Fundamental Limits of Reference-Based Sequence Reordering

TL;DR

This paper investigates the fundamental limits of reconstructing sequences from fragments and a reference, providing exact failure probability scaling in different regimes and analyzing trade-offs between fragment reconstruction accuracy and distortion.

Contribution

It offers a rigorous analysis of sequence reconstruction limits from fragments with and without repeats, including failure probability scaling and reconstruction-distortion trade-offs.

Findings

Failure probability scaling is exactly determined for long, unique fragments.

A trade-off between fragment reconstruction fraction and distortion level is characterized.

Conditions for vanishing failure probability in different regimes are established.

Abstract

The problem of reconstructing a sequence of independent and identically distributed symbols from a set of equal size, consecutive, fragments, as well as a dependent reference sequence, is considered. First, in the regime in which the fragments are relatively long, and typically no fragment appears more than once, the scaling of the failure probability of maximum likelihood reconstruction algorithm is exactly determined for perfect reconstruction and bounded for partial reconstruction. Second, the regime in which the fragments are relatively short and repeating fragments abound is characterized. A trade-off is stated between the fraction of fragments that cannot be adequately reconstructed vs. the distortion level allowed for the reconstruction of each fragment, while still allowing vanishing failure probability