Applications of generalized matrix searching to geometric algorithms

Abstract This paper introduces a generalization of totally monotone matrices, namely totally monotone partial matrices, shows how a number of problems in computational geometry can be reduced to the problem of finding the row maxima and minima in totally monotone partial matrices, and gives an O((m+nlog logn) algorithm for finding row maxima and minima in an n×m totally monotone partial matrix. In particular, if P and Q are nonintersecting n and m vertex convex polygons, respectively, our methods give an O((m+n)log logn) algorithm for finding for each vertex x of P, the farthest vertex of Q which is not visible to x, and the nearest vertex of Q which is not visible to x.

[1]  R. Seidel.  A Method for Proving Lower Bounds for Certain Geometric Problems , 1984 .

[2]  David G. Kirkpatrick,et al.  A Linear Algorithm for Determining the Separation of Convex Polyhedra , 1985, J. Algorithms.

[3]  D. T. Lee,et al.  The All Nearest-Neighbor Problem for Convex Polygons , 1978, Inf. Process. Lett..

[4]  Alok Aggarwal,et al.  Fast algorithms for computing the largest empty rectangle , 1987, SCG '87.

[5]  Leonidas J. Guibas,et al.  Optimal shortest path queries in a simple polygon , 1987, SCG '87.

[6]  Fabrizio Luccio,et al.  On Cahit's Result on Graceful Permutations , 1978, Inf. Process. Lett..

[7]  Robert E. Wilber.  The Concave Least-Weight Subsequence Problem Revisited , 1988, J. Algorithms.

[8]  Daniel J. Kleitman,et al.  An Almost Linear Time Algorithm for Generalized Matrix Searching , 1990, SIAM J. Discret. Math..

[9]  Leonidas J. Guibas,et al.  A linear-time algorithm for computing the voronoi diagram of a convex polygon , 1989, Discret. Comput. Geom..

[10]  D. T. Lee,et al.  On k-Nearest Neighbor Voronoi Diagrams in the Plane , 1982, IEEE Transactions on Computers.

[11]  Alok Aggarwal,et al.  Geometric applications of a matrix-searching algorithm , 1987, SCG '86.

[12]  David G. Kirkpatrick,et al.  Optimal Search in Planar Subdivisions , 1983, SIAM J. Comput..

[13]  Bernard Chazelle,et al.  Computing the Largest Empty Rectangle , 1986, SIAM J. Comput..

[14]  Edward P. F. Chan,et al.  Finding the minimum visible vertex distance between two non-intersecting simple polygons , 1986, SCG '86.