Searching a bitstream in linear time for the longest substring of any given density

TL;DR

This paper presents a linear-time algorithm for finding the longest substring in a bitstream with a specified ratio of ones to zeroes, with applications in cryptography and bioinformatics.

Contribution

It introduces a novel linear-time algorithm for substring density problems by reformulating as a constrained walk and developing a specialized data structure.

Findings

Algorithm solves the problem in linear time

Applicable to cryptography and bioinformatics

Efficiently finds longest substrings with given density constraints

Abstract

Given an arbitrary bitstream, we consider the problem of finding the longest substring whose ratio of ones to zeroes equals a given value. The central result of this paper is an algorithm that solves this problem in linear time. The method involves (i) reformulating the problem as a constrained walk through a sparse matrix, and then (ii) developing a data structure for this sparse matrix that allows us to perform each step of the walk in amortised constant time. We also give a linear time algorithm to find the longest substring whose ratio of ones to zeroes is bounded below by a given value. Both problems have practical relevance to cryptography and bioinformatics.