Maximum-density Segment Research Articles

Fast Algorithms for the Density Finding Problem

We study the problem of finding a specific density subsequence of a sequence arising from the analysis of biomolecular sequences. Given a sequence A=(a 1,w 1),(a 2,w 2),…,(a n ,w n ) of n ordered pairs (a i ,w i ) of numbers a i and width w i >0 for each 1≤i≤n, two nonnegative numbers l, u with l≤u and a number δ, the Density Finding Problem is to find the consecutive subsequence A(i *,j *) over all O(n 2) consecutive subsequences A(i,j) with width constraint satisfying l≤w(i,j)=∑ r=i j w r ≤u such that its density $d(i^{*},j^{*})=\sum_{r=i^{*}}^{j*}a_{r}/w(i^{*},j^{*})$is closest to δ. The extensively studied Maximum-Density Segment Problem is a special case of the Density Finding Problem with δ=∞. We show that the Density Finding Problem has a lower bound Ω(nlog n) in the algebraic decision tree model of computation. We give an algorithm for the Density Finding Problem that runs in optimal O(nlog n) time and O(nlog n) space for the case when there is no upper bound on the width of the sequence, i.e., u=w(1,n). For the general case, we give an algorithm that runs in O(nlog 2 m) time and O(n+mlog m) space, where $m=\min\{\lfloor\frac{u-\ell}{w_{\mathrm{min}}}\rfloor,n\}$and w min=min r=1 n w r . As a byproduct, we give another O(n) time and space algorithm for the Maximum-Density Segment Problem.

Linear-time algorithms for computing maximum-density sequence segments with bioinformatics applications

We study an abstract optimization problem arising from biomolecular sequence analysis. For a sequence A of pairs ( a i , w i ) for i = 1 , … , n and w i > 0 , a segment A ( i , j ) is a consecutive subsequence of A starting with index i and ending with index j. The width of A ( i , j ) is w ( i , j ) = ∑ i ⩽ k ⩽ j w k , and the density is ( ∑ i ⩽ k ⩽ j a k ) / w ( i , j ) . The maximum-density segment problem takes A and two values L and U as input and asks for a segment of A with the largest possible density among those of width at least L and at most U. When U is unbounded, we provide a relatively simple, O ( n ) -time algorithm, improving upon the O ( n log L ) -time algorithm by Lin, Jiang and Chao. We then extend this result, providing an O ( n ) -time algorithm for the case when both L and U are specified.

An Optimal Algorithm for the Maximum-Density Segment Problem

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.

Linear-time algorithm for finding a maximum-density segment of a sequence

Linear-time algorithm for finding a maximum-density segment of a sequence