Selection and sorting in totally monotone arrays
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Thek largest (ork smallest) entries in each row ofA can be computed inO(k(m + n)) time. This result allows us to determine thek farthest (ork nearest) neighbors of each vertex of a convexn-gon inO(kn) time.
Provided the transpose ofA is also totally monotone, thek largest (ork smallest) entries overall inA can be computed inO(m + n + k lg(mn/k)) time. This result allows us to find thek farthest (ork nearest) pairs of vertices from a convexn-gon inO(n + k lg(n2/k)) time.
The rows ofA can be sorted inO(mn) time whenm ≥n and inO(mn(1 + lg(n/m))) time whenm < n. This result allows us to solve the of Ω(S) on the number of combinations of row permutations possible for a totally monotone array would imply an Ω(lgS) lower bound on the time necessary to sort the array's rows in a linear decision tree model.)
In Subsection 4.2 we applied our algorithm for sorting the rows of a totally monotone array to the neighbor-ranking problem for the vertices of a convex polygonP. We then extended this technique to arbitrary point sets. It remains open whether our two selection algorithms for totally monotone arrays, which we also applied to the vertices of a convex polygon, can likewise be applied to arbitrary point sets.
KeywordsDecision Tree Computational Mathematic Tree Model Selection Algorithm Arbitrary Point
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