New Bounds for Circular Trace Reconstruction

TL;DR

This work studies circular trace reconstruction under a deletion channel, establishing a near-tight separation between lower and upper bounds for constant-sparse inputs. The authors introduce cyclic statistics, a family of shift-invariant polynomials linked to Fourier coefficients, and prove that two cyclically distinct sequences differ in some cyclic statistic of order at most 6. They derive a lower bound of tildeΩ(n^5) traces needed to distinguish two constant-sparse strings and an upper bound of tildeO(n^6) traces sufficing to distinguish any constant-sparse pair, implying that further improvements must target non-constant-sparse strings. The results reveal that the circular model inherits increased hardness over the linear one and motivate new algorithmic and lower-bound techniques focused on non-constant sparsity regimes.

Abstract

The ''trace reconstruction'' problem asks, given an unknown binary string $x$ and a channel that repeatedly returns ''traces'' of $x$ with each bit randomly deleted with some probability $p$, how many traces are needed to recover $x$? There is an exponential gap between the best known upper and lower bounds for this problem. Many variants of the model have been introduced in hopes of motivating or revealing new approaches to narrow this gap. We study the variant of circular trace reconstruction introduced by Narayanan and Ren (ITCS 2021), in which traces undergo a random cyclic shift in addition to random deletions. We show an improved lower bound of $\tildeΩ(n^5)$ for circular trace reconstruction. This contrasts with the (previously) best known lower bounds of $\tildeΩ(n^3)$ in the circular case and $\tildeΩ(n^{3/2})$ in the linear case. Our bound shows the indistinguishability of traces from two sparse strings $x,y$ that each have a constant number of nonzeros. Can this technique be extended significantly? How hard is it to reconstruct a sparse string $x$ under a cyclic deletion channel? We resolve these questions by showing, using Fourier techniques, that $\tilde{O}(n^6)$ traces suffice for reconstructing any constant-sparse string in a circular deletion channel, in contrast to the upper bound of $\exp(\tilde{O}(n^{1/3}))$ for general strings in the circular deletion channel. This shows that new algorithms or new lower bounds must focus on non-constant-sparse strings.

Theorems & Definitions (50)

• Lemma 1: Informal
• Theorem 1: Informal version of \ref{['lem:lowerbd']}
• Theorem 2: Informal version of \ref{['thm:upperbd']}
• Definition 1
• Definition 2: Circular deletion channel
• Definition 3: Trace reconstruction problem
• Definition 4: Distinguishing problem
• Remark 1
• Lemma 2
• proof