, Volume 2, Issue 1, pp 195208
First online:
Geometric applications of a matrixsearching 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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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 allfarthestneighbors 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
Allfarthest neighbors Monotone matrix Convex polygon Wire routing Inscribed polygons Circumscribed polygons Title
 Geometric applications of a matrixsearching algorithm
 Journal

Algorithmica
Volume 2, Issue 14 , pp 195208
 Cover Date
 198711
 DOI
 10.1007/BF01840359
 Print ISSN
 01784617
 Online ISSN
 14320541
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Allfarthest neighbors
 Monotone matrix
 Convex polygon
 Wire routing
 Inscribed polygons
 Circumscribed polygons
 Industry Sectors
 Authors

 Alok Aggarwal ^{(1)}
 Maria M. Klawe ^{(2)}
 Shlomo Moran ^{(1)}
 Peter Shor ^{(3)}
 Robert Wilber ^{(2)}
 Author Affiliations

 1. IBM T. J. Watson Research Center, Yorktown Heights, New York, USA
 2. IBM Almaden Research Center, San Jose, California, USA
 3. Mathematical Sciences Research Institute, Berkeley, California, USA