Efficient implementations of Dijkstra’s shortest path algorithm are investigated. A new data structure, called the radix heap, is proposed for use in this algorithm. On a network with n vertices, m edges, and nonnegative integer arc costs bounded by C, a one-level form of radix heap gives a time bound for Dijkstra’s algorithm of O(m + n log C). A two-level form of radix heap gives a bound of O(m + n log C/log log C). A combination of a radix heap and a previously known data structure called a Fibonacci heap gives a bound of O(m + nm). The best previously known bounds are O(m + n log n) using Fibonacci heaps alone and O(m log log C) using the priority queue structure of Van Emde Boas et al. [17].

Shortest Path Problems are among the most studied network flow optimization problems, with interesting applications in various fields. One such field is transportation, where shortest path problems of different kinds need to be solved. Due to the nature of the application, transportation scientists need very flexible and efficient shortest path procedures, both from the running time point of view, and also for the memory requirements. Since no "best" algorithm currently exists for every kind of transportation problem, research in this field has recently moved to the design and implementation of "ad hoc" shortest path procedures, which are able to capture the peculiarities of the problems under consideration. The aim of this work is to present in a unifying framework both the main algorithmic approaches that have been proposed in the past years for solving the shortest path problems arising most frequently in the transportation field, and also some important implementation techniques ...

The quest for a linear-time single-source shortest-path (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 w 1g where w denotes the word length, the currently best time bound for directed sparse graphs on a RAM is O(n + m log log n). In the present paper we study the average-case complexity of SSSP. We give a simple algorithm for arbitrary directed graphs with random edge weights uniformly distributed in [0; 1] and show that it needs linear time O(n + m) with high probability. 1 Introduction The single-source shortest-path problem (SSSP) is a fundamental and well-studied combinatorial optimization problem with many practical and theoretical applications [1]. Let G = (V; E) be a directed graph, jV j = n, jEj = m, let s be a distinguished vertex of the graph, and c be a function assigning a n...

Abstract not found

We consider Fibonacci heap style integer priority queues supporting insert and decrease key operations in constant time. We present a deterministic linear space solution that with n integer keys support delete in O(log log n) time. If the integers are in the range [0,N), we can also support delete in O(log log N) time. Even for the special case of monotone priority queues, where the minimum has to be non-decreasing, the best previous bounds on delete were O((log n) 1/(3−ε) ) and O((log N) 1/(4−ε)). These previous bounds used both randomization and amortization. Our new bounds a deterministic, worst-case, with no restriction to monotonicity, and exponentially faster. As a classical application, for a directed graph with n nodes and m edges with non-negative integer weights, we get single source shortest paths in O(m + n log log n) time, or O(m + n log log C) ifC is the maximal edge weight. The later solves an open problem of Ahuja, Mehlhorn, Orlin, and

and have found that it is complete and satisfactory in all respects,

The quest for a linear-time single-source shortest-path (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 1g where w denotes the word length, the currently best time bound for directed sparse graphs on a RAM is O(n +m log log n).

Algorithms for timetable information systems are usually based on Dijkstra's algorithm. The concrete scenario that we have in mind is a central information server in the realm of public railroad traffic on wide-area networks. Due to the large size of the underlying timetables the efficiency of a naive implementation is not acceptable in practice, so usually heuristics are used to improve the efficiency. Typically, using such heuristics means that the optimality of the solutions can no longer be guaranteed. In contrast, we investigate optimality-preserving speed-up techniques for Dijkstra's algorithm. The basic question is whether algorithms that compute optimal solutions are competitive on contemporary computer technology. Therefore, we present the results of a computational study based on real-world data: the timetable that contains all German trains, and a snapshot of customer queries to an existing information server.

This report describes the methodologies and procedures developed through a contract to the University of Texas at Austin, in collaboration with the University of Maryland, to address these essential needs. Specifically, a simulation-assignment methodology has been developed to describe user's path choices in the network in response to real-time information, and the resulting flow patterns that propagate through the network, yielding information about overall quality of service and effectiveness, as well as localized information pointing to problem spots and opportunities for improvement. This methodology is intended for use off-line for evaluation purposes, or on-line for prediction purpose in support of advanced traffic management functions. In additional, algorithmic procedures have been developed to determine the best paths to which users should be directed so as to optimize overall system performance. Powerful extension

Abstract: In this paper, we present a new labeling algorithm for the shortest path problem with time windows (SPPTW). It is generalized from the threshold algorithm for the unconstrained shortest path problem. Our computational experiments show that this generalized threshold algorithm outperforms a label setting algorithm for the SPPTW on a set of randomly generated test problems. The average running time of the new algorithm is about 40 % less than the label setting algorithm, which istoday the best algorithm based on published experimental evidence. 1 The shortest path problem with time windows (SPPTW) is a generalization of the classical (unconstrained) shortest path problem (SPP) involving the added complexity of time windows. The SPPTW can be described as follows. Let G =(V�A) be a directed graph where V = N [fp � qg is the set of nodes with source node p and sink node q, A is the set of arcs. Each nodei2V has a time window [ai�bi] within which nodeican be visited. Each arc (i � j) has a positive duration tij