DeepAI
Log In Sign Up

Near-Optimal Average-Case Approximate Trace Reconstruction from Few Traces

07/24/2021
โˆ™
by   Xi Chen, et al.
โˆ™
0
โˆ™

In the standard trace reconstruction problem, the goal is to exactly reconstruct an unknown source string ๐—‘โˆˆ{0,1}^n from independent "traces", which are copies of ๐—‘ that have been corrupted by a ฮด-deletion channel which independently deletes each bit of ๐—‘ with probability ฮด and concatenates the surviving bits. We study the approximate trace reconstruction problem, in which the goal is only to obtain a high-accuracy approximation of ๐—‘ rather than an exact reconstruction. We give an efficient algorithm, and a near-matching lower bound, for approximate reconstruction of a random source string ๐—‘โˆˆ{0,1}^n from few traces. Our main algorithmic result is a polynomial-time algorithm with the following property: for any deletion rate 0 < ฮด < 1 (which may depend on n), for almost every source string ๐—‘โˆˆ{0,1}^n, given any number M โ‰คฮ˜(1/ฮด) of traces from Del_ฮด(๐—‘), the algorithm constructs a hypothesis string ๐—‘ that has edit distance at most n ยท (ฮด M)^ฮฉ(M) from ๐—‘. We also prove a near-matching information-theoretic lower bound showing that given M โ‰คฮ˜(1/ฮด) traces from Del_ฮด(๐—‘) for a random n-bit string ๐—‘, the smallest possible expected edit distance that any algorithm can achieve, regardless of its running time, is n ยท (ฮด M)^O(M).

READ FULL TEXT

page 1

page 2

page 3

page 4

โˆ™ 12/12/2020

Approximate Trace Reconstruction

In the usual trace reconstruction problem, the goal is to exactly recons...
โˆ™ 07/20/2021

Approximate Trace Reconstruction via Median String (in Average-Case)

We consider an approximate version of the trace reconstruction problem, ...
โˆ™ 08/04/2018

Lower bounds for trace reconstruction

In the trace reconstruction problem, an unknown bit string xโˆˆ{0,1}^n is...
โˆ™ 11/07/2022

Approximate Trace Reconstruction from a Single Trace

The well-known trace reconstruction problem is the problem of inferring ...
โˆ™ 08/27/2020

Polynomial-time trace reconstruction in the smoothed complexity model

In the trace reconstruction problem, an unknown source string x โˆˆ{0,1}^n...
โˆ™ 09/02/2020

Circular Trace Reconstruction

Trace Reconstruction is the problem of learning an unknown string x from...
โˆ™ 11/27/2020

Limitations of Mean-Based Algorithms for Trace Reconstruction at Small Distance

Trace reconstruction considers the task of recovering an unknown string ...