In this paper we look at the classical set-maxima problem. We give a new result using a sparse lattice of subsets. We then extend this to a geometric setting. Let @math be a set of @math points in a plane and @math be a set of convex regions. Let a key value be assigned to each point. The problems is to determine for each convex region of @math the point having the largest key. We give a parameterized result for the comparison complexity.
@cite_4 showed that this can be generalized using weighted matroids. One of the canonical examples of matroids is the graphic matroid. Generalized to binary matroids (since graphic matroids are also regular) this has been termed by Liberatore as the fundamental path maxima problem over such matroids. A cographic matroid is a dual of a graphic matroid. For a cographic matroid the problem can be solved in @math ( @cite_3 ) comparisons. Liberatore generalized these results to a restricted class of matroids that can be constructed via direct-sums and 2-sums and gave a @math -comparison algorithm ( @cite_4 ). @cite_0 proposed an algorithm that chose a rank sequence randomly. They show that the expected number of comparisons in their algorithm is @math which is optimal according to the comparison tree complexity. The randomized algorithm can solve a more general problem of computing the largest @math elements for each subset @math .
@cite_0 @cite_4 @cite_3
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Randomized algorithms for two sorting problems are presented. In the local sorting problem, a graph is given in which each vertex is assigned an element of a total order, and the task is to determine the relative order of every pair of adjacent vertices. In the set-maxima problem, a collection of sets whose elements are drawn from a total order is given, and the task is to determine the maximum element in each set. Lower bounds for the problems in the comparison model are described and it is shown that the algorithms are optimal within a constant factor. A device for sharpening axe heads comprising a generally U-shaped frame placed over the head of the axe with opposite legs of the frame straddling opposite side faces of the axe head. Securing the frame to the head is a thumb screw threaded through one of the frame legs to engage one side face of the axe head. The other leg of the frame has an elongated aperture therein receiving a threaded bolt which in turn receives an elongated file holder arm to mount the same relative to the main frame for pivotal movement about the bolt. The inner end of the bolt has an enlarged head engaging the adjacent side face of the axe head while the other end of the bolt has a wing nut threaded thereon. A coil compression spring between the leg and the file holder arm yieldably urges the latter against the wing nut which acts as an adjustable stop. The file is mounted to the holder arm as an extention thereof by a U-shaped clamp secured across the file by wing nuts received on threaded studs of the clamp. In use the file may be grasped and oscillated across the cutting edge of the axe head as the file and its associated holder arm pivot about the bolt.