An Optimal Algorithm for the Maximum-Density Segment Problem

TL;DR

This paper presents a linear-time algorithm for the maximum-density segment problem, improving efficiency and enabling online processing, which is crucial for analyzing large biomolecular sequences.

Contribution

The authors develop an O(n) algorithm that bypasses complex decomposition methods, allowing online processing and exploiting sequence sparsity for faster computation.

Findings

Achieves O(n) time complexity for the problem.

Enables online processing of sequences.

Handles sparse input sequences efficiently.

Abstract

We address a fundamental problem arising from analysis of biomolecular sequences. The input consists of two numbers $w_{\min}$ and $w_{\max}$ and a sequence $S$ of $n$ number pairs $(a_i,w_i)$ with $w_i>0$. Let {\em segment} $S(i,j)$ of $S$ be the consecutive subsequence of $S$ between indices $i$ and $j$. The {\em density} of $S(i,j)$ is $d(i,j)=(a_i+a_{i+1}+...+a_j)/(w_i+w_{i+1}+...+w_j)$. The {\em maximum-density segment problem} is to find a maximum-density segment over all segments $S(i,j)$ with $w_{\min}\leq w_i+w_{i+1}+...+w_j \leq w_{\max}$. The best previously known algorithm for the problem, due to Goldwasser, Kao, and Lu, runs in $O(n\log(w_{\max}-w_{\min}+1))$ time. In the present paper, we solve the problem in O(n) time. Our approach bypasses the complicated {\em right-skew decomposition}, introduced by Lin, Jiang, and Chao. As a result, our algorithm has the capability to process the input sequence in an online manner, which is an important feature for dealing with genome-scale sequences. Moreover, for a type of input sequences $S$ representable in $O(m)$ space, we show how to exploit the sparsity of $S$ and solve the maximum-density segment problem for $S$ in $O(m)$ time.