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.
Key wordsAll-farthest neighbors Monotone matrix Convex polygon Wire routing Inscribed polygons Circumscribed polygons
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- A. Aggarwal and R. C. Melville, Fast computation of the modality of polygons,Proceedings of the Conference on Information Sciences and Systems, The Johns Hopkins University. Also appears inJ. Algorithms,7 (1986), 369–381.Google Scholar
- A. Aggarwal, J. S. Chang, and C. K. Yap, Minimum area circumscribing polygons, Technical Report, Courant Institute of Mathematical Sciences, New York University, 1985. Also to appear inVisual Comput. (1986).Google Scholar
- D. Dolev, K. Karplus, A. Siegel, A. Strong, and J. D. Ullman, Optimal wiring between rectangles,Proceedings of the 13th Annual ACM Symposium on the Theory of Computing, Milwaukee, WI, 1981, pp. 312–317.Google Scholar
- M. McKenna, J. O'Rourke, and S. Suri, Finding the largest rectangle in an orthogonal polygon, Technical Report, The Johns Hopkins University, 1985. Also appears inProceedings of the Allerton Conference on Control, Communications, and Computing, Allerton, IL, 1985.Google Scholar
- F. P. Preparata, Minimum spanning circle, inSteps in Computational Geometry (F. P. Preparata, ed.), University of Illinois Press, Urbana, 1977, pp. 3–5.Google Scholar
- M. I. Shamos, Geometric complexity,Proceedings of the Seventh Annual Symposium on Theory of Computing, Albuquerque, NM, 1975, pp. 224–233.Google Scholar
- M. I. Shamos and D. Hoey, Closest-point problems,Proceedings of the 16th Annual IEEE Symposium on Foundations of Computer Science, Berkeley, CA, 1975, pp. 151–162.Google Scholar
- A. Siegel and D. Dolev, The separation for general single-layer wiring barriers,Proceedings of the CMU Conference on VLSI Systems and Computations, Pittsburgh, PA, 1981, pp. 143–152.Google Scholar
- G. T. Toussaint, Complexity, convexity and unimodality,Proceedings of the Second World Conference on Mathematics, Las Palmas, Spain, 1982.Google Scholar
- G. T. Toussaint and B. K. Bhattacharya, On geometric algorithms that use the furthest-neighbor pair of a finite planar set, Technical Report, School of Computer Science, McGill University, 1981.Google Scholar