On the size distribution of the fixed-length Levenshtein balls with radius one

Abstract

The fixed-length Levenshtein (FLL) distance between two words $\varvec{x}, \varvec{y}\in \mathbb {Z}_m^n$ is the smallest integer t such that $\varvec{x}$ can be transformed to $\varvec{y}$ by t insertions and t deletions. The size of a ball in the FLL metric is a fundamental yet challenging problem. Very recently, Bar-Lev, Etzion, and Yaakobi explicitly determined the minimum, maximum and average sizes of the FLL balls with radius one, respectively. In this paper, based on these results, we further prove that the size of the FLL balls with radius one is highly concentrated around its mean by Azuma’s inequality.

Appendices

A Proof of Eq. (18)

We follow the notations in the proof of Theorem 4.

• Case $i = 1$: By symmetry, we have $|Z_1 - Z_0| = 0$.

• Case $1< i <n$:

  By Eq. (16), we have

  $$\begin{aligned}{} & {} Z_i - Z_{i-1} \\{} & {} \quad \ge f_{m,n}(\varvec{x}_{[1,i]}) + g_{m,n}(i) + 1 -n + \frac{n}{m} - \frac{m-1}{m}t(\varvec{x}_{[1,i]})\left( 2 - \frac{1}{m^{n-i-1}}\right) \\{} & {} \qquad - \left[ f_{m,n}(\varvec{x}_{[1,i-1]}) + g_{m,n}(i-1) -n + \frac{n}{m} + \frac{1}{m}\right] \\{} & {} \quad = f_{m,n}(\varvec{x}_{[1,i]}) - f_{m,n}(\varvec{x}_{[1,i-1]}) + g_{m,n}(i) - g_{m,n}(i-1) \\{} & {} \qquad + \frac{m-1}{m} - \frac{m-1}{m}t(\varvec{x}_{[1,i]})\left( 2 - \frac{1}{m^{n-i-1}}\right) ,\\{} & {} Z_i - Z_{i-1} \\{} & {} \quad \le f_{m,n}(\varvec{x}_{[1,i]}) + g_{m,n}(i) - n + \frac{n}{m} + \frac{1}{m} \\{} & {} \qquad - \left[ f_{m,n}(\varvec{x}_{[1,i-1]}) + g_{m,n}(i-1) + 1 -n + \frac{n}{m} - \right. \\{} & {} \quad \left. \frac{m-1}{m}t(\varvec{x}_{[1,i-1]})\left( 2 - \frac{1}{m^{n-i}}\right) \right] \\{} & {} \quad = f_{m,n}(\varvec{x}_{[1,i]}) - f_{m,n}(\varvec{x}_{[1,i-1]}) + g_{m,n}(i) - g_{m,n}(i-1) \\{} & {} \qquad - \frac{m-1}{m} + \frac{m-1}{m}t(\varvec{x}_{[1,i-1]})\left( 2 - \frac{1}{m^{n-i}}\right) . \end{aligned}$$

  By Lemma 5, we have³

  $$\begin{aligned} 0 \le f_{m,n}(\varvec{x}_{[1,i]}) - f_{m,n}(\varvec{x}_{[1,i-1]}) \le mn - n - 1, \end{aligned}$$

  and

  $$\begin{aligned} g_{m,n}(i) - g_{m,n}(i-1) = 1 - n(m + \frac{1}{m} - 2) - \frac{1}{m^{n-i}}. \end{aligned}$$

  Therefore, we have

  $$\begin{aligned} \begin{aligned} Z_i - Z_{i-1}&\ge 1 - n \left( m + \frac{1}{m} - 2 \right) - \frac{1}{m^{n-i}} + \frac{m-1}{m} - \frac{m-1}{m}t(\varvec{x}_{[1,i]})\left( 2 - \frac{1}{m^{n-i-1}}\right) \\&> - n\left( m + \frac{1}{m} - 2\right) - \frac{1}{m^{n-i}} - (n-1) \cdot 2 \\&> - n\left( m + \frac{1}{m}\right) + 1, \end{aligned} \end{aligned}$$

  and also

  $$\begin{aligned} \begin{aligned} Z_i - Z_{i-1}&\le mn - n - 1 + 1 - n\left( m + \frac{1}{m} - 2\right) - \frac{1}{m^{n-i}} - \frac{m-1}{m} \\&\quad + \frac{m-1}{m}t(\varvec{x}_{[1,i-1]})\left( 2 - \frac{1}{m^{n-i}}\right) \\&= n\left( 1 - \frac{1}{m}\right) - \frac{1}{m^{n-i}} - \frac{m-1}{m} + \frac{m-1}{m}t(\varvec{x}_{[1,i-1]})\left( 2 - \frac{1}{m^{n-i}}\right) \\&< n\left( 1 - \frac{1}{m}\right) + 2n = n\left( 3 - \frac{1}{m}\right) . \end{aligned} \end{aligned}$$

  Note that $|-n(m + \frac{1}{m}) + 1| \le n(m + \frac{1}{m})$ and $|n(3 - \frac{1}{m})| \le n(m + \frac{1}{m})$. Hence, we have $|Z_i - Z_{i-1}| \le n(m + \frac{1}{m})$ for $1< i < n$.

• Case $i = n$:

  By Eq. (16) and Lemma 5, we have

  $$\begin{aligned} 0 \le f_{m,n}(\varvec{x}) - f_{m,n}(\varvec{x}_{[1,n-1]}) \le mn -n - 1, \end{aligned}$$

  and

  $$\begin{aligned} \begin{aligned} Z_n - Z_{n-1}&\le f_{m,n}(\varvec{x}) - \left[ f_{m,n}(\varvec{x}_{[1,n-1]}) + \left( 1 - \frac{1}{m} \right) \left( mn -n -t(\varvec{x}_{[1,n-1]})\right) \right] \\&\le mn - n - 1 - \left( 1 - \frac{1}{m} \right) (mn - n -t(\varvec{x}_{[1,n-1]})) \\&= t(\varvec{x}_{[1,n-1]}) \left( 1 - \frac{1}{m} \right) - \frac{n}{m} \\&< n \left( 1 - \frac{1}{m} \right) , \end{aligned} \end{aligned}$$

  and also,

  $$\begin{aligned} \begin{aligned} Z_n - Z_{n-1}&\ge f_{m,n}(\varvec{x}) - \left[ f_{m,n}(\varvec{x}_{[1,n-1]}) + \left( 1 - \frac{1}{m}\right) (mn - n -1) \right] \\&\ge -\left( 1 - \frac{1}{m}\right) (mn - n - 1)\\&= -n\left( m + \frac{1}{m} \right) - \frac{1}{m} + 1. \end{aligned} \end{aligned}$$

  Thus, we have $|Z_n - Z_{n-1}| \le n(m + \frac{1}{m})$.

B Simulation results

We independently pick $x \in \mathbb {Z}_m^n$ uniformly at random and record the value $|L_1(\varvec{x})|$. The distribution of $|L_1(\varvec{x})|$ is then reflected by the frequency that $|L_1(\varvec{x})|$ lies in different intervals. We also compare it with the expected frequency given by the bounds in Theorems 3 and 4. For instance, the expected frequency of the event $|L_1(\varvec{x})| > \tau$ by Eq. (7) is

$$\begin{aligned} N e^{-2 \left( \frac{\tau - \mathbb {E}\left[ |L_1(\varvec{x})| \right] }{n \sqrt{n-1}}\right) ^2}, \end{aligned}$$

where N is the sample size. The simulation results for $n = 100$, $m = 2,3,4,5$ are depicted in Fig. 1.

Fig. 1

$N = 5000$ for each experiment