Faster DBSCAN and HDBSCAN in Low-Dimensional Euclidean Spaces

Faster DBScan and HDBScan in Low-Dimensional Euclidean Spaces

We present a new algorithm for the widely used density-based clustering method dbscan. Our algorithm computes the dbscan-clustering in O(n log n) time in R, irrespective of the scale parameter ε (and assuming the second parameter MinPts is set to a fixed constant, as is the case in practice). Experiments show that the new algorithm is not only fast in theory, but that a slightly simplified vers...

Faster Exact Algorithms for Computing Steiner Trees in Higher Dimensional Euclidean Spaces

The Euclidean Steiner tree problem asks for a network of minimum total length interconnecting a finite set of points in d-dimensional space. For d ≥ 3, only one practical algorithmic approach exists for this problem — proposed by Smith in 1992. A number of refinements of Smith’s algorithm have increased the range of solvable problems a little, but it is still infeasible to solve problem instanc...

Lines in n-Dimensional Euclidean Spaces

In this paper, we define the line of n-dimensional Euclidean space and we introduce basic properties of affine space on this space. Next, we define the inner product of elements of this space. At the end, we introduce orthogonality of lines of this space. provide the terminology and notation for this paper. (4) (a · b) · x = a · (b · x). (6) a · (x 1 + x 2) = a · x 1 + a · x 2. (7) (a + b) · x ...

Euclidean Structure in Finite Dimensional Normed Spaces

Packing hyperspheres in high-dimensional Euclidean spaces.

We present a study of disordered jammed hard-sphere packings in four-, five-, and six-dimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the maximally random jammed (MRJ) states for space dimensions d=4, 5, and 6 to be phi(MRJ) approximately 0.46, 0.31, and 0.20, respectively. To a good approximation, t...