Algorithmica

, Volume 53, Issue 3, pp 298–313

Fast Algorithms for the Density Finding Problem

Authors

  • D. T. Lee
    • Department of Computer Science and Information EngineeringNational Taiwan University
    • Institute of Information ScienceAcademia Sinica
    • Department of Computer Science and Information EngineeringNational Taiwan University
    • Institute of Information ScienceAcademia Sinica
  • Hsueh-I Lu
    • Department of Computer Science and Information EngineeringNational Taiwan University
Article

DOI: 10.1007/s00453-007-9023-8

Cite this article as:
Lee, D.T., Lin, T. & Lu, H. Algorithmica (2009) 53: 298. doi:10.1007/s00453-007-9023-8

Abstract

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≤in, two nonnegative numbers , u with 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 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.

Keywords

Maximum-density segment problem Density finding problem Slope selection problem Convex hull Computational geometry GC content DNA sequence Bioinformatics

Copyright information

© Springer Science+Business Media, LLC 2007