Convex Partition Research Articles

Abstract A decade ago two groups of authors, Karasev, Hubard and Aronov and Blagojević and Ziegler, have shown that the regular convex partitions of a Euclidean space into n parts yield a solution to the generalised Nandakumar and Ramana-Rao conjecture when n is a prime power. This was obtained by parametrising the space of regular equipartitions of a given convex body with the classical configuration space. Now, we repeat the process of regular convex equipartitions many times, first partitioning the Euclidean space into n 1 parts, then each part into n 2 parts, and so on. In this way we obtain iterated convex equipartions of a given convex body into n = n 1 ⋯ n k parts. Such iterated partitions are parametrised by the (wreath) product of classical configuration spaces. We develop a new configuration space–test map scheme for solving the generalised Nandakumar and Ramana-Rao conjecture using the Hausdorff metric on the space of iterated convex equipartions. The new scheme yields a solution to the conjecture if and only if all the n i ’s are powers of the same prime. In particular, for the failure of the scheme outside prime power case we give three different proofs.

This paper addresses the deployment of a multi-functional radar network (MFRN) in complex regions that may exhibit non-connectivity, holes, or concave shapes, utilizing multi-objective particle swarm optimization (MOPSO). Unlike traditional approaches that rely on constraint-handling techniques, the proposed methodology leverages the unique characteristics of polygonal deployment regions to enhance deployment efficiency. Specifically, for the aforementioned complex deployment regions, a region decomposition approach based on convex partitioning is proposed. This approach allows for the decomposition of complex regions into multiple non-overlapping convex subregions. Moreover, for convex deployment regions or subregions, we propose a coordinate transformation approach to eliminate the constraints introduced by the shape of the convex region. By combining the above approaches, we introduce a novel MOPSO based on decomposition and transformation, named MOPSO-DT. This algorithm aims to optimize MFRN deployment in these challenging environments. Experimental results demonstrate the superiority of the MOPSO-DT algorithm over two existing algorithms across a variety of deployment cases, highlighting its enhanced efficiency, effectiveness, and stability. These findings indicate that the proposed algorithm is well suited for optimizing MFRN deployment in complex, irregular regions, offering significant improvements in performance compared to conventional methods.

How to create an efficient and accurate interactive tool for triangular mesh clipping is one of the key problems to be solved in computer-assisted surgical planning. Although the existing algorithms can realize three-dimensional model clipping, problems still remain unsolved regarding the flexibility of clipping paths and the capping of clipped cross-sections. In this study, we propose a mesh clipping algorithm for surgical planning based on polygonal convex partitioning. First, two-dimensional polygonal regions are extended to three-dimensional clipping paths generated from selected reference points. Second, the convex regions are partitioned with a recursive algorithm to obtain the clipped and residual models with closed surfaces. Finally, surgical planning software with the function of mesh clipping has been developed, which is capable to create complex clipping paths by normal vector adjustment and thickness control. The robustness and efficiency of our algorithm have been demonstrated by surgical planning of craniomaxillofacial osteotomy, pelvis tumor resection and cranial vault remodeling.

Many feedforward neural networks (NNs) generate continuous and piecewise-linear (CPWL) mappings. Specifically, they partition the input domain into regions on which the mapping is affine. The number of these so-called linear regions offers a natural metric to characterize the expressiveness of CPWL NNs. The precise determination of this quantity is often out of reach in practice, and bounds have been proposed for specific architectures, including for ReLU and Maxout NNs. In this work, we generalize these bounds to NNs with arbitrary and possibly multivariate CPWL activation functions. We first provide upper and lower bounds on the maximal number of linear regions of a CPWL NN given its depth, width, and the number of linear regions of its activation functions. Our results rely on the combinatorial structure of convex partitions and confirm the distinctive role of depth which, on its own, is able to exponentially increase the number of regions. We then introduce a complementary stochastic framework to estimate the average number of linear regions produced by a CPWL NN. Under reasonable assumptions, the expected density of linear regions along any 1D path is bounded by the product of depth, width, and a measure of activation complexity (up to a scaling factor). This yields an identical role to the three sources of expressiveness: no exponential growth with depth is observed anymore.

The enormous volumes of geospatial data and the need to process and distribute them cry out for a unified framework that enables their efficient storage, analysis, and a high degree of interoperability. Discrete global grid systems provide such a framework by hierarchically tessellating cells to seamlessly partition and address the globe. Since they are usually based on a regular polyhedron, they partition the entire world into as many discrete data sets as the given polyhedron has sides. In this paper, we try to reduce the number of partitions to two, which is a minimum if we want to obtain spatially convex partitions without interruptions. Two approaches are presented, based on an adjusted spherical cube and an equidistant cylindrical projection. The distortions resulting from the application of these projections are compared and guidelines are presented to improve the quality of their implementation by reducing the distortion of the continental plates and making a better mapping to the WGS84 ellipsoid.

This paper considers trajectory planning for a mobile robot which persistently relays data between pairs of far-away communication nodes. Data accumulates stochastically at each source, and the robot must move to appropriate positions to enable data offload to the corresponding destination. The robot needs to minimize the average time that data waits at a source before being serviced. We are interested in finding optimal robotic routing policies consisting of 1) locations where the robot stops to relay (relay positions) and 2) conditional transition probabilities that determine the sequence in which the pairs are serviced. We first pose this problem as a non-convex problem that optimizes over both relay positions and transition probabilities. To find approximate solutions, we propose a novel algorithm which alternately optimizes relay positions and transition probabilities. For the former, we find efficient convex partitions of the non-convex relay regions, then formulate a mixed-integer second-order cone problem. For the latter, we find optimal transition probabilities via sequential least squares programming. We extensively analyze the proposed approach and mathematically characterize important system properties related to the robot’s long-term energy consumption and service rate. Finally, through extensive simulation with real channel parameters, we verify the efficacy of our approach.

Extraction and application of super-smooth cubic B-splines over triangulations

Computing Balanced Convex Partitions of Lines

This work describes the contribution of Artigas, Dantas, Dourado, and Szwarcfiter on the topic of convexity in the context of geodesic convex partitions, covers and geodetic sets from 2008 to 2013.

An infinite linear order with finitely many unary relations (colors), $\langle X,{ \lt }, U_0,\dots ,U_{n-1} \rangle $, is a <em>good colored linear order</em> iff the largest convex partition of the set $X$ refining the partition generated by the sets $U

A new species of green paleosiphonocladаl algae Kamaena gigantea from the Lower–lowermost Middle Mississippian sediments of the Donbas has been described. This species was distinguished from other representatives of the genus Kamaena Antropov by its extremely large size, tortuous shape of the thallus and convex partitions. The species belongs to an artificial taxonomic unit of the Kamaenae Shuysky tribe, 1985 , of the family Palaeoberesellaceae Mamet et Roux , a systematic grouping which is still controversial. The attribution of this family to green siphonocladal algae is controversial and quite conditional, the opinions of different authors being based on personal vision, and varying in range from the plant to the animal kingdom. A characteristic feature of the family is the tubular shape , the segments of which are connected by partitions with a large central pore, sometimes with additional small pores. The thallus wall (fossilized remains of the body) is porous or non-porous and has simple or branched pores. It has been emphasized that study of Paleoberezellides in thin sections, the sometimes fuzzy images of the typical material in publications and ignorance of other researchers’ publications have caused confusion and led to the selection of an unreasonably large number of genera and species within the family. It has been noted that in previous works, representatives of this species were mistakenly identified as Anthracoporellopsis Maslov, a genus characteristic of the Lower–Middle Pennsylvanian sediments. This erroneous definition was based mainly on general external similarity, a poorly illustrated description of the type species, and did not take into account the morphological features that were characteristic for the genus. It has been found that representatives of the new species had a rather limited stratigraphic distribution: the Upper Tournaisian (Dokuchaevskian horizon) and the Lower Visean (Hlybokian–Sukhinskian horizons), and the most similar specimens found in the Ural region in underlying Tournaisian sediments were, unfortunately, poorly illustrated and smaller in size and had a narrower thallus. It has been noted that a characteristic feature of the tribe Kamaenae Shuysky, 1985 was the tubular shape, its inter-segmental partitions were perpendicular to the walls and were at the approximately same interval from each other. It has been pointed out that representatives of the new species were found mainly in grainstones, packstones, and wackstones − organogenic-detrital limestones along with such groups of microfauna as echinoderms and ostracods, isolated spicules of sponges and remains of worms. The material for illustrations was mainly taken from the borehole74 (near the village of Rodnikove, Starobeshiv district, Donetsk region) , which most fully revealed the Lower–Middle Pennsylvanian deposits of the southern part of the Donbas. The knowledge of the systematic composition of the Early Carboniferous algoflora has been expanded. This has helped us to conclude that the tribe Kamaeneae Shuysky, 1985 ,includes 6 genera and at least 22 species that were found in the layers from the Early Devonian to the Early Visean of the Lower Carboniferous.

In this paper, we construct a novel normalized B-spline-like representation for C2-continuous cubic spline space defined on an initial partition refined by inserting two new points inside each sub-interval. The basis functions are compactly supported non-negative functions that are geometrically constructed and form a convex partition of unity. With the help of the control polynomial theory introduced herein, a Marsden identity is derived, from which several families of super-convergent quasi-interpolation operators are defined.

On a partition with a lower expected [formula omitted]-discrepancy than classical jittered sampling

Control architecture of autonomous underwater vehicle for coverage mission in irregular region

Solving the minimum convex partition of point sets with integer programming

Rays are classes of an equivalence relation on a module V over a supertropical semiring. They provide a version of convex geometry, supported by a ‘supertropical trigonometry’ and compatible with quasilinearity, in which the CS-ratio takes the role of the Cauchy–Schwarz inequality. CS-functions that emerge from the CS-ratio are a useful tool that helps to understand the variety of quasilinear stars in the ray space . In particular, these functions induce a partition of into convex sets, and thereby a finer convex analysis which includes the notions of median, minima, glens, and polars.

Suppose that an m -simplex is partitioned into n convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance ε from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant m uses poly( n , log (1/ε)) queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant n uses poly( m , log (1/ε)) queries. We show via Kakutani’s fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.

In this paper, we describe a general class of C1 smooth rational splines that enables, in particular, exact descriptions of ellipses and ellipsoids — some of the most important primitives for CAD and CAE. The univariate rational splines are assembled by transforming multiple sets of NURBS basis functions via so-called design-through-analysis compatible extraction matrices; different sets of NURBS are allowed to have different polynomial degrees and weight functions. Tensor products of the univariate splines yield multivariate splines. In the bivariate setting, we describe how similar design-through-analysis compatible transformations of the tensor-product splines enable the construction of smooth surfaces containing one or two polar singularities. The material is self-contained, and is presented such that all tools can be easily implemented by CAD or CAE practitioners within existing software that support NURBS. To this end, we explicitly present the matrices (a) that describe our splines in terms of NURBS, and (b) that help refine the splines by performing (local) degree elevation and knot insertion. Finally, all C1 spline constructions yield spline basis functions that are locally supported and form a convex partition of unity.

We prove an extension of a ham sandwich theorem for families of lines in the plane by Dujmović and Langerman. Given two sets A, B of n lines each in the plane, we prove that it is possible to partition the plane into r closed convex regions so that the following holds. For each region C of the partition there is a subset of \(c_r n^{1/r}\) lines of A whose pairwise intersections are in C, and the same holds for B. In this statement \(c_r\) only depends on r. We also prove that the dependence on n is optimal. For a single set A of n lines, we prove that there exists a partition of the plane into r parts using \(r-1\) vertical lines such that each region contains the pairwise intersections of a set of \(n^{1/r}\) lines of A. The value \(n^{1/r}\) is optimal.

The irregular forbidden zone (FZ) is a common phenomenon with giant hydropower plants developed in the past two decades in China. Irregular shapes of FZs significantly challenge hydro unit commitment (HUC). This paper proposes a novel MILP based framework for HUC considering irregular FZ related constraints including the FZ constraint, effects of linearization errors in both the net head and the output, and the FZ crossing constraint. In the framework, the FZ constraint is handled by the optimal convex partitioning algorithm and the common structure-based formulation method. Inspired by the planar translating robot placement problem, linearization errors are considered by the Minkowski sum method. To handle the FZ crossing constraint, we then propose a graph theory-based approximate formulation method. The framework is integrated into a HUC model with an objective of peak shaving. The model is then tested with a batch of real-world instances of a cascade hydropower system formed by ten giant units with highly irregular FZs. The results show our framework can effectively consider the irregular FZ related constraints. The major advantage of our framework is its ability to handle the irregular shapes of FZs without any tedious and error-prone manual processing.