Average-Case Fine-Grained Hardness Marshall Ball Alon Rosen Manuel - PowerPoint PPT Presentation

Average-Case Fine-Grained Hardness Marshall Ball Alon Rosen Manuel Sabin Prashant Nalini Vasudevan

Average-Case Fine-Grained Hardness Marshall Ball Alon Rosen Manuel Sabin Prashant Nalini Vasudevan

Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Marshall Ball Alon Rosen Manuel Sabin Prashant Nalini Vasudevan

Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Marshall Ball Alon Rosen Manuel Sabin Prashant Nalini Vasudevan ◮ 3SUM

Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Marshall Ball Alon Rosen Manuel Sabin Prashant Nalini Vasudevan ◮ 3SUM ◮ APSP

Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Marshall Ball Alon Rosen Manuel Sabin Prashant Nalini Vasudevan ◮ 3SUM ◮ APSP ◮ Orthogonal Vectors

Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Marshall Ball Alon Rosen Manuel Sabin Prashant Nalini Vasudevan ◮ 3SUM ◮ APSP ◮ Orthogonal Vectors

Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Marshall Ball Alon Rosen Manuel Sabin Prashant Nalini Vasudevan ◮ 3SUM ◮ APSP ◮ Orthogonal Vectors

Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Marshall Ball Alon Rosen Manuel Sabin Prashant Nalini Vasudevan ◮ 3SUM ◮ Natural object of study ◮ APSP ◮ Orthogonal Vectors

Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Marshall Ball Alon Rosen Manuel Sabin Prashant Nalini Vasudevan ◮ 3SUM ◮ Natural object of study ◮ APSP ◮ Necessary for cryptography ◮ Orthogonal Vectors

Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Average-Case Fine-Grained Hardness Marshall Ball Alon Rosen Manuel Sabin Prashant Nalini Vasudevan ◮ 3SUM ◮ Natural object of study ◮ APSP ◮ Necessary for cryptography ◮ Orthogonal Vectors ◮ Potential use in algorithm design

Plan ◮ Introduce problems ◮ Present average-case reduction ◮ Summarise ◮ Present Proof of Work ◮ ??? ◮ Profit.

Worst-Case: Orthogonal Vectors U V

Worst-Case: Orthogonal Vectors 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 U V 0 0 1 0 0 1 1 0 0

Worst-Case: Orthogonal Vectors 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 n U V 0 0 1 0 0 1 1 0 0 d

Worst-Case: Orthogonal Vectors 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 n U V ∃ u ∈ U , v ∈ V : disjoint? 0 0 1 0 0 1 1 0 0 d

Worst-Case: Orthogonal Vectors 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 n U V ∃ u ∈ U , v ∈ V : disjoint? 0 0 1 0 0 1 1 0 0 d Best known worst-case algorithm [AWY15]: O ( n 2 − 1 / O ( log ( d / log n )) )

Worst-Case: Orthogonal Vectors 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 n U V ∃ u ∈ U , v ∈ V : disjoint? 0 0 1 0 0 1 1 0 0 d Best known worst-case algorithm [AWY15]: O ( n 2 − 1 / O ( log ( d / log n )) ) OV Conjecture (implied by SETH [Wil05]) If d = ω ( log n ) , OV takes n 2 − o ( 1 ) time.

Worst-Case: Orthogonal Vectors 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 n U V ∃ u ∈ U , v ∈ V : disjoint? 0 0 1 0 0 1 1 0 0 log 2 n Best known worst-case algorithm [AWY15]: O ( n 2 − 1 / O ( log ( d / log n )) ) OV Conjecture (implied by SETH [Wil05]) If d = ω ( log n ) , OV takes n 2 − o ( 1 ) time.

Average-Case: A Polynomial for OV (independently featured in [Wil16])       u i 1 u i 2 . . . u id  i      f U V        v j 1 v j 2 . . . v jd  j

Average-Case: A Polynomial for OV (independently featured in [Wil16])       u i 1 u i 2 . . . u id  i      f ( 1 − u i 1 v j 1 )( 1 − u i 2 v j 2 ) · · · ( 1 − u id v jd ) U V        v j 1 v j 2 . . . v jd  j

Average-Case: A Polynomial for OV (independently featured in [Wil16])     1 ⇔ u i , v j disjoint   u i 1 u i 2 . . . u id  i      f ( 1 − u i 1 v j 1 )( 1 − u i 2 v j 2 ) · · · ( 1 − u id v jd ) U V        v j 1 v j 2 . . . v jd  j

Average-Case: A Polynomial for OV (independently featured in [Wil16])     1 ⇔ u i , v j disjoint   u i 1 u i 2 . . . u id  i    = � �   f ( 1 − u i 1 v j 1 )( 1 − u i 2 v j 2 ) · · · ( 1 − u id v jd ) U V     i ∈ [ n ] j ∈ [ n ]    v j 1 v j 2 . . . v jd  j

Average-Case: A Polynomial for OV (independently featured in [Wil16])     1 ⇔ u i , v j disjoint   u i 1 u i 2 . . . u id  i    = � �   f ( 1 − u i 1 v j 1 )( 1 − u i 2 v j 2 ) · · · ( 1 − u id v jd ) U V     i ∈ [ n ] j ∈ [ n ]    v j 1 v j 2 . . . v jd  j p > n 2 f : F 2 nd → F p p

Average-Case: A Polynomial for OV (independently featured in [Wil16])     1 ⇔ u i , v j disjoint   u i 1 u i 2 . . . u id  i    = � �   f ( 1 − u i 1 v j 1 )( 1 − u i 2 v j 2 ) · · · ( 1 − u id v jd ) U V     i ∈ [ n ] j ∈ [ n ]    v j 1 v j 2 . . . v jd  j p > n 2 deg ( f ) = 2 d d = log 2 n f : F 2 nd → F p p

Worst-Case to Average-Case Theorem ∃ A in time n 1 + α : Pr x ← F 2 nd 1 [ A ( x ) = f ( x )] ≥ n o ( 1 ) p ⇓ ∃ B in time n 1 + α + o ( 1 ) that decides OV

Worst-Case to Average-Case Theorem ∃ A in time n 1 + α : Pr x ← F 2 nd 1 [ A ( x ) = f ( x )] ≥ n o ( 1 ) p ⇓ ∃ B in time n 1 + α + o ( 1 ) that decides OV Corollary OV takes n 2 − o ( 1 ) ⇒ f takes n 2 − o ( 1 ) on average

Worst-Case to Average-Case (using ideas from [Lip91, GS92, CPS99]) f : F 2 nd → F p , deg ( f ) = 2 d p

Worst-Case to Average-Case (using ideas from [Lip91, GS92, CPS99]) f : F 2 nd → F p , deg ( f ) = 2 d p p [ A ( x ) = f ( x )] ≥ 0 . 9 Pr x ← F 2 nd Time: t = n 1 + α

Worst-Case to Average-Case (using ideas from [Lip91, GS92, CPS99]) f : F 2 nd → F p , deg ( f ) = 2 d p p [ A ( x ) = f ( x )] ≥ 0 . 9 ∀ x : Pr B [ B ( x ) = f ( x )] ≥ 2 Pr x ← F 2 nd 3 Time: t = n 1 + α Time:

Worst-Case to Average-Case (using ideas from [Lip91, GS92, CPS99]) f : F 2 nd → F p , deg ( f ) = 2 d p p [ A ( x ) = f ( x )] ≥ 0 . 9 ∀ x : Pr B [ B ( x ) = f ( x )] ≥ 2 Pr x ← F 2 nd 3 Time: t = n 1 + α Time: F 2 nd p x

Worst-Case to Average-Case (using ideas from [Lip91, GS92, CPS99]) f : F 2 nd → F p , deg ( f ) = 2 d p p [ A ( x ) = f ( x )] ≥ 0 . 9 ∀ x : Pr B [ B ( x ) = f ( x )] ≥ 2 Pr x ← F 2 nd 3 Time: t = n 1 + α Time: g ( t ) = f ( x + y t ) g ( 0 ) = f ( x ) , deg ( g ) ≤ 2 d F 2 nd p x

Worst-Case to Average-Case (using ideas from [Lip91, GS92, CPS99]) f : F 2 nd → F p , deg ( f ) = 2 d p p [ A ( x ) = f ( x )] ≥ 0 . 9 ∀ x : Pr B [ B ( x ) = f ( x )] ≥ 2 Pr x ← F 2 nd 3 Time: t = n 1 + α Time: g ( t ) = f ( x + y t ) g ( 0 ) = f ( x ) , deg ( g ) ≤ 2 d x + y t F 2 nd p x

Worst-Case to Average-Case (using ideas from [Lip91, GS92, CPS99]) f : F 2 nd → F p , deg ( f ) = 2 d p p [ A ( x ) = f ( x )] ≥ 0 . 9 ∀ x : Pr B [ B ( x ) = f ( x )] ≥ 2 Pr x ← F 2 nd 3 Time: t = n 1 + α Time: g ( t ) = f ( x + y t ) g ( 0 ) = f ( x ) , deg ( g ) ≤ 2 d Error-correct from (noisy) g ( 1 ) , g ( 2 ) , . . . , g ( cd ) x + y t F 2 nd p x

Worst-Case to Average-Case (using ideas from [Lip91, GS92, CPS99]) f : F 2 nd → F p , deg ( f ) = 2 d p p [ A ( x ) = f ( x )] ≥ 0 . 9 ∀ x : Pr B [ B ( x ) = f ( x )] ≥ 2 Pr x ← F 2 nd 3 Time: t = n 1 + α Time: g ( t ) = f ( x + y t ) g ( 0 ) = f ( x ) , deg ( g ) ≤ 2 d Error-correct from (noisy) g ( 1 ) , g ( 2 ) , . . . , g ( cd ) x + y t F 2 nd p Pr y [ too many t ’s : A ( x + y t ) � = g ( t )] < 1 3 x (Markov Bound)

Worst-Case to Average-Case (using ideas from [Lip91, GS92, CPS99]) f : F 2 nd → F p , deg ( f ) = 2 d p p [ A ( x ) = f ( x )] ≥ 0 . 9 ∀ x : Pr B [ B ( x ) = f ( x )] ≥ 2 Pr x ← F 2 nd 3 Time: � Time: t = n 1 + α O ( d · nd + d · t + d 3 ) g ( t ) = f ( x + y t ) g ( 0 ) = f ( x ) , deg ( g ) ≤ 2 d Error-correct from (noisy) g ( 1 ) , g ( 2 ) , . . . , g ( cd ) x + y t F 2 nd p Pr y [ too many t ’s : A ( x + y t ) � = g ( t )] < 1 3 x (Markov Bound)

Worst-Case to Average-Case (using ideas from [Lip91, GS92, CPS99]) f : F 2 nd → F p , deg ( f ) = 2 d p p [ A ( x ) = f ( x )] ≥ 0 . 9 ∀ x : Pr B [ B ( x ) = f ( x )] ≥ 2 Pr x ← F 2 nd 3 Time: � Time: t = n 1 + α O ( d · nd + d · t + d 3 ) � � � f ( U , V ) = ( 1 − u i ℓ v j ℓ ) i ∈ [ n ] j ∈ [ n ] ℓ ∈ [ d ]