The asymptotic number of prefix normal words

TL;DR

This paper establishes the asymptotic count of prefix normal binary words of length n, revealing their exponential growth with a subexponential correction, and analyzes the maximum number of words sharing a fixed prefix normal form.

Contribution

It provides the first precise asymptotic enumeration of prefix normal words and bounds the number of words with a given prefix normal form.

Findings

Number of prefix normal words of length n is 2^{n - Θ((log n)^2)}.

Maximum number of words with a fixed prefix normal form is 2^{n - O(√(n log n))}.

Shows the growth rate and structural properties of prefix normal words.

Abstract

We show that the number of prefix normal binary words of length $n$ is $2^{n-\Theta((\log n)^2)}$. We also show that the maximum number of binary words of length $n$ with a given fixed prefix normal form is $2^{n-O(\sqrt{n\log n})}$.

Equations

$$f_w(k)=\underset{0\le j\le n-k}{\max }|w[j+1,j+k]{|}_1,$$

$$|w[1,k]{|}_1\ge |w[j+1,j+k]{|}_1\text{for }0\le j\le n-k.$$

$$|w[1,k]{|}_1\ge |w[j+1,j+k]{|}_1\text{for }k\le j\le n-k.$$

$$|\tilde{w}[1,k]{|}_1=f_w(k).$$

$$w\sim v⟺f_w=f_v⟺\tilde{w}=\tilde{v}.$$

$$p_k=\{\begin{array}{ll}\frac{1}{2}+c\sqrt{\frac{\log n}{k}},& \text{for }k>16c^2\log n;\\ 1,& \text{for }k\le 16c^2\log n.\end{array}$$

$$\sum _{i=1}^k{p}_i=\frac{k}{2}+2c\sqrt{k\log n}+O(1)$$

$$|w[1,k]{|}_1-|w[j+1,j+k]{|}_1=\sum _{i=1}^k{w}_i+\sum _{i=j+1}^{k+j}(1-w_i)-k$$

$$\mu:=\mathbb{E}\big{(}|w[1,k]|_{1}-|w[j+1,j+k]|_{1}\big{)}=2c\sqrt{k\log n}-2c\sqrt{(j+k)\log n}+2c\sqrt{j\log n}+O(1).$$

$$\mu \ge 2(2-\sqrt{2})c\sqrt{k\log n}+O(1)>c\sqrt{k\log n}$$

$$\mathbb{P}\big{(}X-\mathbb{E}(X)\geq x\big{)}\leq\exp\{-2x^{2}/n\}\quad\text{and}\quad\mathbb{P}\big{(}X-\mathbb{E}(X)\leq-x\big{)}\leq\exp\{-2x^{2}/n\}.$$

$$\displaystyle\mathbb{P}\big{(}|w[1,k]|_{1}<|w[j+1,j+k]|_{1}\big{)}$$

$$\le \mathbb{P}\left(\sum _{i=1}^k{w}_i+\sum _{i=j+1}^{k+j}(1-w_i)-{\mu }^{\ast }<k-{\mu }^{\ast }\right)$$

$$\le \mathbb{P}\left(\sum _{i=1}^k{w}_i+\sum _{i=j+1}^{k+j}(1-w_i)-{\mu }^{\ast }<-\mu \right)$$

$$\displaystyle\stackrel{{\scriptstyle\eqref{eq:hoeffding}}}{{\leq}}\exp\big{\{}-2\mu^{2}/(2k)\big{\}}$$

$$\displaystyle\leq\exp\big{\{}-c^{2}\log n\big{\}}$$

$$H(X):=\sum _x-\mathbb{P}(X=x){\log }_2\mathbb{P}(X=x),$$

$$H_b(p)=1-\frac{1}{2\ln 2}\sum _{n=1}^{\infty }\frac{(1-2p{)}^{2n}}{n(2n-1)}.$$

$$H(X)=H(X\mid X\in \mathcal{B})\mathbb{P}(X\in \mathcal{B})+H(X\mid X\notin \mathcal{B})\mathbb{P}(X\notin \mathcal{B})+H(1_{X\in \mathcal{B}}),$$

$$H(w)=\sum _{k>k_0}^{n}H(w_k)=n-k_0-\Theta \left(\sum _{k=k_0}^n{c}^2\frac{\log n}{k}\right)=n-\Theta ((\log n{)}^2).$$

$$H(w)$$

$$\le {\log }_2(|\mathcal{B}|)\mathbb{P}(w\in \mathcal{B})+n\mathbb{P}(w\notin \mathcal{B})+1$$

$$=n+1-(n-{\log }_2|\mathcal{B}|)(1-o(1)).$$

$$2^{n}\big{(}1-\mathbb{P}\big{(}\mathrm{Bin}(k,\tfrac{1}{2})>d\big{)}\big{)}^{\lfloor\sqrt{n}\rfloor-1}\leq 2^{n-\Omega(\sqrt{n}\,\mathbb{P}(\mathrm{Bin}(k,1/2)>d))}.$$

$$\mathbb{P}\big{(}\mathrm{Bin}(k,\tfrac{1}{2})\geq\lambda k)=\sum_{i=\lambda k}^{k}\binom{k}{i}2^{-k}\geq\frac{2^{kH_{b}(\lambda)-k}}{\sqrt{8k\lambda(1-\lambda)}}\geq\frac{2^{kH_{b}(\lambda)-k}}{\sqrt{2k}},$$

$$\mathbb{P}\big{(}\mathrm{Bin}(k,\tfrac{1}{2})>\tfrac{k}{2}+x\big{)}\geq\frac{1}{\sqrt{2k}}2^{-\Theta(x^{2}/k)},$$

$$|w[1,k]{|}_1\ge \frac{k}{2}+c\sqrt{k\log n}\text{for all k with}\log n\le k\le \sqrt{n}.$$

$$\mathbb{P}\big{(}\mathcal{E}_{\leq t-1}\cap\mathcal{E}_{t,\geq j}\big{)}\leq n^{-2c^{2}(t-t_{0}+1)/3}\beta_{t}^{j}/(1-\beta_{t}),$$

$$\mathbb{P}({\mathcal{E}}_{t_0,\ge j})$$

$$\displaystyle\leq\exp\big{\{}-2d_{0}^{2}-4jd_{0}/2^{t_{0}}\big{\}}=n^{-2c^{2}}\beta_{t_{0}}^{3j/2}<n^{-2c^{2}/3}\beta_{t_{0}}^{j}/(1-\beta_{t_{0}})$$

$$2^{2(t+1)-1}+2^{t+1}{d}_0+j-2^{2t-1}-2^t{d}_0-i=3\cdot 2^{2t-1}+2^t{d}_0+j-i$$

$$\mathbb{P}\big{(}\mathcal{E}_{\leq t}\cap\mathcal{E}_{t+1,\geq j}\big{)}\leq\sum_{i\geq 0}\mathbb{P}(\mathcal{E}_{\leq t-1}\cap\mathcal{E}_{t,i})\mathbb{P}\big{(}|w[4^{t}+1,4^{t+1}]|_{1}\geq 3\cdot 2^{2t-1}+2^{t}d_{0}+j-i\big{)}.$$

$$\frac{3\cdot 4^t}{2}=3\cdot 2^{2t-1}$$

$$\displaystyle\mathbb{P}\big{(}|w[4^{t}+1,4^{t+1}]|_{1}\geq 3\cdot 2^{2t-1}+2^{t}d_{0}+j\big{)}$$

Full text

Keywords: Prefix normal words, random construction

1 Introduction

Given a binary word $w=(w_{i})_{i=1}^{n}\in\{0,1\}^{n}$ of length $n$, denote by $w[j,k]$ the subword of length $k-j+1$ starting at position $j$ and ending at position $k$, that is, $w[j,k]=w_{j}w_{j+1}\dots w_{k}$. Let $|w|_{1}$ be the number of 1s in the word $w$. We define the profile $f_{w}\colon\{0,\dots,n\}\to\{0,\dots,n\}$ of $w$ by

[TABLE]

so that $f_{w}(k)$ is the maximum number of 1s in any subword of $w$ of length $k$. The word $w$ is called prefix normal if for all $0\leq k\leq n$ this number is maximized at $j=0$, so that

[TABLE]

In other words, a word $w$ is called prefix normal if the number of $1$s in any subword is at most the number of $1$s in the prefix of the same length.

If $j<k$ then we can remove the common subword $w[j+1,k]$ of $w[1,k]$ and $w[j+1,j+k]$, so that $|w[1,k]|_{1}\geq|w[j+1,k+j]|_{1}$ iff $|w[1,j]|_{1}\geq|w[k+1,k+j]|_{1}$. Thus to show that $w$ is prefix normal it is enough to check that

[TABLE]

Prefix normal words were introduced by G. Fici and Z. Lipták in [4] because of their connection to binary jumbled pattern matching. Recently, prefix normal words have been used because of their connection to trees with a prescribed number of vertices and leaves in caterpillar graphs [6].

The number of prefix normal words of length $n$ is listed as sequence A194850 in The On-Line Encyclopedia of Integer Sequences (OEIS) [7]. We prove the following result, conjectured in [2] (Conjecture 2) where also weaker upper and lower bounds were shown, see also [3].

Theorem 1**.**

The number of prefix normal words of length $n$ is $2^{n-\Theta((\log n)^{2})}$.

Given an arbitrary binary word $w$ of length $n$, the prefix normal form $\tilde{w}$ of $w$ is the unique binary word of length $n$ that satisfies

[TABLE]

Note that for any $w$, $f_{w}(k)\leq f_{w}(k+1)\leq f_{w}(k)+1$, so $\tilde{w}$ is well-defined. Moreover, we can define an equivalence relation $\sim$ on binary words of length $n$ by

[TABLE]

Indeed, $\tilde{w}$ is just the lexicographically maximal element of the equivalence class $[w]$ of $w$ under this equivalence relation.

In [4] it is asked how large can an equivalence class $[w]$ be. In other words, what is the maximum number of words of length $n$ that have the same fixed prefix normal form. This maximum number is listed in the OEIS as sequence A238110 [7]. From Theorem 1 it is clear that it must be at least $2^{\Theta((\log n)^{2})}$. However, we show that it is much larger.

Theorem 2**.**

For each $n$ there exists a prefix normal word $w$ such that the number of binary words of length $n$ with prefix normal form $w$ is $2^{n-O(\sqrt{n\log n})}$.

2 Proofs

Proof of the lower bound of Theorem 1..

To prove the lower bound we will need to construct $2^{n-\Theta((\log n)^{2})}$ prefix normal words of length $n$. We will do so by giving a random construction and showing that this construction almost always produces a prefix normal word.

Fix a constant $c>\sqrt{2}$ and define

[TABLE]

Write $k_{0}:=\lfloor 16c^{2}\log n\rfloor$ so $p_{k}=1$ if $k\leq k_{0}$, and $p_{k}\in[\frac{1}{2},\frac{3}{4}]$ for $k>k_{0}$. Let $w$ be a random word with each letter $w_{k}$ chosen to be 1 with probability $p_{k}$, independently for each $k=1,\dots,n$. Clearly (1) holds for all $k\leq k_{0}$, so assume $k>k_{0}$. By comparing the integral $\int c\sqrt{\frac{\log n}{k}}\,dk=2c\sqrt{k\log n}+C$ with the corresponding Riemann sum, we note that

[TABLE]

uniformly for $k>k_{0}$ (and uniformly in $c$). Indeed, the approximation of the integral by the Riemann sum has error at most the maximum term, due to the monotonicity of the integrand, and the additive constant is also $O(1)$ by considering the case $k=k_{0}$. From this we estimate the expected difference

[TABLE]

as

[TABLE]

This expression is minimized when $j$ is as small as possible, i.e., $j=k$. Thus

[TABLE]

for sufficiently large $n$. By (2), $|w[1,k]|_{1}-|w[j+1,j+k]|_{1}$ can be considered as the sum of $2k$ independent Bernoulli random variables (with an offset of $-k$).

We recall the Hoeffding bound [5] that states that if $X$ is the sum of $n$ independent random variables in the interval $[0,1]$ then for all $x\geq 0$,

[TABLE]

(Note that these two bounds are essentially the same bound as the second can be easily derived from the first by exchanging the roles of the [math]s and $1$s but we state them both here for convenience.)

Let \mu^{*}=\mathbb{E}\big{(}\sum_{i=1}^{k}w_{i}+\sum_{i=j+1}^{k+j}(1-w_{i})\big{)}. Note that $\mu^{*}=\mu+k$. We have

[TABLE]

Hence if $c$ is large enough ($c>\sqrt{2}$) then $\mathbb{P}(|w[1,k]|_{1}<|w[j+1,j+k]|_{1})=o(n^{-2})$. Taking a union bound over all possible values of $k$ and $j$, we deduce that $w$ is prefix normal with probability $1-o(1)$.

It remains to count the number of such $w$. For any discrete random variable $X$, define the entropy of the distribution of $X$ as

[TABLE]

where the sum is over all possible values $x$ of $X$ and the logarithm is to base 2. If the random variable is a Bernoulli random variable, we call $H(\mathrm{Be}(p))$ the binary entropy function $H_{b}(p)$. We use the following well-known (and easily verified) facts about the entropy.

1. H1)

If $X_{1},\dots,X_{n}$ are independent discrete random variables and $X=(X_{1},\dots,X_{n})$, then $H(X)=\sum_{i=1}^{n}H(X_{i})$. 2. H2)

If $X$ takes on at most $N$ possible values with positive probability then $H(X)\leq\log_{2}N$. 3. H3)

The Taylor series of the binary entropy function in a neighbourhood of $1/2$ is

[TABLE]

In particular, for a Bernoulli random variable with $\mathbb{P}(X=1)=\frac{1}{2}+x$, $H(X)=1-\Theta(x^{2})$. 4. H4)

If $\mathcal{B}$ is subset of possible values of $X$ we have

[TABLE]

where $X\mid\mathcal{E}$ denotes the distribution of $X$ conditioned on the event $\mathcal{E}$ and $1_{\mathcal{E}}$ denotes the indicator function of $\mathcal{E}$.

Applying these results to our random word $w$ we have

[TABLE]

On the other hand, if $\mathcal{B}$ is the set of prefix normal words, then

[TABLE]

We deduce that $n-\log_{2}|\mathcal{B}|\leq\Theta((\log n)^{2})$ and hence $|\mathcal{B}|\geq 2^{n-\Theta((\log n)^{2})}$. ∎

Proof of the upper bound in Theorem 1..

We will prove the upper bound in two parts. Firstly we will show that most prefix normal words have to contain a good number of $1$s in any prefix of reasonable size as we cannot extend a prefix with too few 1s to a prefix normal word in many ways. Secondly, we will show that there are at most $2^{n-\Theta(\log^{2}n)}$ ways to construct a word which has sufficiently many $1$s in all reasonably sized prefixes.

Assume $\log n\leq k\leq\sqrt{n}$ and consider the first $\lfloor\sqrt{n}\rfloor$ blocks of size $k$ of $w$. If $|w[1,k]|_{1}=d$ then the number of choices for the second and subsequent blocks is at most $2^{k}(1-\mathbb{P}(\mathrm{Bin}(k,\tfrac{1}{2})>d))$, and hence the number of choices for $w$ is at most

[TABLE]

If $\mathbb{P}(\mathrm{Bin}(k,\tfrac{1}{2})>d)>n^{-1/3}$, say, then there are far fewer than $2^{n-\Theta((\log n)^{2})}$ choices of such prefix normal words, even allowing for summation over all such $k$ and $d$.

Using Stirling’s formula one can show that for $1/2<\lambda<1$ and $\lambda k$ integral,

[TABLE]

see for example [1] for a detailed proof.

Thus, by H3), we have

[TABLE]

provided $x<k/2$. Thus if $\log n\leq k\leq\sqrt{n}$ and $\mathbb{P}(\mathrm{Bin}(k,\tfrac{1}{2})>d)>n^{-1/3}$ we can deduce that $d\geq\frac{k}{2}+c\sqrt{k\log n}$ for some small universal constant $c>0$. Thus, without loss of generality, we can restrict to prefix normal words with the property that

[TABLE]

Define $d_{0}=c\sqrt{\log n}$, which for simplicity we shall assume is an integer. (One can reduce $c$ slightly to ensure this is the case.) Define $\mathcal{E}_{t}$ to be the event that (4) holds with $k=4^{t}$, i.e., that $|w[1,4^{t}]|_{1}\geq 2^{2t-1}+2^{t}d_{0}$. Let $t_{0}$ be the smallest $t$ such that $4^{t}\geq\log n$ and let $t_{1}$ be the largest $t$ such that $4^{t}\leq\sqrt{n}$. We bound the probability that a uniformly chosen $w\in\{0,1\}^{n}$ satisfies $\mathcal{E}_{t_{0}}\cap\mathcal{E}_{t_{0}+1}\cap\dots\cap\mathcal{E}_{t_{1}}$.

Write $\mathcal{E}_{t,j}$ for the event that $|w[1,4^{t}]|_{1}=2^{2t-1}+2^{t}d_{0}+j$ and $\mathcal{E}_{t,\geq j}$ for the event that $|w[1,4^{t}]|_{1}\geq 2^{2t-1}+2^{t}d_{0}+j$. Thus $\mathcal{E}_{t}$ is just $\mathcal{E}_{t,\geq 0}$. Write $\mathcal{E}_{\leq t}$ for the intersection $\mathcal{E}_{t_{0}}\cap\mathcal{E}_{t_{0}+1}\cap\dots\cap\mathcal{E}_{t}$.

Claim: For $t\in[t_{0},t_{1}]$ and $j\geq 0$,

[TABLE]

where $\beta_{t}:=\exp\{-2^{3-t}d_{0}/3\}$. Note that $\beta_{t}<1$ for all $t\in[t_{0},t_{1}]$. For the case $t=t_{0}$ we simply use the Hoeffding bound (3) to obtain

[TABLE]

as required.

Now assume the claim is true for $t$. We first want to give a bound on $\mathbb{P}(\mathcal{E}_{\leq t}\cap\mathcal{E}_{t+1,\geq j})$. Note that if $\mathcal{E}_{\leq t-1}\cap\mathcal{E}_{t,i}$ holds then in particular $\mathcal{E}_{t,i}$ holds and thus for $\mathcal{E}_{t+1,\geq j}$ to hold we still need at least

[TABLE]

$1$s in the interval $[4^{t}+1,4^{t+1}]$. Thus we get

[TABLE]

Note that there are $4^{t+1}-4^{t}=3\cdot 4^{t}$ elements in the interval $[4^{t}+1,4^{t+1}]$ and that we expect

[TABLE]

$1$s in this interval. Hence by Hoeffding

[TABLE]

Note that the final inequality is even true for negative $j$: for $j\geq-2^{t}d_{0}$ Hoeffding’s bound holds, and for $j\leq-2^{t}d_{0}$ the bound on the probability is larger than $1$. If we let $p_{i}=\mathbb{P}(\mathcal{E}_{\leq t-1}\cap\mathcal{E}_{t,\geq i})$ then we have

[TABLE]

Now by induction, $p_{i}\leq n^{-2c^{2}(t-t_{0}+1)/3}\beta_{t}^{i}/(1-\beta_{t})$. As $\beta_{t}=\beta_{t+1}^{2}$ we have

[TABLE]

as required. Thus the claim is proved.

Now we take $t=t_{1}$ and $j=0$ to deduce that $\mathbb{P}(\mathcal{E}_{\leq t_{1}})\leq n^{-2c^{2}(t_{1}-t_{0}+1)/3}/(1-\beta_{t_{1}})$. Recall $\beta_{t_{1}}=\exp(-2^{3-t_{1}}d_{0}/3)$, $d_{0}=c\sqrt{\log n}$, and that $t_{1}$ was chosen so $\sqrt{n}/4<4^{t_{1}}\leq\sqrt{n}$. Thus, for large $n$, $n^{-1/4}<2^{3-t_{1}}d_{0}/3<1$. Using the inequality $e^{-x}\leq 1-x/2$, which holds for $0\leq x\leq 1$, we deduce that $1-\beta_{t_{1}}\geq n^{-1/4}/2$, and so $1/(1-\beta_{t_{1}})=O(n^{1/4})$. Also, we have $t_{1}-t_{0}+1=\Theta(\log n)$ as $n\to\infty$ and thus $\mathbb{P}(\mathcal{E}_{\leq t_{1}})\leq 2^{-\Omega((\log n)^{2})}$. As the probability that a uniformly chosen word $w$ satisfies $\mathcal{E}_{\leq t_{1}}$ is at most $2^{-\Omega((\log n)^{2})}$, we deduce that the number of prefix normal words is at most $2^{n-\Theta((\log n)^{2})}$. ∎

Proof of Theorem 2..

Fix an integer $t\approx\sqrt{n\log n}$ and assume for simplicity that $n$ is a multiple of $2t$. Define $w=(10)^{t}1^{2t}c_{1}c_{2}\dots c_{(n-4t)/2t}$, where $c_{i}$ are arbitrary Catalan sequences of length $2t$. Here a Catalan sequence is a binary sequence $c$ of length $2t$ such that $|c[1,i]|_{1}\leq i/2$ for all $i=1,\dots,2t$ and $|c|_{1}=t$. It is well-known that the number of choices for $c_{i}$ is the Catalan number

[TABLE]

It is easy to see that the prefix normal form of any $w$ of this form is

[TABLE]

Indeed, there is a subword $1^{k}$ of $w$ for all $k\leq 2t$. For $k>2t$, if we write $k=2tq+r$ with $0\leq r<2t$ then we have a subword $(10)^{r/2}1^{2t}c_{1}\dots c_{q-1}$ or $0(10)^{(r-1)/2}1^{2t}c_{1}\dots c_{q-1}$ which is of length $t$ and has the requisite number $t+\lfloor k/2\rfloor$ of 1s. On the other hand, the definition of a Catalan sequence implies no other subword of length $k$ containing the $1^{2t}$ subword can possibly have more 1s. Any substring intersecting the $1^{2t}$ and of length greater than $2t$ can be replaced by one containing the $1^{2t}$ with at least as many ones. And finally, any subword of $w$ length $k>2t$ not intersecting the $1^{2t}$ subword (so contained within the $c_{1}\dots c_{(n-4t)/2t}$ subword) can have at most $t+\lfloor k/2\rfloor$ 1s as an end-word of $c_{i}$ contains at most $t$ 1s and there are at most $\lfloor k/2\rfloor$ 1s in the initial subword of $c_{i+1}c_{i+2}\dots$ of length $k$.

It remains to count the number of possible $w$’s. This is just

[TABLE]

Taking $t\sim\sqrt{n\log n}$ gives $2^{n-O(\sqrt{n\log n})}$ words $w$ satisfying (5). ∎

Acknowledgement: We would like to thank the anonymous referees for their helpful comments and their quick response.