1 / 28

Towards Polynomial Lower  Bounds for Dynamic Problems

This paper discusses techniques for establishing polynomial lower bounds for various dynamic problems, including reductions related to 3SUM, triangle reporting, and dynamic reachability. It highlights the significance of computational hardness in problems such as 3-party NOF communication games, subgraph connectivity, and range mode queries. The work explores algebraic reductions, the relevance of decision trees, and conjectures regarding dynamic lower bounds. Key paradigms are presented through the lens of empirical performance and theoretical implications, aiming for optimal efficiency in complexity assessments.

cicely
Télécharger la présentation

Towards Polynomial Lower  Bounds for Dynamic Problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related