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21
Faster algorithms for the shortest path problem
, 1990
"... 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, mn edges, and nonnegative integer arc costs bounded by C, a onelevel form of radix heap gives a t ..."
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Cited by 103 (10 self)
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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, mn edges, and nonnegative integer arc costs bounded by C, a onelevel form of radix heap gives a time bound for Dijkstra's algorithm of O(m + n log C). A twolevel 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 + n /log C). 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 Algorithms in Transportation Models: Classical and Innovative Aspects
, 1998
"... 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 ..."
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Cited by 51 (3 self)
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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 ...
Integer Priority Queues with Decrease Key in . . .
 STOC'03
, 2003
"... 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 i ..."
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Cited by 28 (2 self)
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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 nondecreasing, 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, worstcase, with no restriction to monotonicity, and exponentially faster. As a classical application, for a directed graph with n nodes and m edges with nonnegative 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
SingleSource ShortestPaths on Arbitrary Directed Graphs in Linear AverageCase Time
 In Proc. 12th ACMSIAM Symposium on Discrete Algorithms
, 2001
"... The quest for a lineartime singlesource shortestpath (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 ..."
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Cited by 28 (5 self)
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The quest for a lineartime singlesource shortestpath (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 averagecase 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 singlesource shortestpath problem (SSSP) is a fundamental and wellstudied 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...
Averagecase complexity of singlesource shortestpaths algorithms: lower and upper bounds
, 2003
"... ..."
Memoryefficient A*search using sparse embeddings
 IN: PROC. ACM 17TH INTERNATIONAL WORKSHOP ON ADVANCES IN GEOGRAPHIC INFORMATION SYSTEMS (ACM GIS
, 2010
"... When searching for optimal paths in a network, algorithms like A*search need an approximation of the minimal costs between the current node and a target node. A reference node embedding is a universal method for making such an approximation working for any type of positive edge weights. A drawback ..."
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Cited by 1 (1 self)
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When searching for optimal paths in a network, algorithms like A*search need an approximation of the minimal costs between the current node and a target node. A reference node embedding is a universal method for making such an approximation working for any type of positive edge weights. A drawback of the approach is that it is necessary to store the shortest distance to each landmark node for each considered attribute. Thus, the memory consumption of the embedding is linearly increasing with the number of attributes and landmarks. Thus, an embedded graph might not be wellsuited for handheld devices and may significantly increase the loading cost. In this paper, we propose methods for significantly decreasing the memory consumption of embedded graphs and examine the impact of the landmark selection. Furthermore, we propose to limit the number of embedded nodes in the network and propose an algorithm for shortest path computation working on networks for which only a portion of nodes store an embedding. Finally, we propose a heuristic algorithm for finding a suitable subset of nodes that should be embedded in order to guarantee reasonable computation times. Our experimental evaluation examines the tradeoff between embedding memory and processing times on two realworld data sets.
A Reach and Bound Algorithm for Acyclic Dynamic Programming Networks ∗
, 2007
"... Node pruning is a commonly used technique for solution acceleration in a dynamic programming network. In pruning, nodes are adaptively removed from the dynamic programming network when they are determined to not lie on an optimal path. We introduce an εpruning condition that extends pruning to incl ..."
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Node pruning is a commonly used technique for solution acceleration in a dynamic programming network. In pruning, nodes are adaptively removed from the dynamic programming network when they are determined to not lie on an optimal path. We introduce an εpruning condition that extends pruning to include a possible error in the pruning step. This results in a greater reduction of the computation time; however, ∗This work was supported in part by a National Science Foundation GOALI (DMI9900267) and a grant from General Motors 1 as a result of the inclusion of this error, the solution can be suboptimal or possibly infeasible. This condition requires the ability to compare the costs of an optimal path from a node to a terminal node. Therefore, we focus on the class of acyclic dynamic programming networks with monotonically decreasing optimal coststogo. We provide an easily implementable algorithm, Reach and Bound, which maintains feasibility and bounds the solution’s error. We conclude by illustrating the applicability of Reach and Bound on a problem of single location capacity expansion.
von
"... 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 widearea networks. Due to the large size of the underlying timetables the efficiency of a nai ..."
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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 widearea 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 optimalitypreserving speedup 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 realworld data: the timetable that contains all German trains, and a