Notes on computing peaks in k-levels and parametric spanning trees

  • Authors:

  • Naoki Katoh;Takeshi Tokuyama

  • Affiliations:

  • Department of Architecture, Kyoto University, Kyoto, 606-8501, Japan;Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

  • Venue:
  • SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
  • Year:
  • 2001

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Abstract

We give an algorithm to compute all the local peaks in the $k$-level o f an arrangement of $n$ lines in $O(n \log n) + \tilde{O}((kn)^{2/3})$ time. We can also find $\tau$ largest peaks in $O(n \log ^2 n) + \tilde{O}((\tau n)^{2/3})$ time. Moreover, we consider the longest edge in a parametric minimum spanning tree (in other words, a bottleneck edge for connectivity), and give an algorithm to compute the parameter value (within a given interval) maximizing/minimizing the length of the longest edge in MST. The time complexity is $\tilde{O}( n^{8/7}k^{1/7} + n k^{1/3})$.