Reconstruction of Sequences Distorted by Two Insertions

The study of sequence reconstruction problems was initiated by Levenshtein in [1, 2, 3] to combat errors by repeatedly transmitting a message without coding, which is of interest in fields like informatics, molecular biology, and chemistry. The setting of this problem is as follows: the sender transmits a sequence through $N$ different noisy channels and the receiver receives all the channel outputs, called noisy reads; then the receiver aims to reconstruct the transmitted sequence with the help of these $N$ outputs. There are probabilistic and combinatorial versions of this problem. In the probabilistic version, the noisy reads are obtained by transmitting the sequence $\bm{x}$ through probabilistic channels, and the goal is to minimize the value of $N$ required for reconstructing $\bm{x}$ with high probability, see [2, 4, 5, 6, 7, 8, 9] and the references therein. In the combinatorial version, noisy reads are obtained by introducing the maximum number of errors, and the goal is to determine the minimum value of $N$ needed for zero-error reconstruction [2].

In this paper, we focus on the combinatorial version, which requires that all channel outputs are different from each other[2]. Then the problem of determining the minimum value of $N$ is reduced to determining the maximum intersection size of two distinct error balls. Specifically, the maximum intersection size is equal to $N-1$. In [2, 3], Levenshtein solved the problem for several error types, such as substitutions, insertions, deletions, transpositions, and asymmetric errors. In [10, 11, 12], permutation errors were analyzed. For general error graphs, see [2, 13, 14]. Note that these works considered the uncoded case, that is, the transmitted sequences are selected from the entire space. Due to applications in DNA storage, many researchers also investigated this problem under the setting where the transmitted sequences are chosen from a given code with a certain error-correcting property, see for example [15, 16, 17, 18, 19].

Recently, the dual problem of the sequence reconstruction was developed under the scenario that the number of noisy reads is a fixed system parameter. Motivated by applications in DNA-based data storage and racetrack memories, Kiah et al. [20, 21] and Chrisnata et al. [22] initiated the study of designing reconstruction codes, for which the size of intersections between any two error-balls is strictly less than the number of reads $N$, for a given $N$. Note that when $N=1$, reconstruction codes are the classical error-correcting codes. When $N>1$, one can increase the information capacity, or equivalently, reduce the number of redundant bits by leveraging on these multiple reads. The redundancy of a binary code $\mathcal{C}$ of length $n$ is defined to be $n-\log_{2}|\mathcal{C}|$. In [20, 21], the authors focused on channels that introduce single edit error, that is a single substitution, insertion, or deletion, and their variants. For single deletion binary codes, they showed that the number of redundant symbols required can be reduced from $\log_{2}(n)+O(1)$ to $\log_{2}\log_{2}(n)+O(1)$ if one increases the number of noisy reads $N$ from one to two. For two deletions, the best known $2$-deletion correcting code (i.e., $N=1$) has redundancy $4\log_{2}(n)+o(\log_{2}(n))$ [23]. In [22, 24], Chrisnata et al. reduced this redundancy to $2\log_{2}(n)+o(\log_{2}(n))$ by increasing the number of noisy reads $N$ from one to five, while each noisy read suffers exactly two deletions. For $t$-deletions (with $t\geq 2$), it was proved in [24] that the reconstruction code with two reads for a single deletion [21] is able to reconstruct codewords from $N=n^{t-1}/(t-1)!+O(n^{t-2})$ distinct noisy reads.

It is well known that a code can correct $t$-deletions (i.e., $N=1$) if and only if it can correct $t$-insertions. However, it is not the case when $N>1$. When $N>1$, the problem is closely related to the size of the intersection of two error balls. From [17], [18] and [21, Proposition 9], we can see that in general the intersection size of two $t$-deletion balls is different from that of two $t$-insertion balls.

In this paper, we study binary reconstruction codes for channels that cause exactly two insertion errors. We show that the number of redundant symbols required to reconstruct a codeword uniquely can be reduced from $4\log_{2}(n)+o(\log_{2}(n))$ to $\log_{2}(n)+O(\log_{2}\log_{2}(n))$ by increasing the number of noisy reads $N$ from one to five. The redundancy can be further reduced to $\log_{2}\log_{2}(n)+\Theta(1)$ if the number of noisy reads is at least $n+4$. For readers’ easy reference, we summarize the best known results on binary $2$-deletion reconstruction codes and $2$-insertion reconstruction codes in Table I. For general $t$-insertions (with $t>2$), we can show the similar result as in [24]: the reconstruction code with two reads for a single insertion [21] is able to reconstruct codewords from $N=n^{t-1}/(t-1)!+O(n^{t-2})$ distinct noisy reads.

This paper is organized as follows. In Section II, we introduce some notations and the problem statement formally. In Section III, we derive some preliminary results, which help us to solve the trivial cases. In Section IV, we first completely characterize the structures of two sequences when the intersection size of their $2$-insertion balls is exactly $n+4$ or $n+5$. Then by utilizing the structures, we give explicit reconstruction codes for $N=n+4$, which are asymptotically optimal in terms of redundancy. In Section V, by applying higher order parity checks on short subwords, we construct asymptotically optimal codes when $N=5$. Finally, we conclude this paper in Section VI with some open problems.

In this section, we introduce some notations and preliminary results needed throughout this paper. Let $\Sigma_{2}$ denote the alphabet $\{0,1\}$, and $\Sigma_{2}^{n}$ denote the set of all binary sequences of length $n$. Further, let $\Sigma_{2}^{*}\triangleq\mathop{\cup}\limits_{n=0}^{\infty}\Sigma_{2}^{n}$, that is the set consisting of all binary sequences of finite length. Here if $n=0$, we mean the empty sequence, denoted by $\emptyset$.

For a sequence $\bm{x}=x_{1}\cdots x_{n}\in\Sigma_{2}^{n}$ and a sequence $\bm{z}=z_{1}\cdots z_{t}\in\Sigma_{2}^{t}$, where $t$ is an integer satisfying $1\leq t\leq n$, if there exist $1\leq i_{1}<\cdots<i_{t}\leq n$ such that $z_{j}=x_{i_{j}}$ for all $1\leq j\leq t$, we say that $\bm{z}$ is a subsequence of $\bm{x}$, or $\bm{x}$ is a supersequence of $\bm{z}$. In particular, when $i_{j+1}=i_{j}+1$ for all $j$, we say that $\bm{z}$ is a subword of $\bm{x}$, and use the notation $\bm{x}[l:k]$ with $l=i_{1}$ and $k=i_{t}$ to denote $\bm{z}$. Let $|\bm{x}|$ denote the length of $\bm{x}$. For any $\bm{x}\in\Sigma_{2}^{n}$ and $t\geq 0$, let $I_{t}(\bm{x})\triangleq\{\bm{z}\in\Sigma_{2}^{n+t}:\text{ }\bm{z}\text{ is a supersequence of }\bm{x}\}$. We call $I_{t}(\bm{x})$ the $t$-insertion ball centered at $\bm{x}$. It is known that $\left|I_{t}(\bm{x})\right|$ is independent of the choice of $\bm{x}$ by [2] and equal to

The Levenshtein distance between $\bm{x}$ and $\bm{y}$ in $\Sigma_{2}^{n}$, denoted by $d_{L}(\bm{x},\bm{y})$, is the smallest integer $\ell$ such that $I_{\ell}(\bm{x})\cap I_{\ell}(\bm{y})$ is not empty. For example, $d_{L}(10100,01001)=1$. It is easy to see that $0\leq d_{L}(\bm{x},\bm{y})\leq n$, and $d_{L}(\bm{x},\bm{y})=0$ if and only if $\bm{x}=\bm{y}$. Finally, the Hamming distance between $\bm{x}$ and $\bm{y}$, denoted by $d_{H}(\bm{x},\bm{y})$, is the number of coordinates where $\bm{x}$ and $\bm{y}$ differ, and the Hamming weight $wt_{H}(\bm{x})$ of $\bm{x}$ is the number of nonzero coordinates of $\bm{x}$.

The intersection size of two error balls is a key ingredient of the reconstruction problem in coding theory. By Levenshtein’s original framework [2], for channels causing $t$ insertion errors, it is always possible to exactly reconstruct $\bm{x}\in\Sigma_{2}^{n}$ given $N^{+}(n,t)+1$ distinct elements of $I_{t}(\bm{x})$, where

This notation was generalized in [18] for any two sequences with a minimum Levenshtein distance. For integers $n$, $t\geq\ell\geq 0$, let

then the explicit formula is given by

Note that when $\ell=0,1$, the values of $N^{+}(n,t,l)$ reduce to $I(n,t)$ and $N^{+}(n,t)$, respectively. See [18, Corollary 9] and [18, Corollary 10] for details.

Given a code $\mathcal{C}\subseteq\Sigma_{2}^{n}$ with $|\mathcal{C}|\geq 2$, the read coverage $\nu_{t}(\mathcal{C})$ of $\mathcal{C}$ after $t$ insertions is defined as

It is clear that if $\nu_{t}(\mathcal{C})<N$, we can uniquely recover any codeword in $\mathcal{C}$ by its $N$ distinct reads. So we have the following definition.

Given a code $\mathcal{C}\subset\Sigma_{2}^{n}$, the redundancy of $\mathcal{C}$ is defined to be the value $n-\log_{2}\left|\mathcal{C}\right|$. Then we are interested in the following quantity

which is called the optimal redundancy of an $(n,N;B_{2}^{I(t)})$-reconstruction code. By definition, an $(n,N;B^{I(t)})$-reconstruction code is also an $(n,N^{\prime};B^{I(t)})$-reconstruction code for any $N^{\prime}>N$. So we have $\rho(n,N;B^{I(t)})\geq\rho(n,N+1;B^{I(t)})$ for all $N\geq 1$.

When $N=1$, an $(n,1;B^{I(t)})$-reconstruction code is indeed a $t$-insertion correcting code, or equivalently a $t$-deletion correcting code. This class of codes has been extensively studied in recent years, see [25, 26, 27, 28, 23] for more details. For $t=1$, we have the following single deletion correcting code [29]:

where $a$ is a fixed integer between $0$ and $n$. This is the famous Varshamov-Tenengolts (VT) code, which has asymptotically optimal redundancy $\log_{2}(n+1)$ when $a=0$ [30, Corollary 2.3]. In [21, Theorem 24], Cai et al. determined the asymptotic value of $\rho(n,N;B^{I(1)})$ for all $N\geq 1$ as below.

Our work is to extend the result in Theorem II.1 to the case of $t=2$. As will be shown in Corollary III.2 and the paragraph prior to Proposition III.2, the lower bound of the optimal redundancy can be deduced from the case $t=1$. The upper bound needs explicit code constructions. So the main task is to construct an $(n,N;B^{I(2)})$-reconstruction code with redundancy as small as possible, for any given $N$. Before closing this section, we introduce the following notations which will be used later.

For $\bm{x},\bm{y}\in\Sigma_{2}^{*}$, we write $\bm{x}\bm{y}$ for the concatenation of $\bm{x}$ and $\bm{y}$. In particular, if $i\geq 1$, we define $\bm{x}^{i}$ to be the sequence obtained by concatenating $i$ $\bm{x}$’s; if $i=0$, $\bm{x}^{0}$ is the empty sequence. If $\bm{u}=u_{1}\cdots u_{n}\in\Sigma_{2}^{n}$, we denote $\overline{\bm{u}}=(1-u_{1})\cdots(1-u_{n})$ as the complementary sequence of $\bm{u}$. For any $S\subseteq\Sigma_{2}^{*}$ and $a,b\in\Sigma_{2}$, we denote $S^{a}\triangleq\{\bm{x}\in S:\text{ }\bm{x}\text{ starts with }a\}$, $S_{b}\triangleq\{\bm{x}\in S:\text{ }\bm{x}\text{ ends with }b\}$, $S_{b}^{a}\triangleq\{\bm{x}\in S:\text{ }\bm{x}\text{ starts with }a\text{ and ends with }b\}$, $aS\triangleq\{a\bm{y}:\text{ }\bm{y}\in S\}$, and $Sb\triangleq\{\bm{y}b:\text{ }\bm{y}\in S\}$.

Given a sequence $\bm{x}=x_{1}\cdots x_{n}\in\Sigma_{2}^{n}$ with $n\geq 2$, we say that $\bm{x}$ has period $\ell$ ($\leq n$) if $\ell$ is the smallest integer such that $x_{i}=x_{i+\ell}$ for all $1\leq i\leq n-\ell$. A sequence with period $2$ is called alternating. We also view an empty sequence or a length-one sequence as an alternating sequence. For example, $\emptyset$, $1$, $0$, $10$, $01$, $101$ are all alternating sequences.

As mentioned before, the intersection size of two error balls is a key ingredient of the reconstruction problem. In this section, we characterize the structures of two binary sequences whose intersection of $2$-insertion balls are of a certain size.

Let us first recall the results for a single insertion, that is the size of $I_{1}(\bm{x})\cap I_{1}(\bm{y})$. By Equation (2), we know that $\left|I_{1}(\bm{x})\cap I_{1}(\bm{y})\right|\leq 2$ for any two distinct sequences $\bm{x},\bm{y}\in\Sigma_{2}^{n}$. The characterization of $\bm{x},\bm{y}$ when the equality holds was given in [21], which needs the following notations.

For example, the two sequences $110$ and $100$ are Type-A confusable with $\bm{u}=1$, $\bm{w}=1$ and $\bm{v}=0$. In general, if the Hamming distance $d_{H}(\bm{x},\bm{y})=1$, then $\bm{x}$ and $\bm{y}$ are always Type-A confusable by definition. Note that the case where $d_{H}(\bm{x},\bm{y})=1$ was not included in the original definition in [21, Definition 8], where $j\geq 1$ is required when $\bm{w}=(a\bar{a})^{j}a$. However, the following statement is still true, which is just a straightforward combination of the two claims in [21, Proposition 9].

In fact, if two sequences $\bm{x}$ and $\bm{y}$ are Type-A confusable, then

for some $\bm{u},\bm{w}\in\Sigma_{2}^{*}$, $a\in\Sigma_{2}$ and $m\geq 0$. By Lemma III.1, $I_{1}(\bm{x})\cap I_{1}(\bm{y})=\{\bm{u}(\bar{a}a)^{m+1}\bar{a}\bm{w},\bm{u}(a\bar{a})^{m+1}a\bm{w}\}$ for the left case and $I_{1}(\bm{x})\cap I_{1}(\bm{y})=\{\bm{u}(\bar{a}a)^{m+1}\bm{w},\bm{u}(a\bar{a})^{m+1}\bm{w}\}$ for the right case. Next, we define the concept of Type-B confusability, which was introduced in [22, Definition 18] and [24, Definition 11]. This concept will help to characterize the case $\left|I_{1}(\bm{x})\cap I_{1}(\bm{y})\right|=1$.

From Example III.1, we can see that Type-A confusability and Type-B confusability do not contain each other. In general, we have the following proposition, which is easy to verify.

By Lemma III.2, if $\left|I_{1}(\bm{x})\cap I_{1}(\bm{y})\right|=1$, then we must have $\left\{\bm{x},\bm{y}\right\}=\left\{\bm{u}a\bar{a}\bm{v}b\bm{w},\bm{u}\bar{a}\bm{v}b\bar{b}\bm{w}\right\}$ for some $\bm{u}$, $\bm{v}$, $\bm{w}$ $\in\Sigma_{2}^{*}$ and $a,b\in\Sigma_{2}$. Consequently, $\bm{z}=\bm{u}a\bar{a}\bm{v}b\bar{b}\bm{w}$ is the unique sequence in $I_{1}(\bm{x})\cap I_{1}(\bm{y})$. Combining Lemma III.1 and Lemma III.2, we completely characterize the structures of two binary sequences for which the intersection of $1$-insertion balls is of a certain size. See the following corollary and Figure 2.

Now we proceed to estimate the intersection size of two $2$-insertions balls, that is $\left|I_{2}(\bm{x})\cap I_{2}(\bm{y})\right|$, based on the value of $\left|I_{1}(\bm{x})\cap I_{1}(\bm{y})\right|$. By Equation (2), we have that $N^{+}(n,2)=2n+4$. That is, $\left|I_{2}(\bm{x})\cap I_{2}(\bm{y})\right|\leq 2n+4$ for any $\bm{x}\neq\bm{y}\in\Sigma_{2}^{n}$. The following lemma further restricts its value in the case that $\left|I_{1}(\bm{x})\cap I_{1}(\bm{y})\right|=1$.

By Lemma III.3, we are able to give a rough estimate of $\left|I_{2}(\bm{x})\cap I_{2}(\bm{y})\right|$ based on the different values of $\left|I_{1}(\bm{x})\cap I_{1}(\bm{y})\right|$, and consequently based on the confusability of $\bm{x}$ and $\bm{y}$ by Corollary III.1. See the following lemma and Figure 2.