Optimal Randomized Algorithm for the Density Selection Problem

  • Authors:

  • Tien-Ching Lin;D. T. Lee

  • Affiliations:

  • Institute of Information Science, Academia Sinica, Taipei, Taiwan;Institute of Information Science, Academia Sinica, Taipei, Taiwan and Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan

  • Venue:
  • ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
  • Year:
  • 2009

Quantified Score

Hi-index 0.00

Visualization

Abstract

In the paper we consider a generalized version of three well-known problems: Selection Problem in computer science, Slope Selection Problem in computational geometry and Maximum-Density Segment Problem in bioinformatics. 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 real numbers a i and w i 0 for each 1 ≤ i ≤ n, two nonnegative real numbers 驴, u with 驴 ≤ u and a positive integer k, the Density Selection Problem is to find the consecutive subsequence A(i *,j *) over all O(n 2) consecutive subsequences A(i,j) satisfying width constraint $\ell \leq w(i,j) = \sum_{t=i}^j w_t \leq u$ such that the rank of its density $d(i^*,j^*) = \sum_{t=i^*}^{j*} a_t / w(i^*,j^*)$ is k. We will give a randomized algorithm for density selection problem that runs in optimal expected O(n logn) time.