Idempotent permutations

Together with a characteristic function, idempotent permutations uniquely determine idempotent maps, as well as their linearly ordered arrangement simultaneously. Furthermore, in-place linear time transformations are possible between them. Hence, they may be important for succinct data structures, information storing, sorting and searching. In this study, their combinatorial interpretation is given and their application on sorting is examined. Given an array of n integer keys each in [1,n], if it is allowed to modify the keys in the range [-n,n], idempotent permutations make it possible to obtain linearly ordered arrangement of the keys in O(n) time using only 4log(n) bits, setting the theoretical lower bound of time and space complexity of sorting. If it is not allowed to modify the keys out of the range [1,n], then n+4log(n) bits are required where n of them is used to tag some of the keys.Comment: 32 page

On Euclidean Norm Approximations

Euclidean norm calculations arise frequently in scientific and engineering applications. Several approximations for this norm with differing complexity and accuracy have been proposed in the literature. Earlier approaches were based on minimizing the maximum error. Recently, Seol and Cheun proposed an approximation based on minimizing the average error. In this paper, we first examine these approximations in detail, show that they fit into a single mathematical formulation, and compare their average and maximum errors. We then show that the maximum errors given by Seol and Cheun are significantly optimistic.Comment: 9 pages, 1 figure, Pattern Recognitio

Cosine Similarity Measure According to a Convex Cost Function

In this paper, we describe a new vector similarity measure associated with a convex cost function. Given two vectors, we determine the surface normals of the convex function at the vectors. The angle between the two surface normals is the similarity measure. Convex cost function can be the negative entropy function, total variation (TV) function and filtered variation function. The convex cost function need not be differentiable everywhere. In general, we need to compute the gradient of the cost function to compute the surface normals. If the gradient does not exist at a given vector, it is possible to use the subgradients and the normal producing the smallest angle between the two vectors is used to compute the similarity measure