Weighted minimum backward fréchet distance

The minimum backward Fréchet distance (MBFD) problem is a natural optimization problem for the weak Fréchet distance, a variant of the well-known Fréchet distance. In this problem, a threshold " and two polygonal curves, T1 and T2, are given. The objective is to find a pair of walks on T1 and T2, which minimizes the union of the portions of backward movements while the distance between the moving entities, at any time, is at most ". In this paper, we generalize this model to capture scenarios when the cost of backtracking on the input polygonal curves is not homogeneous. More specifically, each edge of T1 and T2 has an associated nonnegative weight. The cost of backtracking on an edge is the Euclidean length of backward movement on that edge multiplied by the corresponding weight. The objective is to find a pair of walks that minimizes the sum of the costs on the edges of the curves, while guaranteeing that the curves remain at weak Fréchet distance ϵ. We propose an exact algorithm whose run time and space complexity is O(n3), where n is the maximum number of the edges of T1 and T2

Weighted region problem in arrangement of lines

In this paper, a geometric shortest path problem in weighted regions is discussed. An arrangement of lines A, a source s, and a target t are given. The objective is to find a weighted shortest path, Πst, from s to t. Existing approximation algorithms for weighted shortest paths work within bounded regions (typically triangulated). To apply these algorithms to unbounded regions, such as arrangements of lines, there is a need to bound the regions. Here, we present a minimal region that contains Πst, called SP-Hull of A. It is a closed polygonal region that only depends on the geometry of the arrangement A and is independent of the weights. It is minimal in the sense that for any arrangement of lines A, it is possible to assign weights to the faces of A and choose s and t such that Πst is arbitrary close to the boundary of SP-Hull of A. We show that SP-Hull can be constructed in O(nlog n) time, where n is the number of lines in the arrangement. As a direct consequence we obtain a shortest path algorithm for weighted arrangements of lines

Weighted minimum backward Fréchet distance

The minimum backward Fréchet distance (MBFD) problem is a natural optimization problem for the weak Fréchet distance, a variant of the well-known Fréchet distance. In this problem, a threshold ε and two polygonal curves, T 1 and T 2 , are given. The objective is to find a pair of walks on T 1 and T 2 , which minimizes the union of the portions of backward movements (backtracking) while maintaining, at any time, a distance between the moving entities of at most ε. In this paper, we generalize this model to capture scenarios when the cost of backtracking on the input polygonal curves is not homogeneous. More specifically, each edge of T 1 and T 2 has an associated non-negative weight. The cost of backtracking on an edge is the Euclidean length of backward movement on that edge multiplied by the corresponding weight. The objective is to find a pair of walks that minimizes the sum of the costs on the edges of the curves, while guaranteeing that the weak traversal of the curves maintains a weak Fréchet distance of at most ε. We propose two exact algorithms, a simple algorithm with O(n 4 ) time and s

Minimum backward fréchet distance

We propose a new measure to capture similarity between polygonal curves, called the minimum backward Fréchet distance. It is a natural optimization on the weak Fréchet distance, a variant of the well-known Fréchet distance. More specifically, for a given threshold ε, we are searching for a pair of walks for two entities on the two input curves, T1 and T2, such that the union of the portions of backward movements is minimized and the distance between the two entities, at any time during the walk, is less than or equal to ". Our algorithm detects if no such pair of walks exists. This natural optimization problem appears in many applications in Geographical Information Systems, mobile networks and robotics. We provide an exact algorithm with time complexity of Ο (n2 log n) and space complexity of Ο (n2), where n is the maximum number of segments in the input polygonal curves

Path Refinement in Weighted Regions

In this paper, we study the weighted region problem (WRP) which is to compute a shortest path in a weighted partitioning of a plane. Recent results show that WRP is not solvable in any algebraic computation model over the rational numbers. Therefore, it is unlikely that WRP can be solved in polynomial time. Research has thus focused on determining approximate solutions for WRP. Approximate solutions for WRP typically show qualitatively different behaviors. We first formulate two qualitative criteria for weighted shortest paths. Then, we show how to produce a path that is quantitatively close-to-optimal and qualitatively satisfactory. More precisely, we propose an algorithm to transform any given approximate linear path into a linear path with the same (or shorter) weighted length for which we can prove that it satisfies the required qualitative criteria. This algorithm has a linear time complexity in the size of the given path. At the end, we explain our experiments on some triangular irregular networks (TINs) from Earth’s terrain. The results show that using the proposed algorithm, on average, 51% in query time and 69% in memory usage could be saved, in comparison with the existing method

Seismic Evaluation of a Tailings Dam Using Uncoupled and Fully Coupled Soil Constitutive Models

The Henderson Tailings Storage Facility (TSF) is an active facility located near Parshall, Colorado, consisting of two dams – 1 Dam and 3 Dam. The upstream method of construction has been used for tailings deposition since the mid-1970s. This paper presents the results of the seismic evaluation of 1 Dam using multiple constitutive modeling methodologies: an uncoupled Mohr-Coulomb approach and a fully coupled critical state-compatible PM4Sand/Silt approach. The analyses were completed to study the seismic response of the tailings embankment under the Maximum Design Earthquake (MDE) with a return period of 10,000 years. The numerical analyses showed that the results of the fully coupled, effective stress PM4 models were generally consistent with the uncoupled Mohr-Coulomb models. The PM4 models have the capability to estimate the generation of excess pore water pressure and onset of soil liquefaction during the application of the input ground motion. Subsequent zones of tailings materials that were prone to soil liquefaction or strength loss were identified based on two criteria: 1) excess pore water pressure ratio; and 2) shear strain in the PM4 modeling. This study provides valuable insights into the methods used to estimate seismic response of the Henderson TSF and highlights the importance of using properly calibrated advanced constitutive modeling methodologies to capture the complex response of tailings materials under seismic loading. The fully coupled models were capable of capturing the hysteretic soil response, including stress-strain response and accumulation of plastic shear strains, providing confidence in the predicted modes of deformation and informing the design and management of the facility.Non UBCUnreviewedOthe

Similarity of polygonal curves in the presence of outliers

The Fréchet distance is a well studied and commonly used measure to capture the similarity of polygonal curves. Unfortunately, it exhibits a high sensitivity to the presence of outliers. Since the presence of outliers is a frequently occurring phenomenon in practice, a robust variant of Fréchet distance is required which absorbs outliers. We study such a variant here. In this modified variant, our objective is to minimize the length of subcurves of two polygonal curves that need to be ignored (MinEx problem), or alternately, maximize the length of subcurves that are preserved (MaxIn problem), to achieve a given Fréchet distance. An exact solution to one problem would imply an exact solution t