Topic

# Dijkstra's algorithm

About: Dijkstra's algorithm is a research topic. Over the lifetime, 4471 publications have been published within this topic receiving 65232 citations. The topic is also known as: Dijkstra algorithm.

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02 Jan 1994

TL;DR: Performance and Scalability of Parallel Systems, General Issues in Mapping Systolic Systems Onto Parallel Computers, and Speedup Anomalies in Parallel Search Algorithms.

Abstract: Introduction. What is Parallel Computing? The Scope of Parallel Computing. Issues in Parallel Computing. Organization and Contents of The Text. Bibliographic Remarks. Problems. References. Models of Parallel Computers. A Taxonomy of Parallel Architectures. An Idealized Parallel Computer. Dynamic Interconnection Networks. Static Interconnection Networks. Embedding Other Networks Into a Hypercube. Routing Mechanisms For Static Networks. Communication Costs in Static Interconnection Networks. Cost-Performance Tradeoffs. Architectural Models For Parallel Algorithm Design. Bibliographic Remarks. References. Basic Communication Operations. Simple Message Transfer Between Two Processors. One-To-All Broadcast. All-To-All Broadcast, Reduction, and Prefix Sums. One-To-All Personalized Communications. All-To-All Personalized Communications. Circular Shift. Faster Methods For Some Communication Operations. Summary. Bibliographic Remarks. Problems. References. Performance and Scalability of Parallel Systems. Performance Metrics For Parallel Systems. The Effect of Granularity and Data Mapping On Performance. The Scalability of Parallel Systems. The Isoefficiency Metric of Scalability. Sources of Parallel Overhead. Minimum Execution Time and Minimum Cost-Optimal Execution Time. Other Scalability Metrics and Bibliographic Remarks. Problems. References. Dense Matrix Algorithms. Mapping Matrices Onto Processors. Matrix Transpositon. Matrix-Vector Multiplication. Matrix Multiplication. Solving a System of Linear Equations. Bibliographic Remarks. Problems. References. Sorting. Issues in Sorting On Parallel Computers. Sorting Networks. Bubble Sort and Its Variants. Quicksort. Other Sorting Algorithms. Bibliographic Remarks. Problems. References. Graph Algorithms. Definitions and Representation. Minimum Spanning Tree: Prim's Algorithm. Single-Source Shortest Paths: Dijkstra's Algorithms. All-Pairs Shortest Paths. Transitive Closure. Connected Components. Algorithms For Sparse Graphs. Bibliographic Remarks. Problems. References. Search Algorithms For Discrete Optimization Problems. Definitions and Examples. Sequential Search Algorithms. Search Overhead Factor. Parallel Depth-First Search. Parallel Best-First Search. Speedup Anomalies in Parallel Search Algorithms. Bibliographic Remarks. Problems. References. Dynamic Programming. Serial Monadic Dp Formulations. Nonserial Monadic Dp Formulations. Serial Polyadic Dp Formulations. Nonserial Polyadic Dp Formulations. Summary and Discussion. Bibliographic Remarks. Problems. References. Fast Fourier Transform. The Serial Algorithm. The Binary-Exchange Algorithm. The Transpose Algorithm. Cost-Effectiveness of Meshes and Hypercubes For Fft. Bibliographic Remarks. Problems. References. Solving Sparse Systems of Linear Equations. Basic Operations. Iterative Methods. Finite Element Method. Direct Methods For Sparse Linear Systems. Multigrid Methods. Bibliographic Remarks. Problems. References. Systolic Algorithms and Their Mapping Onto Parallel Computers. Examples of Systolic Systems. General Issues in Mapping Systolic Systems Onto Parallel Computers. Mapping One-Dimensional Systolic Arrays. Bibliographic Remarks. Problems. References. Parallel Programming. Parallel Programming Paradigms. Primitive For The Message-Passing Programming Paradigm. Data-Parallel Languages. Primitives For The Shared-Address-Space Programming Paradigm. Fortran D. Bibliographic Remarks. References. Appendix A. Complexity of Functions and Order Analysis. Author Index. Subject Index. 0805331700T04062001

1,401 citations

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TL;DR: Two serial and parallel algorithms for solving a system of equations that arises from the discretization of the Hamilton-Jacobi equation associated to a trajectory optimization problem of the following type are presented.

Abstract: We present serial and parallel algorithms for solving a system of equations that arises from the discretization of the Hamilton-Jacobi equation associated to a trajectory optimization problem of the following type. A vehicle starts at a prespecified point x/sub o/ and follows a unit speed trajectory x(t) inside a region in /spl Rscr//sup m/ until an unspecified time T that the region is exited. A trajectory minimizing a cost function of the form /spl int//sub 0//sup T/ r(x(t))dt+q(x(T)) is sought. The discretized Hamilton-Jacobi equation corresponding to this problem is usually solved using iterative methods. Nevertheless, assuming that the function r is positive, we are able to exploit the problem structure and develop one-pass algorithms for the discretized problem. The first algorithm resembles Dijkstra's shortest path algorithm and runs in time O(n log n), where n is the number of grid points. The second algorithm uses a somewhat different discretization and borrows some ideas from a variation of Dial's shortest path algorithm (1969) that we develop here; it runs in time O(n), which is the best possible, under some fairly mild assumptions. Finally, we show that the latter algorithm can be efficiently parallelized: for two-dimensional problems and with p processors, its running time becomes O(n/p), provided that p=O(/spl radic/n/log n). >

816 citations

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30 May 2008TL;DR: CHs can be combined with many other route planning techniques, leading to improved performance for many-to-many routing, transit-node routing, goal-directed routing or mobile and dynamic scenarios, and a hierarchical query algorithm using bidirectional shortest-path search is obtained.

Abstract: We present a route planning technique solely based on the concept of node contraction. The nodes are first ordered by 'importance'. A hierarchy is then generated by iteratively contracting the least important node. Contracting a node υ means replacing shortest paths going through v by shortcuts. We obtain a hierarchical query algorithm using bidirectional shortest-path search. The forward search uses only edges leading to more important nodes and the backward search uses only edges coming from more important nodes. For fastest routes in road networks, the graph remains very sparse throughout the contraction process using rather simple heuristics for ordering the nodes. We have five times lower query times than the best previous hierarchical Dijkstra-based speedup techniques and a negative space overhead, i.e., the data structure for distance computation needs less space than the input graph. CHs can be combined with many other route planning techniques, leading to improved performance for many-to-many routing, transit-node routing, goal-directed routing or mobile and dynamic scenarios.

739 citations

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CA Technologies

^{1}TL;DR: A simple solution to the mutual exclusion problem is presented which allows the system to continue to operate despite the failure of any individual component.

Abstract: A simple solution to the mutual exclusion problem is presented which allows the system to continue to operate despite the failure of any individual component.

737 citations

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TL;DR: An algorithm for determining the shortest path between a source and a destination on an arbitrary (possibly nonconvex) polyhedral surface and generalizes to the case of multiple source points to build the Voronoi diagram on the surface.

Abstract: We present an algorithm for determining the shortest path between a source and a destination on an arbitrary (possibly nonconvex) polyhedral surface. The path is constrained to lie on the surface, and distances are measured according to the Euclidean metric. Our algorithm runs in time O(n log n) and requires O(n2) space, where n is the number ofedges ofthe surface. Afterwe run our algorithm, the distance from the source to any other destination may be determined using standard techniques in time O(log n) by locating the destination in the subdivision created by the algorithm. The actual shortest path from the source to a destination can be reported in time O(k+ log n), where k is the number of faces crossed by the path. The algorithm generalizes to the case of multiple source points to build the Voronoi diagram on the surface, where n is now the maximum of the number of vertices and the number of sources.

705 citations