Cost functionals for large random trees Marion Sciauveau Joint work - PowerPoint PPT Presentation

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Cost functionals for large random trees Marion Sciauveau Joint work with J-F. Delmas and J-S. Dhersin CERMICS (ENPC) and LAGA (Paris 13) es de demain - IH´ Les probabilit´ ES - 11 mai 2017 Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Introduction Trees have lot of applications in various fields such as computer science for data structure or in biology for genealogical or phylogenetic trees of extant species. Here we will consider the class of binary trees (under the Catalan model). Cost functionals are functions defined on the set of trees and described by a recurrence relation. They allow to represent the cost of many divide-and-conquer algorithms and to study the balance of trees. Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Some notations for binary trees T n rooted full binary ordered tree with n internal nodes | T n | = 2 n + 1 : the cardinal of T n L (T n ) : the left-sub-tree of T n R (T n ) : the right-sub-tree of T n the sub-tree T n,v of T n with root v Figure: A binary tree with 5 internal nodes Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Random binary trees A random binary tree is a binary tree selected at random from some probability distribution on binary trees. We often consider two models: Catalan model and Random permutation model . In what follows, we will only take interest in the Catalan model: Catalan model: random tree uniformly distributed among the full binary ordered trees with given number of internals nodes. In other words, the C n where C n is the n th 1 probability that a particular tree occurs is � 2 n � 1 Catalan number: C n = . n +1 n Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Introduction 1 Binary trees and Brownian excursion 2 Results for binary trees 3 Conclusion 4 Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Definition of cost functionals Additive functional A functional F on binary trees is called an additive functional if it satisfies the following recurrence relation: F (T) = F ( L (T)) + F ( R (T)) + b | T | for all trees T such that | T | ≥ 1 and with F ( ∅ ) = 0 . ( b k , k ≥ 1) is called the toll function. Remark: � F (T n ) = b | T n,v | v ∈ T n Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Motivation (1) Goal : study the asymptotics of cost functionals with toll function of type b k = k β for β > 0 . Answer : For β > 0 , � = | T n | − ( β + 1 Z ( n ) 2 ) | T n,v | β − → 2 Z β β n →∞ � �� � v ∈ T n scaling factor � �� � additive functional For β > 0 , Fill and Kapur (2003) showed that Z ( n ) converges in β distribution to Z β . But Z β was only characterized by its moments. Fill and Janson (2007) announced that for β > 1 2 , Z β can be represented as a functional of the normalized Brownian excursion e . Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Some examples of additive functionals 1 Total path length [Aldous (1991) and Tak` acs (1994)] � � 3 2 Z ( n ) P (T n ) = d ( ∅ , v ) = | T n,u | −| T n | ∼ | T n | 1 v ∈ T n u ∈ T n � �� � b k = k � 1 | T n | − 3 a.s. − → P (T n ) n →∞ 2 Z 1 = 2 e ( s ) ds 2 0 � �� � scaling factor 2 Wiener index [Janson (2003) and Chassaing (2004)] � � � | T n,w | 2 W (T n ) = d ( u, v ) = 2 | T n | | T n,w | − 2 u,v ∈ T n w ∈ T n w ∈ T n � �� � � �� � b k = k 2 b k = k � � 5 Z ( n ) − Z ( n ) ∼ 2 | T n | 2 1 2 | T n | − 5 a.s. W (T n ) n →∞ 4 ( Z 1 − Z 2 ) − → 2 � �� � scaling factor Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Motivation (2) We study � | T n,v | � � | T n | − 3 | T n,v | f 2 | T n | � �� � v ∈ T n scaling factor � �� � unnormalized additive functional for f satisfying smooth conditions. Aim : derive an invariance principle for such tree functionals. Model : Catalan model Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Introduction 1 Binary trees and Brownian excursion 2 Results for binary trees 3 Conclusion 4 Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Brownian tree associated to the normalized Brownian excursion Let e be a normalized Brownian excursion on [0 , 1] i.e. a standard Brownian motion on [0 , 1] conditioned on being nonnegative on [0 , 1] and on taking the value 0 at 1 . For s, t ∈ [0 , 1] , s < t , we define d e ( s, t ) = e ( s ) + e ( t ) − 2 inf s<u<t e ( u ) · The Browian tree is defined as T e = [0 , 1] / ∼ e where s ∼ e t ⇔ d e ( s, t ) = 0 and we still denote by d e , the induced distance on the quotient. We denote by p the canonical projection from [0 , 1] to T e . Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Natural embedding of binary trees into the Brownian excursion e (1) Normalized Brownian excursion: e Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Natural embedding of binary trees into the Brownian excursion e (1) Normalized Brownian excursion: e ( U i ) 1 ≤ i ≤ 5 i.i.d. uniform on [0 , 1] and indep. of e Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Natural embedding of binary trees into the Brownian excursion e (1) Normalized Brownian excursion: e ( U i ) 1 ≤ i ≤ 5 i.i.d. uniform on [0 , 1] and indep. of e ( V i ) 1 ≤ i ≤ 4 such that e ( V i ) = u ∈ [ U ( i ) ,U ( i +1) ] e ( u ) min Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Natural embedding of binary trees into the Brownian excursion e (2) Figure: The Brownian excursion and T [ n ] for n = 4 Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Natural embedding of binary trees into the Brownian excursion e (2) Figure: The Brownian excursion, T [ n ] (for n = 4 ) and T n Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Introduction 1 Binary trees and Brownian excursion 2 Results for binary trees 3 Conclusion 4 Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Length of a subexcursion σ r,s = length of the excursion of e above level r straddling s � 1 σ r,s = dt 1 { min e ( s,t ) ≥ r } 0 Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Invariance principle Let � | T n,v | � 2 � A n ( f ) = | T n | − 3 | T n,v | f | T n | v ∈ T n and � 1 � e s Φ e ( f ) = ds dr f ( σ r,s ) 0 0 Theorem A.s., ∀ f ∈ C ((0 , 1]) s.t. lim x ↓ 0 + x a f ( x ) = 0 for some 0 ≤ a < 1 2 , we have: n → + ∞ A n ( f ) = 2 Φ e ( f ) lim Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Application for f ( x ) = x β − 1 For β > 0 and n ∈ N ∗ , we set: � 1 � e s 2 ) � Z ( n ) = | T n | − ( β + 1 dr σ β − 1 | T n,v | β Z β = ds and r,s β 0 0 v ∈ T n Theorem We have a.s., ∀ β > 0 , n → + ∞ Z ( n ) lim = 2 Z β β Marion Sciauveau Cost functionals for large random trees

Framework Introduction Binary trees and Brownian excursion Results for binary trees Conclusion Application for f ( x ) = x β − 1 For β > 0 and n ∈ N ∗ , we set: � 1 � e s 2 ) � Z ( n ) = | T n | − ( β + 1 dr σ β − 1 | T n,v | β Z β = ds and r,s β 0 0 v ∈ T n Theorem We have a.s., ∀ β > 0 , n → + ∞ Z ( n ) lim = 2 Z β β Lemma If β > 1 2 , a.s. Z β < + ∞ and E [ Z β ] < + ∞ Otherwise, a.s. Z β = + ∞ Marion Sciauveau Cost functionals for large random trees