Finding Top-k Shortest Paths with Diversity Liu, Huiping; Jin, Cheqing; Yang, Bin ...
IEEE transactions on knowledge and data engineering,
03/2018, Volume:
30, Issue:
3
Journal Article
Peer reviewed
The classical K Shortest Paths (KSP) problem, which identifies the k shortest paths in a directed graph, plays an important role in many application domains, such as providing alternative paths for ...vehicle routing services. However, the returned k shortest paths may be highly similar, i.e., sharing significant amounts of edges, thus adversely affecting service qualities. In this paper, we formalize the K Shortest Paths with Diversity (KSPD) problem that identifies top-k shortest paths such that the paths are dissimilarwith each other and the total length of the paths is minimized. We first prove that the KSPD problem is NP-hard and then propose a generic greedy framework to solve the KSPD problem in the sense that (1) it supports a wide variety of path similarity metrics which are widely adopted in the literature and (2) it is also able to efficiently solve the traditional KSP problem if no path similarity metric is specified. The core of the framework includes the use of two judiciously designed lower bounds, where one is dependent on and the other one is independent on the chosen path similarity metric, which effectively reduces the search space and significantly improves efficiency. Empirical studies on five real-world and synthetic graphs and five different path similarity metrics offer insight into the design properties of the proposed general framework and offer evidence that the proposed lower bounds are effective.
A Dyck path of semilength n is a path from (0,0) to (2n,0), such that the only allowed steps are U=(1,1) and D=(1,−1). The constraint on a Dyck path is that it never falls below the x-axis. In this ...paper, we give the concept of (a1,b1;a2,b2)-Dyck paths, which generates the recursive matrix corresponding to π=(a1,a2,a1,a2,⋯), σ=(0,0,0,0,⋯) and τ=(b1,b2,b1,b2,⋯). Some new combinatorial identities related to 2×2 minors and 2×2 permanents of the Dπ,σ,τ are investigated. When the weighted parameters (a1,b1;a2,b2) are specialized, several interesting identities are obtained about Catalan numbers and other combinatorial sequences. Based on Motzkin paths, alternating sums of 2×2 minors of the odd Dyck matrix are studied.
Power system restoration after a major blackout is a complex process, in which selection of energising paths is a key issue to realise unit and load restoration safely and efficiently. In general, ...the energising path scheme made beforehand may not be executed successfully due to the possible faults on the related lines under the extreme system condition, so it is necessary to provide alternative path schemes for system restoration. In view of this, the energising path optimisation based on the minimum cost flow model is investigated, then an iterative searching method for alternative path schemes based on mixed integer linear programming is proposed. The iterative method for alternative path schemes could determine more than one scheme with minimal charging reactive power efficiently. In order to make a comprehensive evaluation of the alternative schemes, an evaluation index set is established, and the method based on similarity to ideal grey relational projection is introduced to achieve the final evaluation. The New England 10-unit 39-bus system and the southern Hebei power system of China are employed to demonstrate the effectiveness of the proposed method. The proposed method can provide more efficient and comprehensive decision support for the dispatchers to select reasonable energising paths.
As a challenging high dimension optimization problem, three-dimensional path planning for Uninhabited Combat Air Vehicles (UCAV) mainly centralizes on optimizing the flight route with different types ...of constrains under complicated combating environments. An improved version of Bat Algorithm (BA) in combination with a Differential Evolution (DE), namely IBA, is proposed to optimize the UCAV three-dimensional path planning problem for the first time. In IBA, DE is required to select the most suitable individual in the bat population. By connecting the selected nodes using the proposed IBA, a safe path is successfully obtained. In addition, B-Spline curves are employed to smoothen the path obtained further and make it practically more feasible for UCAV. The performance of IBA is compared to that of the basic BA on a 3-D UCAV path planning problem. The experimental results demonstrate that IBA is a better technique for UCAV three-dimensional path planning problems compared to the basic BA model.
Recently Mansour and Shattuck studied (k,a)-paths and gave formulas that related the total number of humps in all (k,a)-paths to the number of super (k,a)-paths. These results generalized earlier ...results of Regev on Dyck paths and Motzkin paths. Their proofs are based on generating functions and they asked for bijective proofs for their results. In this paper we first give bijective proofs of Mansour and Shattuck’s results, then we extend our study to (n,m)-Dyck paths. We give a bijection that relates the total number of peaks in all (n,m)-Dyck paths to certain free (n,m)-paths when n and m are coprime. From this bijection we get the number of (n,m)-Dyck paths with exactly j peaks, which is a generalization of the well-known result that the number Dyck paths of order n with exactly j peaks is the Narayana number 1kn−1k−1nk−1.
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► Ultrahigh-temperature (UHT) granulites within the Inner Mongolia Suture Zone in the North Chin Craton (NCC) record robust evidence for extreme crustal metamorphism in a ...subduction-collision setting. ► The Paleoproterozoic UHT event broadly coincided with the timing of scissor-like closure of oblique collision during 1.95–1.92Ga between the Yinshan and the Ordos Blocks. ► Plate tectonic models relating asthenospheric upwelling and input of heat and volatiles best account for the formation of the UHT rocks in the NCC.
Ultrahigh-temperature (UHT) metamorphic rocks associated with the ‘Khondalite Belt’ within the Inner Mongolia Suture Zone (IMSZ) provide robust evidence for extreme thermal metamorphism in the North China Craton (NCC). The IMSZ marks the collisional suture between the Yinshan Block to the north and the Ordos Block to the South as the NCC was incorporated within the Columbia supercontinent amalgam during Paleoproterozoic. Here we present a synthesis of the salient features of the UHT rocks from the NCC including petrologic indicators, fluid characteristics, and monazite and zircon chronometry on the extreme crustal metamorphism. The granulites carry diagnostic UHT mineral assemblages including sapphirine+quartz, low Zn/Fe3+ spinel+quartz, high alumina orthopyroxene+sillimanite+quartz and high temperature mesoperthite. The stability fields of the typical mineral assemblages, conventional geothermobarometry and phase equilibria modeling using pseudosections as reported in a number of recent studies converge to indicate that these UHT rocks experienced metamorphic temperatures up to or in excess of 1000°C at ca. 10kbar, followed by an isobaric cooling segment. The rocks were exhumed along a near-isothermal decompression path. Microstructures, mineral reactions and phase equilibria modeling suggest an anti-clockwise P–T path, similar to those displayed by metamorphic orogens developed in subduction-collision settings. The dominant category of fossil fluids preserved within the major UHT minerals is CO2, consistent with the stability of the broadly anhydrous mineral assemblage in these rocks. Both chemical and radiogenic isotopic ages from monazite and zircon chronometry suggest the timing of the UHT event as around 1.92Ga. The Paleoproterozoic high grade metamorphism younging from 1.95Ga in the western domain to 1.92Ga in the eastern domain of the Khondalite Belt might suggest a scissor-like closure of oblique collision between the Yinshan and the Ordos Blocks.
The salient features of the UHT metamorphism in the NCC include: (1) extreme metamorphic temperatures at moderate pressures, (2) dominantly anhydrous nature of the mineral assemblages, typically the stability of orthopyroxene, (3) common presence of CO2-rich fluid inclusions as the trace of the ambient fluid, (4) regional extent of the UHT granulites, and (5) the association of the UHT orogen with an accretionary belt in a continental collisional suture. We evaluate the diverse models on the generation of UHT orogens including their formation in thickened and inverted back-arcs, orogen self-heating through heat producing elements, heat and CO2 input by plume impingement below a carbonated tectosphere, and asthenospheric upwelling through ridge subduction and slab-window process or during post-collisional slab break-off. The ultra-hot and dry UHT rocks in the NCC provide one of the well preserved examples from the Paleoproterozoic globe for investigating extreme metamorphism and related tectonic processes within the plate tectonic paradigm.
This article addresses an important path planning problem for robots and Unmanned Aerial Vehicles (UAVs), which is to find the shortest path of bounded curvature passing through a given sequence of ...target points on a ground plane. Currently, no algorithm exists that can compute an optimal solution to this problem. Therefore, tight lower bounds are vital in determining the quality of any feasible solution to this problem. Novel tight lower bounding algorithms are presented in this article by relaxing some of the heading angle constraints at the target points. The proposed approach requires us to solve variants of an optimization problem called the Dubins interval problem between two points where the heading angles at the points are constrained to be within a specified interval. These variants are solved using tools from optimal control theory. Using these approaches, two lower bounding algorithms are presented and these bounds are then compared with existing results in the literature. Computational results are presented to corroborate the performance of the proposed algorithms; the average reduction in the difference between upper bounds and lower bounds is 80 % to 85 % with respect to the trivial Euclidean lower bounds.
This study addresses various formulations of the optimal reliability path problem on stochastic networks. Robust-Cost is adopted as the measure of reliability, which is defined as a weighted ...combination of the mean and standard deviation of travel time. The principal problem solved is the Minimum Robust-Cost Path (MRCP) problem in the presence of Correlated link travel times. It is shown that the subpath optimality and subpath non-dominance principles, which are conventionally adopted in the literature, cannot be used to solve this problem. In this light, this study proposes an algorithm based on the subpath pruning approach, which eliminates nonoptimal subpaths by using a pruning criterion. We construct the novel pruning criterion, which depends on only two independent objectives, by transforming the network and using efficient shortest path algorithms. The correctness of the algorithm is established, and its good practical performance is demonstrated on real-world networks. Furthermore, the pruning procedure is generalized with suitable modifications to solve three related problems: (i) the MRCP problem with independent link travel times, (ii) the
K
-best Robust-Cost Paths problem, and (iii) the MRCP problem with stochastic nondominance constraints. These extensions demonstrate the potential wider applications of the proposed solution approach.
The signature of a rough path: Uniqueness Boedihardjo, Horatio; Geng, Xi; Lyons, Terry ...
Advances in mathematics (New York. 1965),
04/2016, Volume:
293
Journal Article
Peer reviewed
Open access
In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and ...only if the path is tree-like. We extend Hambly–Lyons' result and their notion of tree-like paths to the setting of weakly geometric rough paths in a Banach space. At the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures.
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