Algorithms for Computing the Length-Constrained Max-Score Segments with Applications to DNA Copy Number Data Analysis

Abstract

Given a sequence of n real numbers A = (a ₁,a ₂,...,a _n ), two integers L and U with 1 ≤ L ≤ U ≤ n, and a score function f:IR⁺×IR→IR, the Length-Constrained Max-Score Segment Problem is to find a segment A[i,j] = (a _i ,a _i + 1,...,a _j ) maximizing $f(j-i+1,\sum_{h=i}^ja_h)$ subject to j − i + 1 ∈ [L,U]. In this paper, we solve the Length-Constrained Max-Score Segment Problem for the case where the given score function $f(\ell,w)=\frac{w}{\sqrt[r]{\ell}}$ for any constant r > 1. Our algorithm runs in $O(n\frac{T(L^{1/2})}{L^{1/2}})$ time, where T(n′) is the time required to solve the all-pairs shortest paths problem on a graph of n′ nodes. By the latest result of Chan [7], $T(n')=O(n'^3 \frac{(\log\log n')^3}{(\log n')^2})$, so our algorithm runs in subquadratic time $O(nL\frac{(\log\log L)^3}{(\log L)^2})$. Lipson et al. [21] studied a more restricted case where the score function $f(\ell,w)=\frac{w}{\sqrt[2]{\ell}}$ and there are no length constraints, i.e., L = 1 and U = n. They also showed how to apply their algorithm to analyzing DNA copy number data. However, their algorithm takes Ω(n ²) time in the worst situation. Since the length lower bound L for the case considered by Lipson et al. is a constant, our algorithm solves it in O(n) time.