In the usual trace reconstruction problem, the goal is to exactly reconstruct an unknown string of length $n$ after it passes through a deletion channel many times independently, producing a set of traces (i.e., random subsequences of the string). We consider the relaxed problem of approximate reconstruction. Here, the goal is to output a string that is close to the original one in edit distance while using much fewer traces than is needed for exact reconstruction. We present several algorithms that can approximately reconstruct strings that belong to certain classes, where the estimate is within $n/\mathrm{polylog}(n)$ edit distance, and where we only use $\mathrm{polylog}(n)$ traces (or sometimes just a single trace). These classes contain strings that require a linear number of traces for exact reconstruction and which are quite different from a typical random string. From a technical point of view, our algorithms approximately reconstruct consecutive substrings of the unknown string by aligning dense regions of traces and using a run of a suitable length to approximate each region. To complement our algorithms, we present a general black-box lower bound for approximate reconstruction, building on a lower bound for distinguishing between two candidate input strings in the worst case. In particular, this shows that approximating to within $n^{1/3 - δ}$ edit distance requires $n^{1 + 3δ/2}/\mathrm{polylog}(n)$ traces for $0< δ< 1/3$ in the worst case.
The coded trace reconstruction problem asks to construct a code $C\subset \{0,1\}^n$ such that any $x\in C$ is recoverable from independent outputs ("traces") of $x$ from a binary deletion channel (BDC). We present binary codes of rate $1-\varepsilon$ that are efficiently recoverable from ${\exp(O_q(\log^{1/3}(\frac{1}{\varepsilon})))}$ (a constant independent of $n$) traces of a $\operatorname{BDC}_q$ for any constant deletion probability $q\in(0,1)$. We also show that, for rate $1-\varepsilon$ binary codes, $\tilde Ω(\log^{5/2}(1/\varepsilon))$ traces are required. The results follow from a …
The deletion channel takes as input a bit string $\mathbf{x} \in \{0,1\}^n$, and deletes each bit independently with probability $q$, yielding a shorter string. The trace reconstruction problem is to recover an unknown string $\mathbf{x}$ from many independent outputs (called "traces") of the deletion channel applied to $\mathbf{x}$. We show that if $\mathbf{x}$ is drawn uniformly at random and $q < 1/2$, then $e^{O(\log^{1/2} n)}$ traces suffice to reconstruct $\mathbf{x}$ with high probability. The previous best bound, established in 2008 by Holenstein-Mitzenmacher-Panigrahy-Wieder, uses $n^…
In the (deletion-channel) trace reconstruction problem, there is an unknown $n$-bit source string $x$. An algorithm is given access to independent traces of $x$, where a trace is formed by deleting each bit of~$x$ independently with probability~$δ$. The goal of the algorithm is to recover~$x$ exactly (with high probability), while minimizing samples (number of traces) and running time. Previously, the best known algorithm for the trace reconstruction problem was due to Holenstein~et~al.; it uses $\exp(\tilde{O}(n^{1/2}))$ samples and running time for any fixed $0 < δ< 1$. It is also what we ca…
The insertion-deletion channel takes as input a bit string ${\bf x}\in\{0,1\}^{n}$, and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover $\bf x$ from many independent outputs (called "traces") of the insertion-deletion channel applied to $\bf x$. We show that if $\bf x$ is chosen uniformly at random, then $\exp(O(\log^{1/3} n))$ traces suffice to reconstruct $\bf x$ with high probability. For the deletion channel with deletion probability $q < 1/2$ the earlier upper bound was $\exp(O(\log^{1/2} n))$. The case of …
The {\em insertion-deletion channel} takes as input a binary string $x \in\{0, 1\}^n$, and outputs a string $\widetilde{x}$ where some of the bits have been deleted and others inserted independently at random. In the {\em trace reconstruction problem}, one is given many outputs (called {\em traces}) of the insertion-deletion channel on the same input message $x$, and asked to recover the input message. Nazarov and Peres (STOC 2017), and De, O'Donnell and Servedio (STOC 2017) showed that any string $x$ can be reconstructed from $\exp(O(n^{1/3}))$ traces. Holden, Pemantle, Peres and Zhai (COLT 2…
We study the problem of learning a node-labeled tree given independent traces from an appropriately defined deletion channel. This problem, tree trace reconstruction, generalizes string trace reconstruction, which corresponds to the tree being a path. For many classes of trees, including complete trees and spiders, we provide algorithms that reconstruct the labels using only a polynomial number of traces. This exhibits a stark contrast to known results on string trace reconstruction, which require exponentially many traces, and where a central open problem is to determine whether a polynomial …
In the beautifully simple-to-state problem of trace reconstruction, the goal is to reconstruct an unknown binary string $x$ given random "traces" of $x$ where each trace is generated by deleting each coordinate of $x$ independently with probability $p<1$. The problem is well studied both when the unknown string is arbitrary and when it is chosen uniformly at random. For both settings, there is still an exponential gap between upper and lower sample complexity bounds and our understanding of the problem is still surprisingly limited. In this paper, we consider natural parameterizations and gene…
Trace reconstruction is the problem of learning an unknown string $x$ from independent traces of $x$, where traces are generated by independently deleting each bit of $x$ with some deletion probability $q$. In this paper, we initiate the study of Circular trace reconstruction, where the unknown string $x$ is circular and traces are now rotated by a random cyclic shift. Trace reconstruction is related to many computational biology problems studying DNA, which is a primary motivation for this problem as well, as many types of DNA are known to be circular. Our main results are as follows. First, …
The well-known trace reconstruction problem is the problem of inferring an unknown source string $x \in \{0,1\}^n$ from independent "traces", i.e. copies of $x$ that have been corrupted by a $δ$-deletion channel which independently deletes each bit of $x$ with probability $δ$ and concatenates the surviving bits. The current paper considers the extreme data-limited regime in which only a single trace is provided to the reconstruction algorithm. In this setting exact reconstruction is of course impossible, and the question is to what accuracy the source string $x$ can be approximately reconstruc…
In the trace reconstruction problem, one seeks to reconstruct a binary string $s$ from a collection of traces, each of which is obtained by passing $s$ through a deletion channel. It is known that $\exp(\tilde O(n^{1/5}))$ traces suffice to reconstruct any length-$n$ string with high probability. We consider a variant of the trace reconstruction problem where the goal is to recover a "density map" that indicates the locations of each length-$k$ substring throughout $s$. We show that $ε^{-2}\cdot \text{poly}(n)$ traces suffice to recover the density map with error at most $ε$. As a result, when…
Tree trace reconstruction aims to learn the binary node labels of a tree, given independent samples of the tree passed through an appropriately defined deletion channel. In recent work, Davies, Rácz, and Rashtchian used combinatorial methods to show that $\exp(\mathcal{O}(k \log_{k} n))$ samples suffice to reconstruct a complete $k$-ary tree with $n$ nodes with high probability. We provide an alternative proof of this result, which allows us to generalize it to a broader class of tree topologies and deletion models. In our proofs, we introduce the notion of a subtrace, which enables us to conn…
We study the trace reconstruction problem for spider graphs. Let $n$ be the number of nodes of a spider and $d$ be the length of each leg, and suppose that we are given independent traces of the spider from a deletion channel in which each non-root node is deleted with probability $q$. This is a natural generalization of the string trace reconstruction problem in theoretical computer science, which corresponds to the special case where the spider has one leg. In the regime where $d\ge \log_{1/q}(n)$, the problem can be reduced to the vanilla string trace reconstruction problem. We thus study t…
In the standard trace reconstruction problem, the goal is to \emph{exactly} reconstruct an unknown source string $\mathsf{x} \in \{0,1\}^n$ from independent "traces", which are copies of $\mathsf{x}$ that have been corrupted by a $δ$-deletion channel which independently deletes each bit of $\mathsf{x}$ with probability $δ$ and concatenates the surviving bits. We study the \emph{approximate} trace reconstruction problem, in which the goal is only to obtain a high-accuracy approximation of $\mathsf{x}$ rather than an exact reconstruction. We give an efficient algorithm, and a near-matching lower…
In the \emph{trace reconstruction problem}, an unknown source string $x \in \{0,1\}^n$ is transmitted through a probabilistic \emph{deletion channel} which independently deletes each bit with some fixed probability $δ$ and concatenates the surviving bits, resulting in a \emph{trace} of $x$. The problem is to reconstruct $x$ given access to independent traces. Trace reconstruction of arbitrary (worst-case) strings is a challenging problem, with the current state of the art for poly$(n)$-time algorithms being the 2004 algorithm of Batu et al. \cite{BKKM04}. This algorithm can reconstruct an arbi…
We consider an \emph{approximate} version of the trace reconstruction problem, where the goal is to recover an unknown string $s\in\{0,1\}^n$ from $m$ traces (each trace is generated independently by passing $s$ through a probabilistic insertion-deletion channel with rate $p$). We present a deterministic near-linear time algorithm for the average-case model, where $s$ is random, that uses only \emph{three} traces. It runs in near-linear time $\tilde O(n)$ and with high probability reports a string within edit distance $O(εp n)$ from $s$ for $ε=\tilde O(p)$, which significantly improves over th…
In this paper, we consider the problem of reconstructing trees from traces in the tree edit distance model. Previous work by Davies et al. (2019) analyzed special cases of reconstructing labeled trees. In this work, we significantly expand our understanding of this problem by giving general results in the case of arbitrary trees. Namely, we give: a reduction from the tree trace reconstruction problem to the more classical string reconstruction problem when the tree topology is known, a lower bound for learning arbitrary tree topologies, and a general algorithm for learning the topology of any …
In the \emph{trace reconstruction problem}, an unknown source string $x \in \{0,1\}^n$ is sent through a probabilistic \emph{deletion channel} which independently deletes each bit with probability $δ$ and concatenates the surviving bits, yielding a \emph{trace} of $x$. The problem is to reconstruct $x$ given independent traces. This problem has received much attention in recent years both in the worst-case setting where $x$ may be an arbitrary string in $\{0,1\}^n$ \cite{DOS17,NazarovPeres17,HHP18,HL18,Chase19} and in the average-case setting where $x$ is drawn uniformly at random from $\{0,1\…
\emph{Population recovery} is the problem of learning an unknown distribution over an unknown set of $n$-bit strings, given access to independent draws from the distribution that have been independently corrupted according to some noise channel. Recent work has intensively studied such problems both for the bit-flip and erasure noise channels. We initiate the study of population recovery under the \emph{deletion channel}, in which each bit is independently \emph{deleted} with some fixed probability and the surviving bits are concatenated and transmitted. This is a far more challenging noise mo…
Spectral graph sparsification aims to find ultra-sparse subgraphs which can preserve spectral properties of original graphs. In this paper, a new spectral criticality metric based on trace reduction is first introduced for identifying spectrally important off-subgraph edges. Then, a physics-inspired truncation strategy and an approach using approximate inverse of Cholesky factor are proposed to compute the approximate trace reduction efficiently. Combining them with the iterative densification scheme in \cite{feng2019grass} and the strategy of excluding spectrally similar off-subgraph edges in…
X-ray Computed Tomography (CT) reconstruction from a sparse number of views is a useful way to reduce either the radiation dose or the acquisition time, for example in fixed-gantry CT systems, however this results in an ill-posed inverse problem whose solution is typically computationally demanding. Approximate Message Passing (AMP) techniques represent the state of the art for solving under-sampling Compressed Sensing problems with random linear measurements but there are still not clear solutions on how AMP should be modified and how it performs with real world problems. This paper investiga…
We study the problem of reconstructing a hidden graph given access to a distance oracle. We design randomized algorithms for the following problems: reconstruction of a degree bounded graph with query complexity $\tilde{O}(n^{3/2})$; reconstruction of a degree bounded outerplanar graph with query complexity $\tilde{O}(n)$; and near-optimal approximate reconstruction of a general graph.
