, Volume 2, Issue 1, pp 195–208

Geometric applications of a matrix-searching algorithm


  • Alok Aggarwal
    • IBM T. J. Watson Research CenterYorktown Heights
  • Maria M. Klawe
    • IBM Almaden Research Center
  • Shlomo Moran
    • IBM T. J. Watson Research CenterYorktown Heights
  • Peter Shor
    • Mathematical Sciences Research Institute
  • Robert Wilber
    • IBM Almaden Research Center

DOI: 10.1007/BF01840359

Cite this article as:
Aggarwal, A., Klawe, M.M., Moran, S. et al. Algorithmica (1987) 2: 195. doi:10.1007/BF01840359


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 ifi1 >i2 implies thatj(i1) ≥J(i2).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) whenmn 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.

Key words

All-farthest neighborsMonotone matrixConvex polygonWire routingInscribed polygonsCircumscribed polygons

Copyright information

© Springer-Verlag New York Inc. 1987