A note on two problems in connexion with graphs

@article{Dijkstra1959ANO,
  title={A note on two problems in connexion with graphs},
  author={E. Dijkstra},
  journal={Numerische Mathematik},
  year={1959},
  volume={1},
  pages={269-271}
}
  • E. Dijkstra
  • Published 1959
  • Mathematics, Computer Science
  • Numerische Mathematik
We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the… Expand
The shortest-path problem for graphs with random arc-lengths
We consider the problem of finding the shortest distance between all pairs of vertices in a complete digraph on n vertices, whose arc-lengths are non-negative random variables. We describe anExpand
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  • U. Zwick
  • Mathematics, Computer Science
  • STOC '99
  • 1999
TLDR
This work presents the following algorithms for exactly solving the AlI Pairs Lightest Shortest Paths (APLSP) problem for directed graphs with integer weights of small absolute value, and presents results obtained recently by the author for the APSP problem. Expand
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  • T. Takaoka
  • Mathematics, Computer Science
  • Theor. Comput. Sci.
  • 2019
TLDR
This work considers a shortest path problem for a directed graph with edges labeled with a cost and a capacity, and gives a simple Floyd-like algorithm with complexity O ( min ⁡ { t, c n } n 3 ) , better than the best known O ( t c 1.843 n 2.2) for certain ranges of t and c. Expand
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Two improvements on time complexities of the all pairs shortest path (APSP) problem for directed graphs that satisfy certain properties are shown and a nearly acyclic graph is considered. Expand
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This work presents a combinatorial algorithm to solve the Single Source SP-AF problem in O ( m n ) worst case time, followed by an algorithm to solved the All Pairs SP-af problem in T ( t n ( ω + 9 ) / 4 ) time, where t is the number of distinct edge capacities and O ( n ω ) is the time taken to multiply two n-by-n matrices over a ring. Expand
On Shortest Disjoint Paths in Planar Graphs
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This work extends recent results by Colin de Verdiere and Schrijver to prove that, for a planar graph and for terminals adjacent to at most two faces, the Min-Sum 2 Disjoint Paths Problem can be solved in polynomial time and presents an algorithm that solves the problem for graphs with tree-width 2 in polynnomial time. Expand
Sharing information for the all pairs shortest path problem
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  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 2014
TLDR
The all pairs shortest path (APSP) problem can be solved in O(mn+n^2log(c/n) time with the data structure of cascading bucket system, and the traditional computational model such that comparison-addition operations on distance data and random access with O(logn) bits can be done in O (1) time. Expand
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Two methods for systematically selecting the shortest connections from a list of possible connections to obtain a minimum total wire length are presented, which will be called a minimum tree. Expand
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