Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2410.11980

The Quasi-probability Method and Applications for Trace Reconstruction

Ittai Rubinstein
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the trace reconstruction problem, one attempts to reconstruct a fixed but unknown string $x$ of length $n$ from a given number of traces $\tilde{x}$ drawn iid from the application of a noisy process (such as the deletion channel) to $x$. The best known algorithm for the trace reconstruction from the deletion channel is due to Chase, and recovers the input string whp given $\exp(\tilde{O}(n^{1/5}))$ traces [Cha21b]. The main component in Chase's algorithm is a procedure for k-mer estimation, which, for any marker $w$ in $\{0, 1\}^k$ of length $k$, computes a "smoothed" distribution of its appearances in the input string $x$ [CGL+23, MS24]. Current k-mer estimation algorithms fail when the deletion probability is above $1/2$, requiring a more complex analysis for Chase's algorithm. Moreover, the only known extension of these approaches beyond the deletion channels is based on numerically estimating high-order differentials of a multivariate polynomial, making it highly impractical [Rub23]. In this paper, we construct a simple Monte Carlo method for k-mer estimation which can be easily applied to a much wider variety of channels. In particular, we solve k-mer estimation for any combination of insertion, deletion, and bit-flip channels, even in the high deletion probability regime, allowing us to directly apply Chase's algorithm for this wider class of channels. To accomplish this, we utilize an approach from the field of quantum error mitigation (the process of using many measurements from noisy quantum computers to simulate a clean quantum computer), called the quasi-probability method (also known as probabilistic error cancellation) [TBG17, PSW22]. We derive a completely classical version of this technique, and use it to construct a k-mer estimation algorithm. No background in quantum computing is needed to understand this paper.

Explore related subjects

cs.DScs.ITmath.IT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ittai Rubinstein. 2024-10-15. The Quasi-probability Method and Applications for Trace Reconstruction. https://arxiv.org/abs/2410.11980

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗