Improved algorithms for the continuous tree edge-partition problems and a note on ratio and sorted matrices searches

  • Authors:

  • Jyh-Jye Lin;Chi-Yuan Chan;Biing-Feng Wang

  • Affiliations:

  • Department of Computer Science, National Tsing Hua University, Hsinchu, 30043, Taiwan, ROC;Department of Computer Science, National Tsing Hua University, Hsinchu, 30043, Taiwan, ROC;Department of Computer Science, National Tsing Hua University, Hsinchu, 30043, Taiwan, ROC

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

Quantified Score

Hi-index 0.04

Visualization

Abstract

Let p=2 be an integer and T be an edge-weighted tree. A cut on an edge of T is a splitting of the edge at some point on it. A p-edge-partition of T is a set of p subtrees induced by p-1 cuts. Given p and T, the max-min continuous tree edge-partition problem is to find a p-edge-partition that maximizes the length of the smallest subtree; and the min-max continuous tree edge-partition problem is to find a p-edge-partition that minimizes the length of the largest subtree. In this paper, O(n^2)-time algorithms are proposed for these two problems, improving the previous upper bounds by a factor of log (min{p,n}). Along the way, we solve a problem, named the ratio search problem. Given a positive integer m, a (non-ordered) set B of n non-negative real numbers, a real valued non-increasing function F, and a real number t, the problem is to find the largest number z in {b/a|a@?[1,m],b@?B} such that F(z)=t. We give an O(n+t"Fx(logn+logm))-time algorithm for this problem, where t"F is the time required to evaluate the function value F(z) for any real number z.