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Geometric applications of a matrix-searching algorithm

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Abstract

LetA be a matrix with real entries and letj(i) be the index of the leftmost column containing the maximum value in rowi ofA.A is said to bemonotone ifi 1 >i 2 implies thatj(i 1) ≥J(i 2).A istotally monotone if all of its submatrices are monotone. We show that finding the maximum entry in each row of an arbitraryn xm monotone matrix requires Θ(m logn) time, whereas if the matrix is totally monotone the time is Θ(m) whenm≥n and is Θ(m(1 + log(n/m))) whenm<n. The problem of finding the maximum value within each row of a totally monotone matrix arises in several geometric algorithms such as the all-farthest-neighbors problem for the vertices of a convex polygon. Previously only the property of monotonicity, not total monotonicity, had been used within these algorithms. We use the Θ(m) bound on finding the maxima of wide totally monotone matrices to speed up these algorithms by a factor of logn.

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Author information

Authors and Affiliations

  1. IBM T. J. Watson Research Center, Yorktown Heights, New York, USA

    Alok Aggarwal & Shlomo Moran

  2. IBM Almaden Research Center, San Jose, California, USA

    Maria M. Klawe & Robert Wilber

  3. Mathematical Sciences Research Institute, Berkeley, California, USA

    Peter Shor

Authors
  1. Alok Aggarwal
  2. Maria M. Klawe
  3. Shlomo Moran
  4. Peter Shor
  5. Robert Wilber

Additional information

Communicated by Bernard Chazelle.

On leave from the Technion, Haifa, Israel.

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Aggarwal, A., Klawe, M.M., Moran, S. et al. Geometric applications of a matrix-searching algorithm. Algorithmica 2, 195–208 (1987). https://doi.org/10.1007/BF01840359

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  • DOI: https://doi.org/10.1007/BF01840359

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