[Paper Review] Approximate Trace Reconstruction

Sami Davies, Miklós Z. Rácz+2 more|arXiv (Cornell University)|Dec 12, 2020

DNA and Biological Computing38 references7 citations

TL;DR

This paper introduces efficient algorithms for approximate trace reconstruction, where the goal is to reconstruct a string within edit distance $\varepsilon n$ using $\mathrm{polylog}(n)$ traces instead of the $\exp(\widetilde{O}(n^{1/5}))$ traces required for exact reconstruction. It presents methods for classes of strings with long runs, using trace alignment and concentration bounds, and establishes a lower bound showing that $n^{1+3\delta/2}/\mathrm{polylog}(n)$ traces are necessary to achieve $n^{1/3-\delta}$-approximate reconstruction for $0 < \delta < 1/3$.

ABSTRACT

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 $0&lt; δ&lt; 1/3$ in the worst case.

Motivation & Objective

• Address the challenge of reducing trace complexity in string reconstruction by relaxing exact recovery to approximate reconstruction within edit distance $\varepsilon n$.
• Develop efficient algorithms that use $\mathrm{polylog}(n)$ traces or even a single trace for specific string classes.
• Establish theoretical limits on the number of traces required for approximate reconstruction, particularly in the worst case.
• Provide a framework for approximate reconstruction that is computationally efficient and applicable to DNA data storage.
• Bridge the gap between approximate and exact reconstruction by showing that approximation can be significantly more efficient for certain string classes.

Proposed method

• Use alignment of dense regions in traces to estimate substrings of the original string, focusing on strings with long runs of identical bits.
• Apply concentration bounds and empirical averages of trace patterns to estimate the structure of the original string with high confidence.
• Design algorithms that output a string within $n/\mathrm{polylog}(n)$ edit distance using $O(\log n / \varepsilon^2)$ traces for large classes of strings.
• Introduce a black-box lower bound technique based on distinguishing between two candidate strings, extending existing trace reconstruction lower bounds.
• Use Chernoff bounds and probabilistic analysis to show that incorrect reconstructions are unlikely when alignment fails on indistinguishable blocks.
• Construct a worst-case lower bound by reducing the problem to differentiating between two strings with long zero runs, requiring $\Omega(n)$ traces.

Experimental results

Research questions

• RQ1 Can approximate trace reconstruction be achieved with significantly fewer traces than required for exact reconstruction?
• RQ2 Are there classes of strings for which $\mathrm{polylog}(n)$ traces suffice to achieve $\varepsilon n$-approximate reconstruction?
• RQ3 What is the fundamental trade-off between approximation error and trace complexity in the worst case?
• RQ4 Can techniques from concentration of measure and trace alignment be used to design efficient approximate reconstruction algorithms?
• RQ5 Is there a provable separation in trace complexity between approximate and exact reconstruction for any class of strings?

Key findings

• Algorithms can achieve $n/\mathrm{polylog}(n)$ edit distance approximation using only $O(\log n / \varepsilon^2)$ traces for strings with long runs.
• Single-trace approximate reconstruction is possible for certain structured string classes, such as those with long runs of 1s and isolated 0s.
• A lower bound shows that $n^{1+3\delta/2}/\mathrm{polylog}(n)$ traces are necessary to achieve $n^{1/3-\delta}$-approximate reconstruction for $0 < \delta < 1/3$.
• The lower bound is derived by reducing the problem to distinguishing two strings with long zero runs, which requires $\Omega(n)$ traces.
• Approximate reconstruction is strictly easier than exact reconstruction for some string classes, such as those with Hamming weight $n-1$, which require $\Omega(n)$ traces for exact recovery.
• Approximate reconstruction algorithms may provide insight into exact reconstruction, especially if the unknown string lies within a small edit ball of radius $k$, where $n^{O(k)}$ traces may suffice.