Article

Algorithmica

, Volume 2, Issue 1, pp 195-208

First online:

Geometric applications of a matrix-searching algorithm

  • Alok AggarwalAffiliated withIBM T. J. Watson Research Center, Yorktown Heights
  • , Maria M. KlaweAffiliated withIBM Almaden Research Center
  • , Shlomo MoranAffiliated withIBM T. J. Watson Research Center, Yorktown Heights
  • , Peter ShorAffiliated withMathematical Sciences Research Institute
  • , Robert WilberAffiliated withIBM Almaden Research Center

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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) 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 neighbors Monotone matrix Convex polygon Wire routing Inscribed polygons Circumscribed polygons