26,388 research outputs found

    The energy-constrained quickest path problem

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    This paper addresses a variant of the quickest path problem in which each arc has an additional parameter associated to it representing the energy consumed during the transmission along the arc while each node is endowed with a limited power to transmit messages. The aim of the energy-constrained quickest path problem is to obtain a quickest path whose nodes are able to support the transmission of a message of a known size. After introducing the problem and proving the main theoretical results, a polynomial algorithm is proposed to solve the problem based on computing shortest paths in a sequence of subnetworks of the original network. In the second part of the paper, the bi-objective variant of this problem is considered in which the objectives are the transmission time and the total energy used. An exact algorithm is proposed to find a complete set of efficient paths. The computational experiments carried out show the performance of both algorithms

    Quickest Paths: Faster Algorithms and Dynamization

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    Given a network N=(V,E,c,l)N=(V,E,{c},{l}), where G=(V,E)G=(V,E), V=n|V|=n and E=m|E|=m, is a directed graph, c(e)3˘e0{c}(e) \u3e 0 is the capacity and l(e)0{l}(e) \ge 0 is the lead time (or delay) for each edge eEe\in E, the quickest path problem is to find a path for a given source--destination pair such that the total lead time plus the inverse of the minimum edge capacity of the path is minimal. The problem has applications to fast data transmissions in communication networks. The best previous algorithm for the single--pair quickest path problem runs in time O(rm+rnlogn)O(r m+r n \log n), where rr is the number of distinct capacities of NN \cite{ROS}. In this paper, we present algorithms for general, sparse and planar networks that have significantly lower running times. For general networks, we show that the time complexity can be reduced to O(rm+rnlogn)O(r^{\ast} m+r^{\ast} n \log n), where rr^{\ast} is at most the number of capacities greater than the capacity of the shortest (with respect to lead time) path in NN. For sparse networks, we present an algorithm with time complexity O(nlogn+rn+rγ~logγ~)O(n \log n + r^{\ast} n + r^{\ast} \tilde{\gamma} \log \tilde{\gamma}), where γ~\tilde{\gamma} is a topological measure of NN. Since for sparse networks γ~\tilde{\gamma} ranges from 11 up to Θ(n)\Theta(n), this constitutes an improvement over the previously known bound of O(rnlogn)O(r n \log n) in all cases that γ~=o(n)\tilde{\gamma}=o(n). For planar networks, the complexity becomes O(nlogn+nlog3γ~+rγ~)O(n \log n + n\log^3 \tilde{\gamma}+ r^{\ast} \tilde{\gamma}). Similar improvements are obtained for the all--pairs quickest path problem. We also give the first algorithm for solving the dynamic quickest path problem

    Dealing with residual energy when transmitting data in energy-constrained capacitated networks

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    This paper addresses several problems relating to the energy available after the transmission of a given amount of data in a capacitated network. The arcs have an associated parameter representing the energy consumed during the transmission along the arc and the nodes have limited power to transmit data. In the first part of the paper, we consider the problem of designing a path which maximizes the minimum of the residual energy remaining at the nodes. After formulating the problem and proving the main theoretical results, a polynomial time algorithm is proposed based on computing maxmin paths in a sequence of non-capacitated networks. In the second part of the paper, the problem of obtaining a quickest path in this context is analyzed. First, the bi-objective variant of this problem is considered in which we aim to minimize the transmission time and to maximize the minimum residual energy. An exact polynomial time algorithm is proposed to find a minimal complete set of efficient solutions which amounts to solving shortest path problems. Second, the problem of computing an energy-constrained quickest path which guarantees at least a given residual energy at the nodes is reformulated as a variant of the energy-constrained quickest path problem. The algorithms are tested on a set of benchmark problems providing the optimal solution or the Pareto front within reasonable computing times

    Optimizing Emergency Transportation through Multicommodity Quickest Paths

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    In transportation networks with limited capacities and travel times on the arcs, a class of problems attracting a growing scientific interest is represented by the optimal routing and scheduling of given amounts of flow to be transshipped from the origin points to the specific destinations in minimum time. Such problems are of particular concern to emergency transportation where evacuation plans seek to minimize the time evacuees need to clear the affected area and reach the safe zones. Flows over time approaches are among the most suitable mathematical tools to provide a modelling representation of these problems from a macroscopic point of view. Among them, the Quickest Path Problem (QPP), requires an origin-destination flow to be routed on a single path while taking into account inflow limits on the arcs and minimizing the makespan, namely, the time instant when the last unit of flow reaches its destination. In the context of emergency transport, the QPP represents a relevant modelling tool, since its solutions are based on unsplittable dynamic flows that can support the development of evacuation plans which are very easy to be correctly implemented, assigning one single evacuation path to a whole population. This way it is possible to prevent interferences, turbulence, and congestions that may affect the transportation process, worsening the overall clearing time. Nevertheless, the current state-of-the-art presents a lack of studies on multicommodity generalizations of the QPP, where network flows refer to various populations, possibly with different origins and destinations. In this paper we provide a contribution to fill this gap, by considering the Multicommodity Quickest Path Problem (MCQPP), where multiple commodities, each with its own origin, destination and demand, must be routed on a capacitated network with travel times on the arcs, while minimizing the overall makespan and allowing the flow associated to each commodity to be routed on a single path. For this optimization problem, we provide the first mathematical formulation in the scientific literature, based on mixed integer programming and encompassing specific features aimed at empowering the suitability of the arising solutions in real emergency transportation plans. A computational experience performed on a set of benchmark instances is then presented to provide a proof-of-concept for our original model and to evaluate the quality and suitability of the provided solutions together with the required computational effort. Most of the instances are solved at the optimum by a commercial MIP solver, fed with a lower bound deriving from the optimal makespan of a splittable-flow relaxation of the MCQPP

    Quickest paths: faster algorithms and dynamization

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    Given a network N=(V,E,c,l)N=(V,E,{c},{l}), where G=(V,E)G=(V,E), V=n|V|=n and E=m|E|=m, is a directed graph, c(e)>0{c}(e) > 0 is the capacity and l(e)0{l}(e) \ge 0 is the lead time (or delay) for each edge eEe\in E, the quickest path problem is to find a path for a given source--destination pair such that the total lead time plus the inverse of the minimum edge capacity of the path is minimal. The problem has applications to fast data transmissions in communication networks. The best previous algorithm for the single pair quickest path problem runs in time O(rm+rnlogn)O(r m+r n \log n), where rr is the number of distinct capacities of NN. In this paper, we present algorithms for general, sparse and planar networks that have significantly lower running times. For general networks, we show that the time complexity can be reduced to O(rm+rnlogn)O(r^{\ast} m+r^{\ast} n \log n), where rr^{\ast} is at most the number of capacities greater than the capacity of the shortest (with respect to lead time) path in NN. For sparse networks, we present an algorithm with time complexity O(nlogn+rn+rγ~logγ~)O(n \log n + r^{\ast} n + r^{\ast} \tilde{\gamma} \log \tilde{\gamma}), where γ~\tilde{\gamma} is a topological measure of NN. Since for sparse networks γ~\tilde{\gamma} ranges from 11 up to Θ(n)\Theta(n), this constitutes an improvement over the previously known bound of O(rnlogn)O(r n \log n) in all cases that γ~=o(n)\tilde{\gamma}=o(n). For planar networks, the complexity becomes O(nlogn+nlog3γ~+rγ~)O(n \log n + n\log^3 \tilde{\gamma}+ r^{\ast} \tilde{\gamma}). Similar improvements are obtained for the all--pairs quickest path problem. We also give the first algorithm for solving the dynamic quickest path problem

    Arc Routing with Time-Dependent Travel Times and Paths

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    Vehicle routing algorithms usually reformulate the road network into a complete graph in which each arc represents the shortest path between two locations. Studies on time-dependent routing followed this model and therefore defined the speed functions on the complete graph. We argue that this model is often inadequate, in particular for arc routing problems involving services on edges of a road network. To fill this gap, we formally define the time-dependent capacitated arc routing problem (TDCARP), with travel and service speed functions given directly at the network level. Under these assumptions, the quickest path between locations can change over time, leading to a complex problem that challenges the capabilities of current solution methods. We introduce effective algorithms for preprocessing quickest paths in a closed form, efficient data structures for travel time queries during routing optimization, as well as heuristic and exact solution approaches for the TDCARP. Our heuristic uses the hybrid genetic search principle with tailored solution-decoding algorithms and lower bounds for filtering moves. Our branch-and-price algorithm exploits dedicated pricing routines, heuristic dominance rules and completion bounds to find optimal solutions for problem counting up to 75 services. Based on these algorithms, we measure the benefits of time-dependent routing optimization for different levels of travel-speed data accuracy

    The Quick Time Dependent Quickest Flow Problem: A Lesson in Zero-Sum Cycles

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    A quick solution technique for the integral time-dependent quickest flow problem with no waiting is presented. The proposed technique is based on the successive shortest path approach and modifies an existing algorithm to improve its average performance. At each iteration, a reoptimization procedure is employed to determine the augmenting path given updates to the residual graph. The residual graph, by construction, almost always contains zero-sum cycles when employed in this context. These zero-sum cycles pose a unique problem for the reoptimization technique. A heuristic that can be embedded in the reoptimization algorithm to provide path solutions in the presence of zero-sum cycles has been proposed. In the computational experiments, the heuristic provided an optimal solution nearly 100% of the times. Further, a modified implementation of an existing path-finding algorithm has been used to solve the time-dependent quickest flow problem with source waiting

    Quality of service routing on wide area networks.

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    Moore [20] introduced the quickest path problem and it has been studied extensively in recent times. The quickest path problem is to determine a routing path to minimize end-to-end delay from the source to the destination node taking into account message size, and propagation delay and bandwidth on the links of the network. Thus the quickest path is a path with minimum end-to-end delay time required to send sigma units of message from a source node to the destination node.The main theme of this dissertation is to investigate unicast and multicast routing algorithms in wide area networks. Towards this end, first we present a unifying quickest path algorithm for different message transfer modes at intermediate nodes. The source to destination path varies with message sizes. Quickest path algorithms build a table called the Path-Table that when searched with message size gives the minimum end-to-end delay path for that message size. Our second result deals with efficient construction of the Path-Table when a link or path bandwidth changes, where path bandwidth is defined as the minimum of the bandwidths on the links of the path. Third, we present efficient algorithms for all-to-all quickest path problems in the presence of unreliable links in the network. By assigning probability of link failure to each link we can cast two problems namely, quickest most reliable path and most reliable quickest path.Routing is the process of sending a message from a source node to the destination node and the routing algorithm is a method to determine links that a message should be transmitted in order to reach the destination. The routing algorithm can be classified into the following three categories: unicast, multicast, and broadcast. Unicast involves sending a message from a given source to a destination; multicasting is a mechanism to send a message from a given source to a chosen set of destinations; broadcasting is sending a message from a given source to all the nodes in the network. Clearly, unicast and broadcast are special cases of multicast. The path selected by a routing algorithm depends on the application's Quality-of-Service (QoS) demands such as, end-to-end delay time, cost, delay jitter, and other factors.Our fourth result deals with multicast routing in wide area networks. We have developed several heuristics for the construction of a multicast tree that minimizes end-to-end delay time taking into account message size, and propagation delay and bandwidths on links. We consider different modes of message transfers at intermediate nodes and for each type of intermediate node architecture we present heuristics for the multicast tree construction. The heuristics are simulated on large networks that are generated using different network generation models including Waxman I and II, Locality, and Transit-Stub. Our heuristics are shown to outperform existing heuristics that are based on shortest path and minimum spanning tree for multicast tree construction. Finally, we introduce a novel heuristic for the construction of a multicast tree with minimum cost in Internet like topologies. Our algorithm on directed asymmetric networks is shown to have a performance gain in terms of tree costs over existing algorithms

    The variational nature of the gentlest ascent dynamics and the relation of a variational minimum of a curve and the minimum energy path

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    It is shown that the path described by the gentlest ascent dynamics to nd transition states [W. E and X. Zhou, Nonlinearity 24, 1831 (2011)] is an example of a quickest nautical path for a given stationary wind or current, the so-called Zermelo navigation variational problem. In the present case the current is the gradient of the potential energy surface. The result opens the possibility to propose new curves based on Zermelo's theory for two tasks: locate transition states and de ne reaction paths. The relation between a minimal variational character, that some former reaction pathways possess, and the minimum energy path is discussed

    Route Planning in Road Networks

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    We present various speedup techniques for route planning in road networks. After performing some preprocessing steps, we can compute accurate quickest-path lengths in a few microseconds on a 2.0 GHz machine, using real-world road networks with several million nodes. In addition to dealing with the static point-to-point problem, we also handle dynamic scenarios (like traffic jams) and many-to-many instances
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