Search

In this paper, we consider enumeration of geodesics on a polyhedron, where a geodesic means locally-shortest path between two points. Particularly, we consider the following preprocessing problem: given a point s on a polyhedral surface and a positive real number r , to build a data structure that enables, for any point t on the surface, to enumerate all geodesics from s to t whose length is less than r. First, we present a naive algorithm by removing the trimming process from the MMP algorithm (1987). Next, we present an improved algorithm which is practically more efficient on a non-convex polyhedron, in terms of preprocessing time and memory consumption. Moreover, we introduce a single-pair geodesic graph to succinctly encode a result of geodesic query. Lastly, we compare these naive and improved algorithms by some computer experiments. [ABSTRACT FROM AUTHOR]

Various angle characteristics are used (e.g. in finite element methods or computer graphics) when evaluating the quality of computational meshes which may consist, in the three-dimensional case, of tetrahedra, prisms, hexahedra and pyramids. Thus, it is of interest to derive (preferably tight) bounds for dihedral angle sums, i.e. sums of angles between faces, of such mesh elements. For tetrahedra this task was solved by Gaddum in 1952. For pyramids, this was resolved by Korotov, Lund and Vatne in 2022. In this paper, we compute tight bounds for the remaining two cases, hexahedra and prisms. [ABSTRACT FROM AUTHOR]

Chvátal and Klincsek (1980) gave an O (n 3) -time algorithm for the problem of finding a maximum-cardinality convex subset of an arbitrary given set P of n points in the plane. This paper examines a generalization of the problem, the Bottleneck Convex Subsets problem: given a set P of n points in the plane and a positive integer k , select k pairwise disjoint convex subsets of P such that the cardinality of the smallest subset is maximized. Equivalently, a solution maximizes the cardinality of k mutually disjoint convex subsets of P of equal cardinality. We give an algorithm that solves the problem exactly, with running time polynomial in n when k is fixed. We then show the problem to be NP-hard when k is an arbitrary input parameter, even for points in general position. Finally, we give a fixed-parameter tractable algorithm parameterized in terms of the number of points strictly interior to the convex hull. [ABSTRACT FROM AUTHOR]

Orthogonal range search is ubiquitous nowadays, with natural applications in databases, data mining, and text indexing. Very recently, yet another application was discovered, which is to maintain a DFS forest in a dynamic graph. In this paper, we want to extend the above recent study, by applying orthogonal range search to efficient maintenance of a BFS-like forest, called no-back-edge-traversal (NBET) forest, which refers to a spanning forest obtained from a traversal that does not create any back edge. The study of such a problem is motivated by the fact that NBET forest can be used as a strong certificate of 2-connectivity of an undirected graph, which is more general than a spanning forest obtained from a scan-first search traversal. [ABSTRACT FROM AUTHOR]

In 1985 Hopcroft, Joseph and Whitesides showed it is NP-complete to decide whether a carpenter's ruler with segments of given positive lengths can be folded into an interval of at most a given length, such that the folded hinges alternate between 180 degrees clockwise and 180 degrees counter-clockwise. At the open-problem session of 33rd Canadian Conference on Computational Geometry (CCCG '21), O'Rourke proposed a natural variation of this problem called ruler wrapping, in which all folded hinges must be folded the same way. In this paper we show O'Rourke's variation has a linear-time solution. [ABSTRACT FROM AUTHOR]

Given a set C of points and a horizontal line L in the plane and a set F of points on L , we want to find a set of disks such that (1) each disk has the center at a point in F (but with arbitrary radius), (2) each point in C is covered by at least one disk, and (3) the cost of the set of disks is minimized. Here the (transmission) cost of a disk with radius r is r α , where α is a positive constant depending on the power consumption model, and the cost of a set of disk is the sum of the cost of disks in the set. In this paper we first give an algorithm based on dynamic programming method to solve the problem in L 1 metric. A naive dynamic programming algorithm runs in O (| C | 3 | F | 2) time. We design an algorithm which runs in O (| C | | F | 2 + | C | log | C |) time. Then we design another algorithm to solve the problem in L 1 metric based on a reduction to a shortest path problem in a directed acyclic graph. The running time of the algorithm is O (| C | 2 + | C | | F |). [ABSTRACT FROM AUTHOR]

Robust optimization problems have attracted considerable attention in recent years. In this paper, we focus on two fundamental robust optimization problems: SVM with outliers and k -center clustering with outliers. The key obstacle is that the outliers can be located arbitrarily in the space (e.g., by an attacker), and thus they are actually quite challenging combinatorial optimization problems. Their computational complexities can be very high especially in high dimensional spaces. The Johnson–Lindenstrauss (JL ) Transform is a popular random projection method for dimension reduction. Though the JL transform has been widely studied in the past decades, its effectiveness for dealing with high-dimensional optimizations with outliers has never been investigated before (to the best of our knowledge). Based on some novel insights from the geometry, we prove that the complexities of these two problems can be significantly reduced through the JL transform. Moreover, we prove that the solution in the dimensionality-reduced space can be efficiently recovered in the original ℝ d space while the quality is still preserved. To study its performance in practice, we compare different JL transform methods with several other well known dimension reduction methods in our experiments. [ABSTRACT FROM AUTHOR]

We consider a random variable expressed as the Euclidean distance between an arbitrary point and a random variable uniformly distributed in a closed and bounded set of a three-dimensional Euclidean space. Four cases are considered for this set: a union of disjoint disks, a union of disjoint balls, a union of disjoint line segments, and the boundary of a polyhedron. In the first three cases, we provide closed-form expressions of the cumulative distribution function and the density. In the last case, we propose two algorithms with complexity O (n ln n) , n being the number of edges of the polyhedron, that computes exactly the cumulative distribution function. An application of these results to probabilistic seismic hazard analysis and extensions are discussed. Finally, we present an open source library, available at https://github.com/vguigues/Areas%5fLibrary , that implements the algorithms presented in this paper. [ABSTRACT FROM AUTHOR]

In the present paper, we propose a novel generalization of the celebrated MMP algorithm in order to find the wavefront propagation and the cut-locus on a convex polyhedron with an emphasis on actual implementation for instantaneous visualization and numerical computation. [ABSTRACT FROM AUTHOR]