We study the problem of computing the diameter of a set of n points in d-dimensional Euclidean space for a  xed dimension d, and propose a new (1 + ")-approximation algorithm with O(n + 1="d􀀀 1) time and O(n) space, where 0 < " 6 1. We also show that the proposed algorithm can be modi ed to a (1 + O("))-approximation algorithm with O(n + 1=" 2d 3 􀀀 1 2 ) running time. Our proposed algorithms are di erent with the previous algorithms in terms of computational technique and data structures. These results provide some improvements in comparison with existing algorithms in terms of simplicity and data structure.

Graphs are common data structures which widely used for information storage and retrieval. Occasionally some vertices of a graph contain specific features or information, which we value in their effect. We consider modeling this effect formally, and we devise two super-fast algorithms to approximate the colored average degree. In the first method, we assume the information of each vertex is available; hence, the provided algorithm works with a 2+ϵ approximation factor. Eventually, we waive this assumption and find another algorithm with the same approximation factor, which computes the answer in the sublinear expected time.

‎‎‎Given a fuzzy normed space, we will introduce the notion of fuzzy near best approximation as a generalization of the notion of fuzzy best approximation. Some basic properties are characterized and also many examples for illustration are presented. Also, the hereditary properties of the fuzzy near best approximation on direct sum and tensor product of linear spaces are discussed.