A new bijection on m -Dyck paths and application to random sampling - - PowerPoint PPT Presentation

A new bijection on m-Dyck paths and application to random sampling

Axel Bacher

LIPN, Université Paris 13

June 3rd, 2016

Outline

1

Introduction

2

The folding bijection m-Łukasiewicz paths and m-Dyck prefixes Folding and unfolding

3

Random sampling Main algorithm Complexity and limit law

4

Perspectives

m-Dyck paths and (m+1)-ary trees

m-Dyck path: path in N from 0 to 0 with steps in {+1, −m}. The number of paths of length n = (m+1) n′ is 1 mn′+1 n n′

• (Fuß-Catalan number).

Random sampling of general classes of trees in time O(n log n), based on the cycle lemma. [Devroye ’12]

The folding bijection for Dyck paths

One can sample a Dyck path in time O(n) by sampling a Dyck prefix and use the folding bijection to get a pointed Łukasiewicz path.

[Barcucci, Pinzani, Sprugnoli ’92; B., Bodini, Jacquot ’15]

m-Łukasiewicz paths

m-Łukasiewicz path: non-negative path except at its end. Paths of length n = (m+1) n′ + r have height h = r − (m+1). Their number is r n n n′

• (Raney number).

m-Łukasiewicz paths

m-Łukasiewicz path: non-negative path except at its end. Paths of length n = (m+1) n′ + r have height h = r − (m+1). Their number is r n n n′

• (Raney number).

A pointed path has an associated factorization: w = pq0d · · · qkd, where

• 0 ≤ h(qi) ≤ m − 1,

i < k 0 ≤ h(qk) ≤ r − 1.

Decorated m-Dyck prefixes

Paths of length n = (m+1) n′ + r have height h = (m+1) h′ + r.

Decorated m-Dyck prefixes

a0 = 1 a1 = 3 a2 = 2 Paths of length n = (m+1) n′ + r have height h = (m+1) h′ + r. A decoration of an m-Dyck prefix is defined as a sequence: (a0, . . . , ah′), where

• 1 ≤ ai ≤ m,

i < h′ 1 ≤ ak ≤ r. (a path has mh′r possible decorations).

Decorated m-Dyck prefixes

a0 = 1 a1 = 3 a2 = 2 Paths of length n = (m+1) n′ + r have height h = (m+1) h′ + r. A decoration of an m-Dyck prefix is defined as a sequence: (a0, . . . , ah′), where

• 1 ≤ ai ≤ m,

i < h′ 1 ≤ ak ≤ r. (a path has mh′r possible decorations). A decorated m-Dyck prefix has an associated factorization: w = puq0 · · · uqh′, where h(qi) = ai.

Folding and unfolding

puq0 · · · uqk pq0d · · · qkd

Theorem

The folding operation is a bijection between decorated m-Dyck prefixes and pointed m-Łukasiewicz paths. Folding or unfolding only requires reading the part of the path after the point.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

If we go in the negatives, we randomly point the path and unfold.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

If we go in the negatives, we randomly point the path and unfold.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

If we go in the negatives, we randomly point the path and unfold.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

If we go in the negatives, we randomly point the path and unfold.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

If we go in the negatives, we randomly point the path and unfold.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

If we go in the negatives, we randomly point the path and unfold.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

If we go in the negatives, we randomly point the path and unfold.

Random m-Dyck prefix

We draw u and d steps with probabilities

m m+1 and 1 m+1.

If we go in the negatives, we randomly point the path and unfold. At all times, paths of height (m+1) h′ + r appear with probability proportional to mh′.

Random m-Łukasiewicz path

We draw a random m-Dyck prefix of length n = (m+1) n′ + r and height h = (m+1) h′ + r. We randomly decorate this prefix and we fold. The result is a uniform pointed m-Łukasiewicz path.

Complexity

Random m-Łukasiewicz path

w ← ε for i = 1, . . . , n do s ← u with probability

m m+1, d otherwise

w ← ws if h(w) < 0 then randomly point w unfold w (forget the decoration) end if end for randomly decorate w fold w (forget the point) return w

Complexity

Random m-Łukasiewicz path

w ← ε for i = 1, . . . , n do s ← u with probability

m m+1, d otherwise

w ← ws if h(w) < 0 then randomly point w unfold w (forget the decoration) end if end for randomly decorate w fold w (forget the point) return w We consider complexity in random bits and memory accesses.

Complexity

Random m-Łukasiewicz path

w ← ε for i = 1, . . . , n do s ← u with probability

m m+1, d otherwise

β w ← ws 1 if h(w) < 0 then randomly point w O(log i) unfold w (forget the decoration) Unif{1, . . . , i} end if end for randomly decorate w O(√n) fold w (forget the point) Unif{1, . . . , n} return w We consider complexity in random bits and memory accesses.

Complexity

Random m-Łukasiewicz path

w ← ε for i = 1, . . . , n do s ← u with probability

m m+1, d otherwise

β w ← ws 1 if h(w) < 0 then randomly point w O(log i) unfold w (forget the decoration) Unif{1, . . . , i} end if end for randomly decorate w O(√n) fold w (forget the point) Unif{1, . . . , n} return w We consider complexity in random bits and memory accesses. The if branches are independent with probability ∼ 1

2i.

Complexity (cont.)

Theorem

The cost in random bits and memory accesses satisfies: Bn n

d

− → β; Mn n

d

− → 1 + X + Unif[0, 1]. The number β is the cost in random bits of Bernoulli

• 1

m+1

• .

According to [Knuth, Yao ’76], we can take: β ∼ −

1 m+1 log2

• 1

m+1

• −

m m+1 log2

• m

m+1

• .

Complexity (cont.)

Theorem

The cost in random bits and memory accesses satisfies: Bn n

d

− → β; Mn n

d

− → 1 + X + Unif[0, 1]. The number β is the cost in random bits of Bernoulli

• 1

m+1

• .

According to [Knuth, Yao ’76], we can take: β ∼ −

1 m+1 log2

• 1

m+1

• −

m m+1 log2

• m

m+1

• .

The law X is defined by: X =

• x∈S

Unif[0, x], where S is a Poisson point process of density λ(x) =

1 2x on (0, 1].

Properties of the limit law

The distribution X =

• Poissonx∈(0,1](1−x

2x )

Cumulant generating function K(z) = log E(ezX): K(z) = z ey − 1 − y 2y2 dy, κn(X) = 1 2n(n + 1).

Properties of the limit law

The distribution X =

• Poissonx∈(0,1](1−x

2x )

Cumulant generating function K(z) = log E(ezX): K(z) = z ey − 1 − y 2y2 dy, κn(X) = 1 2n(n + 1). Distribution function F(x) = P(X ≤ x): F(x) + F ′(x) + 2xF ′′(x) = F(x − 1)

Properties of the limit law

The distribution X =

• Poissonx∈(0,1](1−x

2x )

Cumulant generating function K(z) = log E(ezX): K(z) = z ey − 1 − y 2y2 dy, κn(X) = 1 2n(n + 1). Distribution function F(x) = P(X ≤ x): F(x) + F ′(x) + 2xF ′′(x) = F(x − 1) F(x) = sin √ 2x, 0 ≤ x ≤ 1.

Properties of the limit law

The distribution X =

• Poissonx∈(0,1](1−x

2x )

Cumulant generating function K(z) = log E(ezX): K(z) = z ey − 1 − y 2y2 dy, κn(X) = 1 2n(n + 1). Distribution function F(x) = P(X ≤ x): F(x) + F ′(x) + 2xF ′′(x) = F(x − 1) F(x) =

• 2e1−γ

π sin √ 2x, 0 ≤ x ≤ 1.

Properties of the limit law

The distribution X =

• Poissonx∈(0,1](1−x

2x )

Cumulant generating function K(z) = log E(ezX): K(z) = z ey − 1 − y 2y2 dy, κn(X) = 1 2n(n + 1). Distribution function F(x) = P(X ≤ x): F(x) + F ′(x) + 2xF ′′(x) = F(x − 1) F(x) =

• 2e1−γ

π sin √ 2x, 0 ≤ x ≤ 1. Tail distribution asymptotics: 1 − F(x) = x−x(log x)−2x(e/2)x+o(x).

Graph of the distribution function

1 1 x F(x) 1 1 x

• 2e1−γ

π sin √ 2x

Perspectives

Can we use a similar method for other paths (+a, −b)?