The 1-Steiner tree problem, the problem of constructing a Steiner minimum tree containing at most one Steiner point, has been solved in the Euclidean plane by Georgakopoulos and Papadimitriou using plane subdivisions called oriented Dirichlet cell partitions. Their algorithm produces an optimal solution within O(n 2 ) time. In this paper we generalise their approach in order to solve the k-Steiner tree problem, in which the Steiner minimum tree may contain up to k Steiner points for a given constant k. We also extend their approach further to encompass other normed planes, and to solve a much wider class of problems, including the k-bottleneck Steiner tree problem and other generalised k-Steiner tree problems. We show that, for any fixed k, such problems can be solved in O(n 2k ) time.
A disordered and heterogeneous, quasi-brittle granular material can withstand certain levels of internal damage before global failure. This robustness depends not just on the bond strengths but also on the topology and redundancy of the bonded contact network, through which forces and damage propagate. Despite extensive studies on quasi-brittle failure, there still lacks a unified framework that can quantitatively characterize and model the interdependent evolution of damage and force transmission. Here we develop a framework to do so. It is data-driven, multiscale and relies solely on the contact strengths and topology of the contact network for material properties. The discrete element method (DEM) was used to directly simulate quasi-brittle materials like concrete under uniaxial tension. Concrete was modeled as a random heterogeneous 2-phase and 3-phase material composed of aggregate particles, cement matrix and interfacial transitional zones with experimental-based meso-structure from X-ray micro-CT-images of real concrete. We uncover evidence of an optimized force transmission, characterized by two novel transmission patterns that predict and explain the coupled evolution of force and damage pathways from the microstructural to the macroscopic level. The first comprises the shortest possible percolating paths that can transmit the global force transmission capacity. These paths reliably predict tensile force chains. The second pattern is the flow bottleneck, a path in the optimized route that is prone to congestion and is where the macrocrack emerges. The cooperative evolution of preferential pathways for damage and force casts light on why sites of highest concentrations of stress and damage in the nascent stages of pre-failure regime do not provide a reliable indicator of the ultimate location of the macrocrack.
Heterogeneous quasibrittle composites like concrete, ceramics and rocks comprise grains held together by bonds. The question on whether or not the path of the crack that leads to failure can be predicted from known microstructural features, viz. bond connectivity, size, fracture surface energy and strength, remains open. Many fracture criteria exist. The most widely used are based on a postulated stress and/or energy extremal. Since force and energy share common transmission paths, their flow bottleneck may be the precursory failure mechanism to reconcile these optimality criteria in one unified framework. We explore this in the framework of network flow theory, using microstructural data from 3D discrete element models of concrete under uniaxial tension. We find the force and energy bottlenecks emerge in the same path and provide an early and accurate prediction of the ultimate macrocrack path $${\mathcal {C}}$$ C . Relative to all feasible crack paths, the Griffith’s fracture surface energy and the Francfort–Marigo energy functional are minimum in $${\mathcal {C}}$$ C ; likewise for the critical strain energy density if bonds are uniformly sized. Redundancies in transmission paths govern prefailure dynamics, and predispose $${\mathcal {C}}$$ C to cascading failure during which the concomitant energy release rate and normal (Rankine) stress become maximum along $${\mathcal {C}}$$ C .
Given a graph G with edge lengths, the minimum bottleneck spanning tree (MBST) problem is to find a spanning tree where the length of the longest edge in tree is minimum. It is a well-known fact that every minimum spanning tree (MST) is a minimum bottleneck spanning tree. In this article, we introduce the δ-MBST problem, which is the problem of finding an MBST such that every vertex in the tree has degree at most δ. We show that optimal solutions to the similarly defined δ-MST problem are not necessarily optimal solutions to the δ-MBST, and we establish that the δ-MBST problem is NP-complete for any δ ≥ 2. We show that when edge lengths of the graph are Euclidean distances between points in the plane, the problem is NP-hard for δ = 2 and 3, and tractable for δ ≥ 5. We give a dual approximation scheme for the general graph version of the problem which is the best possible with respect to feasibility. For the Euclidean version, we give a √ 3-factor approximation algorithm for the 4-MBST. We also give a 2-factor algorithm for the Euclidean 3-MBST and a 3-factor approximation algorithm for the general Euclidean δ-MBST, both of which can be generalized to metric spaces.
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