1. Towards Polynomial Lower Bounds for Dynamic Problems STOC 2010 Mihai P ă trașcu

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

3. Complexity inside P MaxFlow: O(m 1.5 ) time 3SUM: O(n 2 ) time “S = {n numbers}, ( ∃ )x,y,z ∈ S with x+y+z=0?” Wouldn’t it be nice… “If 3SUM requires Ω*(n 2 )  MaxFlow requires Ω*(m 1.5 )” Is this optimal?

4. 3SUM O(n 2 ) [Gajentaan, Overmars’95] FFT  O(U lg U) if S ⊆ [U] smart hashing  roughly O(n 2 / lg 2 n) [Baran, Demaine, P.’06] Hardness: Ω(n 2 ) for low-degree decision tree [Erickson’95,’99] [Ailon-Chazelle’04] n o(d) for d-SUM => 2 o(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,a 3 ) (a,a 3 ) – (b,b 3 ) – (c,c 3 ) collinear  a+b+c=0 “require” Ω*(n 2 ) time

6. A “Fancier” Reduction Theorem: If 3SUM requires Ω*(n 2 )  reporting m triangles in a graph requires Ω*(m 4/3 ) Assuming FMM takes O(n 2 ) : Triangle detection: O(m 4/3 ) Reporting m triangles: O(m 1.4 ) [Pagh] Recently, more reductions by [Vassilevska, Williams] inc. reporting triangles  triangle detection

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

8. Convolution-3SUM Convolution:A[1..n], B[1..n]  C[k] = Σ A[i+k]·B[i] A[1]A[2]A[3]A[4]A[5]A[6]A[7]A[8] B[1]B[2]B[3]B[4]B[5]B[6]B[7]B[8] k=2 C[0]C[1]C[2]C[3]C[4]C[5]C[6]C[7]

9. Convolution-3SUM Convolution-3SUM: ( ∃ ) i, k ? C[k] = A[i+k] + B[i] A[1]A[2]A[3]A[4]A[5]A[6]A[7]A[8] B[1]B[2]B[3]B[4]B[5]B[6]B[7]B[8] k=2 C[0]C[1]C[2]C[3]C[4]C[5]C[6]C[7]

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*(n 2 ) time iffConvolution-3SUM requires Ω*(n 2 ) 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 aebdc x + y = z  h(x)+h(y) = h(z)  A[h(x)]+B[h(y)]=C[h(z)] a b c d e h(a) h(e) h(b) h(d) h(c)

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

14. Conv-3SUM  Triangle Reporting Hash A, B, C ↦ range [√n] O(n 1.5 ) false positives  can check all if reported fast h 1 + h 2 = h(C[i√n + j] { k | h(B[k]) = h 2 } shifted by j { k | h(A[k]) = h 1 } shifted by i√n (h 1, i) (h 2, j) k

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

16. Dynamic Lower Bounds [Fredman, Saks STOC’89] t q =Ω(lg n / lg (t u lg n)) tqtq tutu max =Ω(lg n/lglg n) n 1-o(1) nεnε lg n lg n/lglg n lg n nεnε lg n/lglg n

17. Dynamic Lower Bounds [Fredman, Saks STOC’89] t q =Ω(lg n / lg (t u lg n)) [Alstrup, Husfeldt, Rauhe FOCS’98] t q =Ω(lg n / lg t u ) tqtq tutu max =Ω(lg n/lglg n) n 1-o(1) nεnε lg n nεnε lg n/lglg n

18. Dynamic Lower Bounds [Fredman, Saks STOC’89] t q =Ω(lg n / lg (t u lg n)) [Alstrup, Husfeldt, Rauhe FOCS’98] t q =Ω(lg n / lg t u ) [P., Demaine STOC’04] t q =Ω(lg n / lg (t u /lg n)) tqtq tutu max =Ω(lg n) n 1-o(1) nεnε lg n lg n/lglg n lg n nεnε lg n/lglg n

19. Dynamic Lower Bounds [Fredman, Saks STOC’89] t q =Ω(lg n / lg (t u lg n)) [Alstrup, Husfeldt, Rauhe FOCS’98] t q =Ω(lg n / lg t u ) [P., Demaine STOC’04] t q =Ω(lg n / lg (t u /lg n)) [P., Tarnita ICALP’05] t u =Ω(lg n / lg (t q /lg n)) tqtq tutu n 1-o(1) nεnε lg n lg n/lglg n lg n nεnε lg n/lglg n tutu max =Ω(lg n)

20. Dynamic Lower Bounds [Fredman, Saks STOC’89] t q =Ω(lg n / lg (t u lg n)) [Alstrup, Husfeldt, Rauhe FOCS’98] t q =Ω(lg n / lg t u ) [P., Demaine STOC’04] t q =Ω(lg n / lg (t u /lg n)) [P., Tarnita ICALP’05] t u =Ω(lg n / lg (t q /lg n)) [P., Thorup ’10] t u =o(lg n)  t q ≥ n 1-o(1) tqtq tutu n 1-o(1) nεnε lg n lg n/lglg n lg n nεnε lg n/lglg n

21. Dynamic Lower Bounds [Fredman, Saks STOC’89] t q =Ω(lg n / lg (t u lg n)) [Alstrup, Husfeldt, Rauhe FOCS’98] t q =Ω(lg n / lg t u ) [P., Demaine STOC’04] t q =Ω(lg n / lg (t u /lg n)) [P., Tarnita ICALP’05] t u =Ω(lg n / lg (t q /lg n)) [P., Thorup ’10] t u =o(lg n)  t q ≥ n 1-o(1) tqtq tutu n 1-o(1) nεnε lg n lg n/lglg n lg n nεnε lg n/lglg n Bold conjecture

22. The Multiphase Problem Conjecture: if u∙X << k, must have X=Ω(u ε ) S 1, …, S k ⊆ [u] time T ⊆ [u] time O(k∙u∙X)time O(u∙X)time O(X) S i ∩T?

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

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

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

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

27. 3-Party, Number-on-Forehead S 1, …, S k ⊆ [u] time T ⊆ [u] time O(k∙u∙X)time O(u∙X)time O(X) S i ∩T? i S 1, …, S k T

28. The End