Finding Monotone Patterns in Sublinear Time Erik Waingarten - PowerPoint PPT Presentation

Finding Monotone Patterns in Sublinear Time Erik Waingarten (Columbia University) Cl´ ement Canonne (Stanford University) Omri Ben-Eliezer (Tel-Aviv University) Shoham Letzter (ETH Zurich) 1 / 14

Testing Monotonicity of an Array: Sortedness 1 3 2 3 6 7 1 9 4 2 5 6 7 Given query access to an unknown f : [ n ] → R and a parameter ε > 0: If f monotone, accept w.p. > 2 / 3; If f is ε -far from monotone, reject w.p. > 2 / 3; Minimum query complexity in terms of n and ε ? 2 / 14

Testing Monotonicity of an Array: Sortedness 1 3 2 3 6 7 1 9 4 2 5 6 7 Given query access to an unknown f : [ n ] → R and a parameter ε > 0: If f monotone, accept w.p. > 2 / 3; If f is ε -far from monotone, reject w.p. > 2 / 3; Minimum query complexity in terms of n and ε ? If f is ε -far from monotone, find evidence: ◮ i , j ∈ [ n ] where i < j and f ( i ) > f ( j ). 2 / 14

Testing Monotonicity of an Array: Sortedness 1 3 2 3 6 7 1 9 4 2 5 6 7 Given query access to an unknown f : [ n ] → R and a parameter ε > 0: If f monotone, accept w.p. > 2 / 3; If f is ε -far from monotone, reject w.p. > 2 / 3; Minimum query complexity in terms of n and ε ? If f is ε -far from monotone, find evidence: ◮ i , j ∈ [ n ] where i < j and f ( i ) > f ( j ). one-sided error. 2 / 14

ε -Far-From-Monotone Sequences f is ε -far from monotone: for any monotone g : [ n ] → R , n 1 � 1 { f ( i ) � = g ( i ) } ≥ ε. n i =1 3 / 14

ε -Far-From-Monotone Sequences f is ε -far from monotone: for any monotone g : [ n ] → R , n 1 � 1 { f ( i ) � = g ( i ) } ≥ ε. n i =1 Lemma Let f : [ n ] → R be ε -far from monotone. There exists set of disjoint pairs, ( i , j ) ∈ [ n ] 2 : i < j and f ( i ) > f ( j ) � � T = of size | T | ≥ ε n / 2 . 3 / 14

Monotonicity Testing Theorem (Erg¨ un, Kannan, Kumar, Rubinfeld, Viswanathan 99) There exists a non-adaptive, one-sided algorithm for testing monotonicity of f : [ n ] → R making O ((log n ) /ε ) queries. 4 / 14

Monotonicity Testing Theorem (Erg¨ un, Kannan, Kumar, Rubinfeld, Viswanathan 99) There exists a non-adaptive, one-sided algorithm for testing monotonicity of f : [ n ] → R making O ((log n ) /ε ) queries. Ω((log n ) /ε ) queries needed for non-adaptive algorithms. 4 / 14

Monotonicity Testing Theorem (Erg¨ un, Kannan, Kumar, Rubinfeld, Viswanathan 99) There exists a non-adaptive, one-sided algorithm for testing monotonicity of f : [ n ] → R making O ((log n ) /ε ) queries. Ω((log n ) /ε ) queries needed for non-adaptive algorithms. Ω((log n ) /ε ) queries needed for adaptive algorithms too! [Fischer 04]. 4 / 14

Testing Forbidden Order Patterns [Newman, Rabinovich, Rajendraprasad, Sohler 17] 5 / 14

Testing Forbidden Order Patterns [Newman, Rabinovich, Rajendraprasad, Sohler 17] Definition Let k ∈ N and π = ( π 1 , . . . , π k ) be a permutation of [ k ]. Given f : [ n ] → R , the k -tuple ( i 1 , . . . , i k ) has order pattern π if: f ( i ℓ ) < f ( i m ) whenever π ℓ < π m . 5 / 14

Testing Forbidden Order Patterns [Newman, Rabinovich, Rajendraprasad, Sohler 17] Definition Let k ∈ N and π = ( π 1 , . . . , π k ) be a permutation of [ k ]. Given f : [ n ] → R , the k -tuple ( i 1 , . . . , i k ) has order pattern π if: f ( i ℓ ) < f ( i m ) whenever π ℓ < π m . If f : [ n ] → R is ε -far from π -free, then there exists T ⊂ [ n ] k of disjoint violating k -tuples ( i 1 , . . . , i k ) with order pattern π of size at least ε n / k . 5 / 14

Testing Forbidden Order Patterns [Newman, Rabinovich, Rajendraprasad, Sohler 17] Definition Let k ∈ N and π = ( π 1 , . . . , π k ) be a permutation of [ k ]. Given f : [ n ] → R , the k -tuple ( i 1 , . . . , i k ) has order pattern π if: f ( i ℓ ) < f ( i m ) whenever π ℓ < π m . If f : [ n ] → R is ε -far from π -free, then there exists T ⊂ [ n ] k of disjoint violating k -tuples ( i 1 , . . . , i k ) with order pattern π of size at least ε n / k . For fixed k and π , query complexity of testing π -freeness? 5 / 14

Testing Forbidden Order Patterns [Newman, Rabinovich, Rajendraprasad, Sohler 17] Definition Let k ∈ N and π = ( π 1 , . . . , π k ) be a permutation of [ k ]. Given f : [ n ] → R , the k -tuple ( i 1 , . . . , i k ) has order pattern π if: f ( i ℓ ) < f ( i m ) whenever π ℓ < π m . If f : [ n ] → R is ε -far from π -free, then there exists T ⊂ [ n ] k of disjoint violating k -tuples ( i 1 , . . . , i k ) with order pattern π of size at least ε n / k . For fixed k and π , query complexity of testing π -freeness? Some sublinear in n upper bounds for general π . 5 / 14

Testing Forbidden Order Patterns [Newman, Rabinovich, Rajendraprasad, Sohler 17] Definition Let k ∈ N and π = ( π 1 , . . . , π k ) be a permutation of [ k ]. Given f : [ n ] → R , the k -tuple ( i 1 , . . . , i k ) has order pattern π if: f ( i ℓ ) < f ( i m ) whenever π ℓ < π m . If f : [ n ] → R is ε -far from π -free, then there exists T ⊂ [ n ] k of disjoint violating k -tuples ( i 1 , . . . , i k ) with order pattern π of size at least ε n / k . For fixed k and π , query complexity of testing π -freeness? Some sublinear in n upper bounds for general π . π = (132) requires Ω( √ n ) queries for non-adaptive, one-sided algorithms. 5 / 14

Testing Forbidden Order Patterns [Newman, Rabinovich, Rajendraprasad, Sohler 17] Definition Let k ∈ N and π = ( π 1 , . . . , π k ) be a permutation of [ k ]. Given f : [ n ] → R , the k -tuple ( i 1 , . . . , i k ) has order pattern π if: f ( i ℓ ) < f ( i m ) whenever π ℓ < π m . If f : [ n ] → R is ε -far from π -free, then there exists T ⊂ [ n ] k of disjoint violating k -tuples ( i 1 , . . . , i k ) with order pattern π of size at least ε n / k . For fixed k and π , query complexity of testing π -freeness? Some sublinear in n upper bounds for general π . π = (132) requires Ω( √ n ) queries for non-adaptive, one-sided algorithms. [Ben-Eliezer, Canonne 18] Many π have complexity a n 1 − 1 / ( k − Θ(1)) . 5 / 14

Finding Monotone Patterns: π = (12 . . . k ) 6 / 14

Finding Monotone Patterns: π = (12 . . . k ) Given query access to f : [ n ] → R and a parameter ε > 0 where: 6 / 14

Finding Monotone Patterns: π = (12 . . . k ) Given query access to f : [ n ] → R and a parameter ε > 0 where: There exists T ⊂ [ n ] k of disjoint violating k -tuples T = { ( i 1 , . . . , i k ) : i 1 < · · · < i k and f ( i 1 ) < · · · < f ( i k ) } of size | T | ≥ ε n / k . 6 / 14

Finding Monotone Patterns: π = (12 . . . k ) Given query access to f : [ n ] → R and a parameter ε > 0 where: There exists T ⊂ [ n ] k of disjoint violating k -tuples T = { ( i 1 , . . . , i k ) : i 1 < · · · < i k and f ( i 1 ) < · · · < f ( i k ) } of size | T | ≥ ε n / k . Find i 1 < · · · < i k where f ( i 1 ) < · · · < f ( i k ). 6 / 14

Finding Monotone Patterns: π = (12 . . . k ) Given query access to f : [ n ] → R and a parameter ε > 0 where: There exists T ⊂ [ n ] k of disjoint violating k -tuples T = { ( i 1 , . . . , i k ) : i 1 < · · · < i k and f ( i 1 ) < · · · < f ( i k ) } of size | T | ≥ ε n / k . Find i 1 < · · · < i k where f ( i 1 ) < · · · < f ( i k ). Theorem (NRRS17) There is a non-adaptive algorithm with query complexity ((log n ) /ε ) O ( k 2 ) . 6 / 14

Finding Monotone Patterns: π = (12 . . . k ) 7 / 14

Finding Monotone Patterns: π = (12 . . . k ) Theorem (Upper bound) For k ∈ N , there exists a non-adaptive algorithm with query complexity (log n ) ⌊ log 2 k ⌋ · poly (1 /ε ) . 7 / 14

Finding Monotone Patterns: π = (12 . . . k ) Theorem (Upper bound) For k ∈ N , there exists a non-adaptive algorithm with query complexity (log n ) ⌊ log 2 k ⌋ · poly (1 /ε ) . Theorem (Lower bound) � (log n ) ⌊ log 2 k ⌋ � Any non-adaptive algorithm needs to make Ω queries. 7 / 14

The Case of k = 2 8 / 14

The Case of k = 2 Let f : [ n ] → R and disjoint subset of pairs T of size ε n / 2 with T = { ( i , j ) ∈ [ n ] 2 : i < j and f ( i ) < f ( j ) } . 8 / 14

The Case of k = 2 Let f : [ n ] → R and disjoint subset of pairs T of size ε n / 2 with T = { ( i , j ) ∈ [ n ] 2 : i < j and f ( i ) < f ( j ) } . [Erg¨ un, Kannan, Kumar, Rubinfeld, Viswanathan 99] Find increasing pair: 8 / 14

The Case of k = 2 Let f : [ n ] → R and disjoint subset of pairs T of size ε n / 2 with T = { ( i , j ) ∈ [ n ] 2 : i < j and f ( i ) < f ( j ) } . [Erg¨ un, Kannan, Kumar, Rubinfeld, Viswanathan 99] Find increasing pair: Sample ℓ ∼ [ n ] uniformly and repeat the following for t iterations: 8 / 14

The Case of k = 2 Let f : [ n ] → R and disjoint subset of pairs T of size ε n / 2 with T = { ( i , j ) ∈ [ n ] 2 : i < j and f ( i ) < f ( j ) } . [Erg¨ un, Kannan, Kumar, Rubinfeld, Viswanathan 99] Find increasing pair: Sample ℓ ∼ [ n ] uniformly and repeat the following for t iterations: ◮ Sample s ∼ { 0 , . . . , log 2 n } and i ∼ [ ℓ − 2 s , ℓ ] and query f ( i ). 8 / 14