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This paper presents a method to reduce a set of n 2D points to a smaller set of s 2D points with the property that the convex hull on the smaller set is the same as the convex hull of the original bigger set. The paper shows, experimentally, that such reduction accelerates computations; the time it takes to reduce from n down to s points plus the time of computing the convex hull on the s points is less than the time to compute the convex hull on the original set of n points. The method accepts 2D points expressed as real numbers and thus extends our previous method that required points as integers. The method achieves a percentage of reduction of points of over 90% in a collections of four datasets. This amount of reduction provides speedup factors of at least two for various common convex hull algorithms. Theoretically, the reduction method executes in time within O(n) and thus is suitable for preprocessing 2D data before computing the convex hull by any known algorithm. [ABSTRACT FROM AUTHOR]

Classical separability problem involving multi-color point sets is an important area of study in computational geometry. In this paper, we study different separability problems for bichromatic point set P = P r ∪ P b in R 2 and R 3 , where P r and P b represent a set of n red points and a set of m blue points respectively, and the objective is to compute a monochromatic object of a desired type and of maximum size. We propose in-place algorithms for computing (i) an arbitrarily oriented monochromatic rectangle of maximum size in R 2 , and (ii) an axis-parallel monochromatic cuboid of maximum size in R 3. The time complexities of the algorithms for problems (i) and (ii) are O (m (m + n) (m n + m log m + n log n)) and O (m 3 n + m 2 n log n) , respectively. As a prerequisite, we propose an in-place construction of the classic data structure the k-d tree , that was originally invented by J. L. Bentley in 1975. Our in-place variant of the k -d tree for a set of n points in R k supports orthogonal range counting query using O (1) extra workspace, and with O (n 1 − 1 ∕ k) query time complexity. The construction time of this data structure is O (n log n). Both the construction and query algorithms are non-recursive in nature that do not need O (log n) size recursion stack compared to the previously known construction algorithm for in-place k -d tree and query in it. We believe that this result is of independent interest. We also propose an algorithm for the problem of computing an arbitrarily oriented rectangle of maximum weight among a point set P = P r ∪ P b , where each point in P b (resp. P r) is associated with a negative (resp. positive) real-valued weight that runs in O (m 2 (n + m) log (n + m)) time using O (n) extra space. [ABSTRACT FROM AUTHOR]

With the emergence and widespread application of cloud computing, the use of cloud platforms to solve the problem of secure multi-party computation has emerged as a new research direction. The traditional computation of a solid geometry is performed through mutual interactions between two parties, which is not suitable in an untrusted cloud computing environment. In this paper, we first design a basic protocol for a secure Euclidean distance calculation that is suitable for cloud platforms and can serve as a building block for other protocols on cloud platforms. Using the solution of the Euclidean distance problem as such a building block, we provide a new method that converts the problems of calculating solid triangular areas and solid tetrahedral volumes into the calculation of distances and determinants in three-dimensional space. Then, we discuss solid point-line distance calculations, which extent the idea of the spatial geometry security problem. We present protocols for the above problems and prove that the proposed protocols can resist conspiracy among users and the untrusted cloud platform so that they can effectively ensure the privacy of the users. We also analyze the performances of these solutions. The analysis results show that our scheme is more versatile. [ABSTRACT FROM AUTHOR]

Recent research shows that significant energy saving can be achieved in wireless sensor networks (WSNs) with a mobile base station that collects data from sensor nodes via short-range communications. However, a major performance bottleneck of such WSNs is the significantly increased latency in data collection due to the low movement speed of mobile base stations. In this paper we study the problem of finding a data collection path for a mobile base station moving along a fixed track in a wireless sensor network to minimize the latency of data collection. The main contribution is an O ( m n log ⁡ n ) expected time algorithm, where n is the number of sensors in the networks and m is the complexity of the fixed track. [ABSTRACT FROM AUTHOR]