Mean asymptotic behaviour of radix-rational sequences and dilation - PowerPoint PPT Presentation

Mean asymptotic behaviour of radix-rational sequences and dilation equations Philippe Dumas, Algorithms, INRIA Paris-Rocquencourt September 5th, 2011

What is a radix-rational sequence (aka a k -regular sequence)? A very simple example � 1 if n is a power of 2 u n = 0 otherwise u n � 1 � 0 � ❅ � � 0 0 A 0 = A 1 = � ❅ 0 1 1 0 � ❅ � ❅ � 1 u 2 n = u n u 2 n +1 � � ❅ � � L = 0 1 C = 0 � ❅ u 4 n +1 = u 2 n +1 u 4 n +3 = 0

Formal definition A (complex) sequence is rational with respect to radix B if there is a finite dimensional vector space which contains the sequence and is left stable by the B -section operators. (Allouche and Shallit, 1992) Linear representation A sequence is B -rational if and only if it admits a linear representation A 0 , A 1 , . . . , A B − 1 , L , C .

Domains ◮ binary coding of integers (sum of digits, Thue-Morse sequence, Rudin-Shapiro sequence . . . ) ◮ theory of numbers (Pascal triangle reduced modulo a power of 2, sum of three squares) ◮ divide-and-conquer algorithms (binary powering, Euclidean matching . . . ) A more natural example Cost of mergesort in the worst case u n = u ⌈ n/ 2 ⌉ + u ⌊ n/ 2 ⌋ + n − 1 , u 0 = 0 , u 1 = 0  2 1 0 0 0   1 0 0 0 0  0 1 0 0 0 1 2 0 0 0         B = 2 A 0 = 0 1 2 1 1 A 1 = 1 0 0 1 − 1         1 0 0 1 0 0 0 0 0 0     0 0 0 0 1 0 1 2 1 3 � tr � � � L = 0 0 0 0 1 C = 1 0 0 0 0 13 = (1101) 2 u 13 = LA 1 A 1 A 0 A 1 C Classical case B = 1, u n = LA n 0 C

Asymptotic behaviour of radix-rational sequences Theorem Each B -rational sequence admits an asymptotic expansion of the form n α log ℓ � � ω ⌊ log B n ⌋ Ψ α,ℓ,ω (log B n ) + O ( n α ∗ ) u n = B ( n ) n → + ∞ α>α ∗ ,ℓ ≥ 0 ω ω modulus 1 complex number, Ψ 1-periodic function Example Worst mergesort: u n = n log 2 n + n Ψ(log 2 n ) + 1, with Ψ( t ) = 1 − { t } − 2 1 −{ t } (Flajolet and Golin, 1994) � Average or not? Study of u n 0 ≤ n ≤ N Tools ◮ rational formal power series ◮ dilation equation ◮ joint spectral radius ◮ Jordan reduction ◮ numeration system

Rational formal power series Radix-rational sequence and rational formal power series ◮ Every radix-rational sequence hides a rational formal power series. alphabet B = { 0 , 1 , . . . , B − 1 } � formal power series S = ( S, w ) w w ∈B ∗ here ( S, w ) = LA w 1 A w 2 · · · A w K C = LA w C if w = w 1 w 2 · · · w K ◮ Every rational formal power series defines a radix-rational sequence. n = ( w ) B , u n = ( S, w ) ◮ The rational formal power series is the essential object.

Running sum � S K ( x ) = A w C Q = A 0 + A 1 + · · · + A B − 1 | w | = K (0 .w ) B ≤ x � � � A r 1 Q K − 1 C + A x 1 A r 2 Q K − 2 C + A x 1 A x 2 A r 3 Q K − 3 C S K ( x ) = r 1 <x 1 r 2 <x 2 r 3 <x 3 � + · · · + A x 1 A x 2 · · · A r K C r K ≤ x K 0 (0 .x 1 ) 3 (0 .x 1 x 2 ) 3 B = 3 (0 .x 1 x 2 x 3 ) 3 x = (0 . 121 . . . ) 3 ❄ ❄ x ❄ ❄ ❄ ✲

Lemma With Q = A 0 + A 1 + · · · + A B − 1 , the sequence of running sums ( S K ) satisfies the recursion � A r 1 Q K C + A x 1 S K ( Bx − x 1 ) , S K +1 ( x ) = r 1 <x 1 where x 1 is the first digit in the radix- B expansion of x in [0 , 1) , with S 0 ( x ) = C .

Dilation equation Basic case � 1 � �� Hypothesis: Q K C K → + ∞ R ( K ) = V + O K for some nonzero vector V with R ( K + 1) /R ( K ) = ρω (1 + O (1 /K )) and ρ > 0, | ω | = 1. 1 F K ( x ) = R ( K ) S K ( x ) , F K +1 = L K F K 1 R ( K ) � A r 1 Q K C + L K Φ( x ) = R ( K + 1) A x 1 Φ( Bx − x 1 ) R ( K + 1) r 1 <x 1 Basic dilation equation: ◮ Φ(0) = 0, Φ(1) = V , ◮ for every digit r of the radix- B system and for x in [ r/B, ( r + 1) /B ), 1 A r 1 V + 1 � Φ( x ) = ρω A r Φ( Bx − r ) . ρω r 1 <r

Wavelets ◮ Scaling function ϕ (or father wavelet), data ( c k ) with hypotheses Daubechies, 1988 : There exists a unique function ϕ ∈ L 2 ( R ) such that K − 1 � ϕ ( x ) = c k ϕ (2 x − k ) 1 k =0 � ϕ ( x ) dx = 1 2 R supp ϕ ⊂ [0 , K − 1] 3 � ( − 1) k c g − 1 − k ϕ (2 x − k ) ◮ Mother wavelet ψ ( x ) = k ψ j,k ( x ) = 2 − j/ 2 ψ (2 − j x − k ) Wavelets ϕ k ( x ) = ϕ ( x − k ), ◮ Expansion f = � � � � f, ψ j,k � ψ j,k for f ∈ L 2 ( R ) � f, ϕ k � ϕ k + k j k ◮ Example: iteration from the box function (contracting operator)

c 0 = 1 / 2, c 1 = 1, hat c 2 = 1 / 2 c 0 = 1 / 8, c 1 = 4 / 8, cubic c 2 = 6 / 8, c 3 = 4 / 8, B-spline c 4 = 1 / 8 c 0 = (1 + α ) / 4, c 1 = (3 + α ) / 4, c 2 = (3 − α ) / 4, c 3 = (1 − α ) / 4, √ Daubechies α = 3

Refinement schemes (Deslauriers and Dubuc, 1986) ◮ Interpolation scheme (gliding Lagrange interpolation) data: ( v k ) k ∈ Z and L > 0, output: f function such that f ( k ) = v k if f defined on 1 2 j Z and x j,k = k 1 2 j + 2 j +1 then f ( x j,k ) = π j,k ( x j,k ) where π j,k is the Lagrange interpolation polynomial at p/ 2 j with k − L ≤ p ≤ k + L + 1 ◮ Correction If ( v k ) bounded, f extends to R as a continuous function ◮ Scaling function ϕ v 0 = 1, v k = 0 for k � = 0 � f ( x ) = v k ϕ ( x − k ) k ∈ Z � ϕ ( x ) = c k ϕ (2 x − k ) k ◮ Example with L = 2 (cascade algorithm)

Basic result Theorem Let L , ( A r ) 0 ≤ r<B , C be a linear representation of dimension d for the radix B . It is assumed that � 1 � �� ◮ Q K C K → + ∞ R ( K ) = V + O for some nonzero vector V with K R ( K + 1) /R ( K ) = ρω (1 + O (1 /K )) and ρ > 0 , | ω | = 1 ◮ there exists and induced norm � � and a constant λ , with 0 < λ < ρ such that all matrices A r , 0 ≤ r < B , satisfy � A r � ≤ λ . Then ◮ the basic dilation equation has a unique solution F , which is continuous from [0 , 1] into C d , ◮ the sequence ( F K ) converges uniformly towards F , with speed essentially O (( λ/ρ ) K ) . K → + ∞ R ( K ) F ( x ) + O ( λ K ) Concretely S K ( x ) =

Contribution of dilation equations ◮ Contracting operator ◮ Cascade algorithm ◮ Regularity F is H¨ older with exponent log B ( ρ/λ ) ◮ Form of the dilation equation ( B = 2) ⋆ Piecewise equation, non homogeneous � T 0 F (2 x ) if 0 ≤ x ≤ 1 / 2 F ( x ) = T 0 V + T 1 F (2 x − 1) if 1 / 2 ≤ x ≤ 1 ⋆ Global equation, homogeneous F ( x ) = T 0 F (2 x ) + T 1 F (2 x − 1) for x real with F constant on the left of 0 and on the right of 1 ◮ Example Billingsley’s distribution functions (Billingsley, 1995) X n � X = 2 n , X n = Bernoulli( p ), 0 < p < 1 n ≥ 0

� 1 � � 1 − p � � p � � 1 � L = , A 0 = , A 1 = , C = � 1 � � 1 � Q = , ρ = 1, ω = 1 V = , R ( K ) = 1, F distribution function, F ( x ) = (1 − p ) F (2 x ) + pF (2 x − 1), F (0) = 0, F (1) = 1 λ = max( p, 1 − p ) H¨ older exponent α = log 2 (1 / max( p, 1 − p )) ≃ 0 . 62 (here p = 13 / 20) if 1 / 2 < p < 1 essentially best H¨ older exponent on the right α + = log 2 (1 / (1 − p )) ≃ 1 . 51 > 1 essentially best H¨ older exponent on the left α − = α

Joint spectral radius Controlling products A w Rota and Strang, 1960: λ T = max | w | = T � A w � 1 /T joint spectral radius λ ∗ = lim T → + ∞ λ T example with worstmergesort 1 = ⋄ ∞ = ⋄ 2 = ⋄

Changing the radix ◮ From B to B T If a sequence is B -rational, it is B T -rational forall T ∈ N > 0 . Linear representation L , A r , C with 0 ≤ r < B for B becomes L , A w , C with | w | = T for B T . A r becomes Q ( T ) = ◮ Eigenvalues Q = � � A w = Q T 0 ≤ r<B | w | = T ρ becomes ρ T (and S K becomes S KT ) ◮ Dichotomy (desired) ρ ✲ • • • • • • • 0 λ ∗ λ T λ 1 ❄ ❄ ❄ ❄ ❄ ❄ ❄ error term ? expansion

Jordan reduction ◮ Idea Jordan reduction of matrix Q and processing of each generalized eigenspace Vector valued functions become matrix valued functions, but same arguments ◮ Qualitative result Theorem Let L , ( A r ) 0 ≤ r<B , C be a linear representation of a formal power series S . The sequence of running sums � S K ( x ) = L S K ( x ) = LA w C | w | = K (0 .w ) B ≤ x admits an asymptotic expansion with error term O ( λ K ) for every λ > λ ∗ , where λ ∗ is the joint spectral radius of the family ( A r ) 0 ≤ r<B . The used asymptotic scale is the family of sequences ρ K � K � , ρ > 0 , ℓ ∈ N ≥ 0 . The coefficients are related to solutions of ℓ dilation equations. The error term is uniform with respect to x ∈ [0 , 1] .

Numeration system N � We return to u n . n =0 ◮ Idea N = B K + t , K = ⌊ log B N ⌋ , t = { log B N } sum up to N = [sum of u n up to B K − 1] plus [sum of u n from B k to N ]= [(sum of ( S, w ) for | w | ≤ K ) minus (sum for those words beginning with 0)] plus [(running sum of ( S, w ) up to N for words of length K + 1) minus (sum for those words beginning with 0)]