r(10) r(11) Earning Profit s(ε) s(0) s(1) Claim: E(a) T(a) Result:E(a) gives free upgrade to E(a){S(a)}.Profit = S(a). s(00) s(01) s(10) s(11) a q(10) q(01) q(11) q(00) t(00)

  42. r(00) r(00) r(01) r(01) r(10) r(10) r(11) r(11) s(ε) s(0) s(1) aL s(00) s(01) s(10) s(11) E(aL){S(aL)} q(10) q(01) q(11) q(00) aR s(ε) s(0) s(1) s(00) s(01) s(10) s(11) E(aR){S(aR)} q(10) q(01) q(11) q(00)

  43. r(00) r(00) r(01) r(01) r(10) r(10) r(11) r(11) Applying submodularity s(ε) s(0) s(1) s(00) s(01) s(10) s(11) (E(aL){S(aL)})  (E(aR){S(aR)}) q(10) q(01) q(11) q(00) s(ε) s(0) s(1) s(00) s(01) s(10) s(11) (E(aL){S(aL)})  (E(aR){S(aR)}) q(10) q(01) q(11) q(00)

  44. r(00) r(01) r(10) r(11) New Investment s(ε) a s(0) s(1) • Union term has more edges  Can use downstreamnessor informational dominance again! • (E(aL){S(aL)})  (E(aR){S(aR)}) = E(a) s(00) s(01) s(10) s(11) (E(aL){S(aL)})  (E(aR){S(aR)}) q(10) q(01) q(11) q(00)

  45. r(00) r(01) r(10) r(11) Debt Consolidation • Intersection term has only sources  Cannot earn new profit. Used for later “debt repayment” • (E(aL){S(aL)}) (E(aR){S(aR)}) = F(a) s(ε) a s(0) s(1) s(00) s(01) s(10) s(11) (E(aL){S(aL)})  (E(aR){S(aR)}) q(10) q(01) q(11) q(00)

  46. What have we shown? • Let aL,aR be sibling leaves; a is their parent. • H(E(aL)) + H(E(aR))  H(E(a)) + H(F(a)) • Iterate and sum over all nodes in tree where r is the root. • Note: E(v) = F(v)  {(q(v),r(v))} when vis a leaf

  47. Debt Repayment Claim: Pf: Simple counting argument. 

  48. = = = 1 (where α = rate of solution) Finishing up  Rate <1/n

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