On Contraction Method in function spaces and the partial match - PowerPoint PPT Presentation

On Contraction Method in function spaces and the partial match problem Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. INRIA Paris, October 18, 2010 joint work with N. Broutin & R. Neininger Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Contraction Method - Example Given a sequence of random variables ( X n ) that contains a recursive structure, contraction method is a tool to obtain asymptotic results for the distribution and moments of ( X n ). Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Contraction Method - Example Given a sequence of random variables ( X n ) that contains a recursive structure, contraction method is a tool to obtain asymptotic results for the distribution and moments of ( X n ). Example - Quickselect Task: Given a list of n different numbers, find the element of rank k , for simplicity assume k = 1. Algorithm: ◮ Choose one element x uniformly at random among all ( pivot ) ◮ Comparing all elements with x gives sublists S < and S > ◮ If I n = | S < | = 1 return x otherwise search recursively in S < Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Analysis Let X n be the number of key comparisons and I n = | S < | . Then d X n = X I n + n − 1 for ( X j ) , I n independent and I n uniformly distributed on { 0 , . . . , n − 1 } . Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Analysis Let X n be the number of key comparisons and I n = | S < | . Then d X n = X I n + n − 1 for ( X j ) , I n independent and I n uniformly distributed on { 0 , . . . , n − 1 } . E [ X n ] ≈ 2 n suggests the scaling Y n := X n n Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Analysis Let X n be the number of key comparisons and I n = | S < | . Then d X n = X I n + n − 1 for ( X j ) , I n independent and I n uniformly distributed on { 0 , . . . , n − 1 } . E [ X n ] ≈ 2 n suggests the scaling n Y I n + 1 − 1 Y n := X n = I n d n . n Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Analysis A possible limit for n Y I n + 1 − 1 Y n := X n = I n d n . n Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Analysis A possible limit for n Y I n + 1 − 1 Y n := X n = I n d n . n should satisfy d Y = UY + 1 for independent U , Y , U uniform on [0 , 1]. Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Analysis A possible limit for n Y I n + 1 − 1 Y n := X n = I n d n . n should satisfy d Y = UY + 1 for independent U , Y , U uniform on [0 , 1]. Observe that Y (or rather L ( Y )) satisfies this if L ( Y ) is a fixed-point of the following map : M ( R ) → M ( R ) F F ( µ ) = L ( UY + 1) , with L ( Y ) = µ , U uniform on [0 , 1] and U , Y independent. Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Analysis A possible limit for n Y I n + 1 − 1 Y n := X n = I n d n . n should satisfy d Y = UY + 1 for independent U , Y , U uniform on [0 , 1]. Observe that Y (or rather L ( Y )) satisfies this if L ( Y ) is a fixed-point of the following map : M ( R ) → M ( R ) F F ( µ ) = L ( UY + 1) , with L ( Y ) = µ , U uniform on [0 , 1] and U , Y independent. Idea: Use Banach fixed point theorem. Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Contraction For µ, ν ∈ M ( R ) let ℓ 1 ( µ, ν ) = X , Y : L ( X )= µ, L ( Y )= ν E [ | X − Y | ] . inf Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Contraction For µ, ν ∈ M ( R ) let ℓ 1 ( µ, ν ) = X , Y : L ( X )= µ, L ( Y )= ν E [ | X − Y | ] . inf ℓ 1 is a complete metric on the subset M ′ ( R ) of M ( R ) consisting of probability measures with finite first moment and w ℓ 1 ( µ n , µ ) → 0 ⇒ µ − → µ . Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Contraction For µ, ν ∈ M ( R ) let ℓ 1 ( µ, ν ) = X , Y : L ( X )= µ, L ( Y )= ν E [ | X − Y | ] . inf ℓ 1 is a complete metric on the subset M ′ ( R ) of M ( R ) consisting of probability measures with finite first moment and w ℓ 1 ( µ n , µ ) → 0 ⇒ µ − → µ . Show: F is a contraction according to ℓ 1 in M ′ ( R ) . Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Contraction Proof: Let X , Y s.t. L ( X ) = µ and L ( Y ) = ν and E [ | X − Y | ] ≤ ℓ 1 ( µ, ν ) + ε. Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Contraction Proof: Let X , Y s.t. L ( X ) = µ and L ( Y ) = ν and E [ | X − Y | ] ≤ ℓ 1 ( µ, ν ) + ε. Then ℓ 1 ( F ( µ ) , F ( ν )) ≤ E [ | UX + 1 − ( UY + 1) | ] = E U E [ | X − Y | ] ≤ E U ℓ 1 ( µ, ν ) + ε Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Contraction Proof: Let X , Y s.t. L ( X ) = µ and L ( Y ) = ν and E [ | X − Y | ] ≤ ℓ 1 ( µ, ν ) + ε. Then ℓ 1 ( F ( µ ) , F ( ν )) ≤ E [ | UX + 1 − ( UY + 1) | ] = E U E [ | X − Y | ] ≤ E U ℓ 1 ( µ, ν ) + ε which gives ℓ 1 ( F ( µ ) , F ( ν )) ≤ E U ℓ 1 ( µ, ν ) . � Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Contraction The stochastic fixed-point equation d Y = UY + 1 has a unique solution in M ′ ( R ) and it is easy to show that ℓ 1 ( Y n , Y ) → 0 . Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Quickselect - Contraction The stochastic fixed-point equation d Y = UY + 1 has a unique solution in M ′ ( R ) and it is easy to show that ℓ 1 ( Y n , Y ) → 0 . Typically the number of subproblems is larger than one. For example, if X n denotes the number of key comparisons performed by Quicksort sorting a list of n elements, then d = X ′ I n + X ′′ n − 1 − I n + n − 1 X n with independent copies ( X ′ j ) , ( X ′′ j ) independent of I n . Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Contraction method for recursive stochastic processes Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Contraction method for recursive stochastic processes Usual situation of an affine recursion after scaling: K � d A ( n ) ◦ X r + b ( n ) = X n r I ( n ) r r =1 with r . v . ( X n ) , b ( n ) taking values in some space S , A ( n ) random r operators from S to S , and independent copies ( X 1 n ) , . . . , ( X K n ) of ( X n ) . Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Contraction method for recursive stochastic processes Usual situation of an affine recursion after scaling: K � d A ( n ) ◦ X r + b ( n ) = X n r I ( n ) r r =1 with r . v . ( X n ) , b ( n ) taking values in some space S , A ( n ) random r operators from S to S , and independent copies ( X 1 n ) , . . . , ( X K n ) of ( X n ) . If A ( n ) → A r and b ( n ) → b for some S valued processes A r , b , this r suggests X n → X , = � K r =1 A r ◦ X ( r ) + b (uniquely). where X solves X d Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Applications - The R d case Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Applications - The R d case ◮ K = 1 : Quickselect ◮ K = 2 : BST, RRT: Pathlength, Profile,. . . , Size of random Tries ◮ K = m : m -ary search trees ◮ K = K ( n ) random: Galton-Watson trees ◮ d = 2 : Wiener Index Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Example in C ([0 , 1]) - Donsker’s Theorem Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match

Example in C ([0 , 1]) - Donsker’s Theorem Let X 1 , X 2 , . . . be iid random variables with E X 1 = 0 , E X 2 1 = 1. The process   ⌊ nt ⌋ � 1 S n  ,  t = √ n X k + ( nt − ⌊ t ⌋ ) X ⌊ nt ⌋ +1 t ∈ [0 , 1] k =1 Henning Sulzbach J. W. Goethe-Universit¨ at Frankfurt a. M. On Contraction Method in function spaces and the partial match