DeepAI
Log In Sign Up

Average-Case to (shifted) Worst-Case Reduction for the Trace Reconstruction Problem

07/23/2022
by   Ittai Rubinstein, et al.
0

The insertion-deletion channel takes as input a binary string x ∈{0, 1}^n, and outputs a string x where some of the bits have been deleted and others inserted independently at random. In the trace reconstruction problem, one is given many outputs (called traces) of the insertion-deletion channel on the same input message x, and asked to recover the input message. Nazarov and Peres (STOC 2017), and De, O'Donnell and Servedio (STOC 2017) showed that any string x can be reconstructed from exp(O(n^1/3)) traces. Holden, Pemantle, Peres and Zhai (COLT 2018) adapt the techniques used to prove this upper bound, to an algorithm for the average-case trace reconstruction with a sample complexity of exp(O(log^1/3 n)). However, it is not clear how to apply their techniques more generally and in particular for the recent worst-case upper bound of exp(O(n^1/5)) shown by Chase (STOC 2021) for the deletion-channel. We prove a general reduction from the average-case to smaller instances of a problem similar to worst-case. Using this reduction and a generalization of Chase's bound, we construct an improved average-case algorithm with a sample complexity of exp(O(log^1/5 n)). Additionally, we show that Chase's upper-bound holds for the insertion-deletion channel as well.

READ FULL TEXT

page 1

page 2

page 3

page 4

01/15/2018

Subpolynomial trace reconstruction for random strings and arbitrary deletion probability

The deletion-insertion channel takes as input a bit string x∈{0,1}^n, a...
08/27/2020

Polynomial-time trace reconstruction in the smoothed complexity model

In the trace reconstruction problem, an unknown source string x ∈{0,1}^n...
02/18/2021

Mean-Based Trace Reconstruction over Practically any Replication-Insertion Channel

Mean-based reconstruction is a fundamental, natural approach to worst-ca...
05/23/2019

On the Average Case of MergeInsertion

MergeInsertion, also known as the Ford-Johnson algorithm, is a sorting a...
11/07/2022

Approximate Trace Reconstruction from a Single Trace

The well-known trace reconstruction problem is the problem of inferring ...
04/11/2019

Beyond trace reconstruction: Population recovery from the deletion channel

Population recovery is the problem of learning an unknown distribution o...
08/12/2019

Coded trace reconstruction in a constant number of traces

The coded trace reconstruction problem asks to construct a code C⊂{0,1}^...