Quickest Path Research Articles

Finding the k quickest simple paths in a network

Let N be a network with n nodes and m arcs, and let σ be the amount of data to be transmitted. The quickest path problem is to find a routing path in N such that the time required to ship σ units of data from the source to the sink is minimum. The problem considered in this paper is to find the first k quickest simple paths for a given pair of nodes, and an algorithm is designed. The complexity required by the proposed algorithm is O( kmn 3 + kmn log k) if the network is directed and is O( kmn 2 + kmn log k) if the network is undirected.

Algorithms for the constrained quickest path problem and the enumeration of quickest paths

The quickest path problem, which was originally proposed by Chen and Chin, is a variant of the shortest path problem. Its objective is to find quickest paths in a network to transmit a given amount of data such that the transmission time is minimized. If the quickest paths are required to go through a specified path, then the restricted problem is called the constrained quickest path problem. In this paper, for all pairs of nodes in a network N, an O( mn 2) time algorithm is first presented to find constrained quickest paths, and then an O( k 2 mn 2) time algorithm is presented to enumerate the first k quickest paths. Our results improve previous results by Rosen, Sun and Xue.

Classical differential geometry solution of the brachistochrone tunnel problem

The Frenet-Serret equations of classical differential geometry are used to describe the quickest descent tunneling path problem. The optimal tunnel is shown to have a constant turn rate with zero torsion and is equivalent to Edelbaum's hypocycloid solution. The solutions are obtained using the maximum principle and singular arc conditions. The optimal curvature is a first-order singular arc and the optimal torsion is a second-order singular arc. Our treatment includes both the normal and the abnormal optimal control problems. Our problem is abnormal for the case where the final speed is zero. Analytical solutions for the optimal time histories are derived for all states and all adjoint states. One of Leitmann's sufficiency field theorems is used to establish optimality of the solutions.

Quick and Easy Optimal Approach Paths for Nonlinear Natural Resource Models

AbstractAn analytical optimal time‐path solution method for nonlinear natural resource models based on backward induction is discussed. A method for estimating a time‐independent current period decision rule from the discrete‐point time‐path solution set generated with backward induction is then reviewed. A model of the northern anchovy fishery of California is used to compare the performance of this approach against previously reported results based on linear quadratic analytical dynamic programming and approximately optimal analytical current period decision rule solution methods. The approach provides an analytical shortcut alternative to numerical dynamic programming, without sacrificing precision.

On the quickest path problem

Let N be an input network and σ be the amount of data to be transmitted. We present an O( mn 2) time algorithm that finds all-pairs quickest paths, for a given value of σ, and show that the quickest path between any two nodes for any value of σ can be found in O(log m) time, provided O( mn 2) preprocessing time is spent.

The all-pairs quickest path problem

In this paper we discuss the quickest path problem and proposed an θ(λn 2 ) space data structure which allows to answer quickest path queries in O(log r) time and whose construction requires θ(min {mn 2 , rn 2 log n+rnm}) time

An algorithm for finding the k quickest paths in a network

Let N be a network with n nodes and m arcs, and let σ be the amount of data to be transmitted. The quickest path problem is to find a routing path in N such that the time required to ship σ units of data from the source to the sink is minimum. The problem considered in this paper is to find the first k quickest looping paths from the source to the sink, and an algorithm with time complexity of O(m 2 + (m + k)n log n + k 3 2 log k) is developed.

Distributed algorithms for the quickest path problem

Let N = ( V, A, C, L) be a network with node set V, arc set A, positive arc capacity function C, and nonnegative arc lead time function L. The quickest path problems is to find paths in N to transmit a given amount of data such that the transmission time is minimized. In this paper, distributed algorithms are developed for the quickest path problem in an asynchronous communication network. For the one-source quickest path problem, we present three algorithms that require O( rn 2) messages and O( 2 ) time, O(n) messages and O( rn) time, and O 1+ ε log w) messages and O( rn 1+ ε log w) time for any ε, 0< ε<1, respectively, where m = | A|, n = | V|, r is the number of distinct capacity values of N, and w is the maxima arc weight of N. For the all-pairs quickest path problem, we present an algorithm that requires O( mn) messages and O( m) time.

Algorithms for the quickest path problem and the enumeration of quickest paths

Let N = ( V, A, c, l) be an input network with node set V, arc set A, positive arc weight function c and nonnegative arc weight function l. Let σ be the amount of data to be transmitted. The quickest path problem is to find a routing path in N to transmit the given amount of data in minimum time. In a recent paper, Chen and Chin proposed this problem and developed algorithms for the single pair quickest path problem with time complexity O( re + rn log n), where n = ¦V¦, e = ¦A¦, and r is the number of distinct capacity values of N. In this paper, we first develop an alternative algorithm for the single pair quickest path problem with same time complexity and less space requirement. We then study the constrained quickest path problem and propose an O( re + rn log n) time algorithm. Finally, we develop an algorithm to enumerate the first m quickest paths to send a given amount of data from one node to another with time complexity O( rmne + rmn 2 log n).

New proofs of the consistency of the normal Moore space conjecture II

This article continues our first paper on this topic. In this article we first continue our review of the technique of iterated forcing and reflection by describing the machinery developed for reflection from a weakly compact cardinal. Next we present more applications of the technique to show conditions under which nonmetrizability, nonparacompactness, or nondevelopability reflect. In the final section we present yet another proof of the normal Moore space conjecture which avoids elementary embeddings by using filter combinatorics and provides a quick path to the solution for those less interested in generally applicable techniques.

The quickest path problem

Let N be an input network and σ be the amount of data to be transmitted. The quickest path problem is to find a routing path in N such that the time required to ship σ units of data from the source to the sink is minimum. When the value of σ is given, an O( m 2 + nm log m) algorithm is designed to find the quickest path, where m and n are the numbers of arcs and nodes in the network N, respectively. By generalizing this algorithm properly, a more general algorithm with similar time complexity is developed to find quickest paths for all possible values of σ. Using the algorithm, the values of σ can be divided into O( m) ordered regions such that each region has a corresponding quickest path; therefore, if the general algorithm is used as a preprocessing step to handle the input network N, then the problem of finding a quickest path for a given value of σ can be solved in time O(log m).

The mathematical technique in Fermat's deduction of the law of refraction

This article attempts to explain Fermat's not quite obvious calculations connected with his deduction of the law of refraction in Analysis ad refractiones (1662), and to describe the development which led to these calculations. In 1657 Fermat tried to deduce a law of refraction based on the principle that light follows the quickest path between two given points. He did not succeed because he found that the calculations were too long and tedious. The calculations are indeed complicated, but if Fermat, in 1657, had been willing to accept Descartes' law of refraction he would probably also have seen that it solved his problem. However, Fermat was of the opinion that Descartes' law was wrong and, therefore, he did not expect that solution. Only in 1662, when he succeeded in reducing the calculations substantially, did he realize that they led to the sine law of Descartes.

Designing considerate systems (abstract only)

Ease of Use can be thought of as consisting of two components: Ease of Learning and Ease of Doing. In the past, most of the attention in discussions of Ease of Use has focused on Ease of Learning. This is the motivation behind consideration for the "naive" or "casual" user. The most common approach has been to allow trading computing functionality for Ease of Learning. This makes the most commonly performed tasks very simple to perform, but prevents a wide range of other tasks from being performed at all. This affects Ease of Doing.Ease of Doing is a concept that has been primarily associated with expert users of computing systems. A task is only Easy to Do on a computer if the proper tools have been provided for doing it. Since there is an enormous range of tasks to apply systems to, there must also be a large collection of tools. A great variety of software tools that are finely tuned to particular applications should be made available to users. In addition, the system should be extensible to allow for ready customization.We feel that a sophisticated personal computing environment must provide a quick path for casual users to be able to operate parts of the system, and yet allow more habitual users a path to gain mastery over the more esoteric components of the system with time.