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Towards Polynomial Lower  Bounds for Dynamic Problems

Towards Polynomial Lower  Bounds for Dynamic Problems. Mihai P ătrașcu. STOC 2010. Reduction Roadmap. 3SUM. Convolution-3SUM. 3-party NOF communication game. triangle reporting. Multiphase Problem. …. dyn . reachability. subgraph conn. range mode. Complexity inside P.

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Towards Polynomial Lower  Bounds for Dynamic Problems

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  1. Towards Polynomial Lower Bounds for Dynamic Problems Mihai Pătrașcu STOC 2010

  2. Reduction Roadmap 3SUM Convolution-3SUM 3-party NOF communication game triangle reporting Multiphase Problem … dyn. reachability subgraphconn. range mode

  3. Complexity inside P MaxFlow: O(m1.5) time 3SUM: O(n2) time “S = {n numbers}, (∃)x,y,z ∈ S with x+y+z=0?” Wouldn’t it be nice…“If 3SUM requires Ω*(n2) MaxFlow requires Ω*(m1.5)” Is this optimal?

  4. 3SUM • O(n2) [Gajentaan, Overmars’95] • FFT  O(UlgU) if S⊆ [U] • smart hashing roughly O(n2/ lg2n) [Baran, Demaine, P.’06] Hardness: • Ω(n2) for low-degree decision tree[Erickson’95,’99] [Ailon-Chazelle’04] • no(d) for d-SUM => 2o(n) for k-SAT, ∀k=O(1) [P.-Williams’10]

  5. 3SUM-hardness • ∃ 3 collinear points? • ∃ line separating n segments in two? • minimum area triangle • Do n triangles cover given triangle? • Does this polygon fit into this polygon? • Motion planning: robot, obstacles = segments Algebraic reductions… E.g. a ↦ (a,a3) (a,a3) – (b,b3) – (c,c3) collinear a+b+c=0 “require” Ω*(n2) time

  6. A “Fancier” Reduction Theorem: If 3SUM requires Ω*(n2)  reporting m triangles in a graph requires Ω*(m4/3) Assuming FMM takes O(n2) : • Triangle detection: O(m4/3) • Reporting m triangles: O(m1.4) [Pagh] Recently, more reductions by [Vassilevska, Williams]inc. reporting triangles  triangle detection

  7. Reduction Roadmap 3SUM Convolution-3SUM 3-party NOF communication game triangle reporting Multiphase Problem … dyn. reachability subgraphconn. range mode

  8. Convolution-3SUM Convolution: A[1..n], B[1..n]  C[k] = ΣA[i+k]·B[i] k=2

  9. Convolution-3SUM Convolution-3SUM: (∃) i, k ? C[k] = A[i+k] + B[i] k=2

  10. Convolution-3SUM Convolution-3SUM: (∃) i, k ? C[k] = A[i+k] + B[i] 3SUM: (∃) i, j, k ? C[i] = A[i] + B[j] Theorem: 3SUM requires Ω*(n2) timeiff Convolution-3SUM requires Ω*(n2) time

  11. Linear Hashing Want: x ↦ h(x) linear & few collisions • almost linear (±1) • surprisingly good load balancing x random odd # h(x)

  12. 3SUM  Conv-3SUM b d e x + y = z  h(x)+h(y) = h(z)  A[h(x)]+B[h(y)]=C[h(z)] c a h(a) h(c) h(e) h(d) h(b)

  13. Reduction Roadmap 3SUM Convolution-3SUM 3-party NOF communication game triangle reporting Multiphase Problem … dyn. reachability subgraphconn. range mode

  14. Conv-3SUM  Triangle Reporting Hash A, B, C ↦ range [√n] • O(n1.5) false positives  can check all if reported fast k { k | h(B[k]) = h2 } shifted by j { k | h(A[k]) = h1 } shifted by i√n (h2, j) (h1, i) h1 + h2 = h(C[i√n + j]

  15. Reduction Roadmap 3SUM Convolution-3SUM 3-party NOF communication game triangle reporting Multiphase Problem … dyn. reachability subgraphconn. range mode

  16. Dynamic Lower Bounds tq [Fredman, Saks STOC’89]tq=Ω(lg n / lg (tulg n)) n1-o(1) nε max=Ω(lgn/lglg n) lg n tu lg n/lglg n nε lg n lg n/lglg n

  17. Dynamic Lower Bounds tq [Fredman, Saks STOC’89]tq=Ω(lg n / lg (tulg n)) [Alstrup, Husfeldt, Rauhe FOCS’98]tq=Ω(lg n / lgtu) n1-o(1) nε max=Ω(lgn/lglg n) lg n tu lg n/lglg n nε lg n

  18. Dynamic Lower Bounds tq [Fredman, Saks STOC’89]tq=Ω(lg n / lg (tulg n)) [Alstrup, Husfeldt, Rauhe FOCS’98]tq=Ω(lg n / lgtu) [P., Demaine STOC’04]tq=Ω(lg n / lg (tu/lg n)) n1-o(1) nε max=Ω(lgn) lg n tu lg n/lglg n nε lg n lg n/lglg n

  19. Dynamic Lower Bounds tq [Fredman, Saks STOC’89]tq=Ω(lg n / lg (tulg n)) [Alstrup, Husfeldt, Rauhe FOCS’98]tq=Ω(lg n / lgtu) [P., Demaine STOC’04]tq=Ω(lg n / lg (tu/lg n)) [P., Tarnita ICALP’05]tu=Ω(lg n / lg (tq/lg n)) n1-o(1) nε max=Ω(lgn) lg n tu tu lg n/lglg n nε lg n lg n/lglg n

  20. Dynamic Lower Bounds tq [Fredman, Saks STOC’89]tq=Ω(lg n / lg (tulg n)) [Alstrup, Husfeldt, Rauhe FOCS’98]tq=Ω(lg n / lgtu) [P., Demaine STOC’04]tq=Ω(lg n / lg (tu/lg n)) [P., Tarnita ICALP’05]tu=Ω(lg n / lg (tq/lg n)) [P., Thorup ’10]tu=o(lg n)  tq ≥ n1-o(1) n1-o(1) nε lg n tu lg n/lglg n nε lg n lg n/lglg n

  21. Dynamic Lower Bounds tq [Fredman, Saks STOC’89]tq=Ω(lg n / lg (tulg n)) [Alstrup, Husfeldt, Rauhe FOCS’98]tq=Ω(lg n / lgtu) [P., Demaine STOC’04]tq=Ω(lg n / lg (tu/lg n)) [P., Tarnita ICALP’05]tu=Ω(lg n / lg (tq/lg n)) [P., Thorup ’10]tu=o(lg n)  tq ≥ n1-o(1) n1-o(1) Bold conjecture nε lg n tu lg n/lglg n nε lg n lg n/lglg n

  22. The Multiphase Problem Conjecture: if u∙X << k, must have X=Ω(uε) time Si ∩T? S1, …, Sk ⊆[u] T ⊆[u] time O(k∙u∙X) time O(u∙X) time O(X)

  23. Reduction Roadmap 3SUM Convolution-3SUM 3-party NOF communication game triangle reporting Multiphase Problem … dyn. reachability subgraphconn. range mode

  24. The Multiphase Problem Conjecture: if u∙X << k, must have X=Ω(uε) Sample application: maintain array A[1..n] under updates Query: what’s the must frequent element in A[i..j]? Conjecture max{tu,tq} = Ω(nε) time Si ∩T? S1, …, Sk ⊆[u] T ⊆[u] time O(k∙u∙X) time O(u∙X) time O(X) [u]\S1 ; S1 ; … ; [u]\Si ; Si ; … ; [u]\Sk ; Sk ; T query(i)

  25. Reduction Roadmap For every edge (u,v) … test whether N(u) ∩N(v) 3SUM Convolution-3SUM 3-party NOF communication game triangle reporting Multiphase Problem … dyn. reachability subgraphconn. range mode

  26. Reduction Roadmap 3SUM Convolution-3SUM 3-party NOF communication game triangle reporting Multiphase Problem … dyn. reachability subgraphconn. range mode

  27. 3-Party, Number-on-Forehead time Si ∩T? S1, …, Sk ⊆[u] T ⊆[u] time O(k∙u∙X) time O(u∙X) time O(X) T S1, …, Sk i

  28. The End

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