towards polynomial lower bounds for dynamic problems
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Towards Polynomial Lower Bounds for Dynamic Problems
STOC 2010
Mihai Pătrașcu
Reduction Roadmap3SUM
Convolution-3SUM
triangle reporting
Multiphase Problem
dyn. reachability subgraph conn. range mode …
3-party NOF communication game
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?
3SUM• O(n2) [Gajentaan, Overmars’95]
• FFT O(U lg U) 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]
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
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
Reduction Roadmap
Multiphase Problem
dyn. reachability subgraph conn. range mode …
3SUM
Convolution-3SUM
triangle reporting
3-party NOF communication game
Convolution-3SUM
Convolution: A[1..n], B[1..n] C[k] = ΣA[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]
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]
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
Linear Hashing
Want: x h(x) linear & few collisions↦
• almost linear (±1)• surprisingly good load balancing
x
random odd #
h(x)
3SUM Conv-3SUM
a e b d c
x + y = z
h(x)+h(y) = h(z)
A[h(x)]+B[h(y)]=C[h(z)]
abc
de
h(a)h(e
)h(b)
h(d)h(c)
Reduction Roadmap
Multiphase Problem
dyn. reachability subgraph conn. range mode …
3SUM
Convolution-3SUM
triangle reporting
3-party NOF communication game
Conv-3SUM Triangle Reporting
Hash A, B, C range [√n]↦• O(n1.5) false positives can check all if reported fast
h1 + h2 = h(C[i√n
+ j]
{ k | h(B[k]) = h2 } shifted by j
{ k | h(A[k]) = h1 }shifted by i√n
(h1, i) (h2, j)
k
Reduction Roadmap3SUM
Convolution-3SUM
triangle reporting
dyn. reachability subgraph conn. range mode …Multiphase Problem
3-party NOF communication game
Dynamic Lower Bounds[Fredman, Saks STOC’89]
tq=Ω(lg n / lg (tulg n))tq
tu
max=Ω(lg n
/lglg n)
n1-o(1)
nε
lg n
lg n/lglg
nlg
n nεlg n/lglg
n
Dynamic Lower Bounds[Fredman, Saks STOC’89]
tq=Ω(lg n / lg (tulg n))
[Alstrup, Husfeldt, Rauhe FOCS’98]tq=Ω(lg n / lg tu)
tq
tu
max=Ω(lg n
/lglg n)
n1-o(1)
nε
lg n
lg n nεlg
n/lglg
n
Dynamic Lower Bounds[Fredman, Saks STOC’89]
tq=Ω(lg n / lg (tulg n))
[Alstrup, Husfeldt, Rauhe FOCS’98]tq=Ω(lg n / lg tu)
[P., Demaine STOC’04]tq=Ω(lg n / lg (tu/lg n))
tq
tu
max=Ω(lg n
)
n1-o(1)
nε
lg n
lg n/lglg
nlg
n nεlg n/lglg
n
Dynamic Lower Bounds[Fredman, Saks STOC’89]
tq=Ω(lg n / lg (tulg n))
[Alstrup, Husfeldt, Rauhe FOCS’98]tq=Ω(lg n / lg tu)
[P., Demaine STOC’04]tq=Ω(lg n / lg (tu/lg n))
[P., Tarnita ICALP’05]tu=Ω(lg n / lg (tq/lg n))
tq
tu
n1-o(1)
nε
lg n
lg n/lglg
nlg
n nεlg n/lglg
ntu
max=Ω(lg n
)
Dynamic Lower Bounds[Fredman, Saks STOC’89]
tq=Ω(lg n / lg (tulg n))
[Alstrup, Husfeldt, Rauhe FOCS’98]tq=Ω(lg n / lg tu)
[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)
tq
tu
n1-o(1)
nε
lg n
lg n/lglg
nlg
n nεlg n/lglg
n
Dynamic Lower Bounds[Fredman, Saks STOC’89]
tq=Ω(lg n / lg (tulg n))
[Alstrup, Husfeldt, Rauhe FOCS’98]tq=Ω(lg n / lg tu)
[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)
tq
tu
n1-o(1)
nε
lg n
lg n/lglg
nlg
n nεlg n/lglg
n
Bold conjecture
The Multiphase Problem
Conjecture: if u X << k∙ , must have X=Ω(uε)
S1, …, Sk [u]⊆ timeT [u]⊆
time O(k u X)∙ ∙ time O(u X)∙ time O(X)
Si ∩T?
Reduction Roadmap3SUM
Convolution-3SUM
triangle reporting
dyn. reachability subgraph conn. range mode …Multiphase Problem
3-party NOF communication game
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)
S1, …, Sk [u]⊆ timeT [u]⊆
time O(k u X)∙ ∙ time O(u X)∙ time O(X)
[u]\S1 ; S1 ; … ; [u]\Si ; Si ; … ; [u]\Sk ; Sk ; T
Si ∩T?
query(i)
Reduction Roadmap3SUM
Convolution-3SUM
triangle reporting
dyn. reachability subgraph conn. range mode …Multiphase Problem
For every edge (u,v)… test whether N(u) ∩N(v)
3-party NOF communication game
Reduction Roadmap3SUM
Convolution-3SUM
triangle reporting
dyn. reachability subgraph conn. range mode …Multiphase Problem
3-party NOF communication game
3-Party, Number-on-Forehead
S1, …, Sk [u]⊆ timeT [u]⊆
time O(k u X)∙ ∙ time O(u X)∙ time O(X)
Si ∩T?
i S1, …, Sk T
The End