# 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} }

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

#### Topics from this paper

#### 20,955 Citations

The shortest-path problem for graphs with random arc-lengths

- Computer Science, Mathematics
- Discret. Appl. Math.
- 1985

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 an… Expand

All pairs lightest shortest paths

- Mathematics, Computer Science
- STOC '99
- 1999

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

Shortest Shortest Path Trees of a Network

- Mathematics, Computer Science
- Discret. Appl. Math.
- 1996

This work presents an O(mn log n) algorithm to find a shortest shortest path tree, and gives an algorithm with the same complexity to determine a maximum set of non-equivalent efficient points of N for the two criteria cited above. Expand

Shortest Path Problems with Time Constraints

- Mathematics, Computer Science
- MFCS
- 1996

A new version of the shortest path problem is studied, where all transit times b(e, u) are positive integers and zero waiting times are assumed, namely, waiting at any vertex is strictly prohibited. Expand

Some Extensions of the Bottleneck Paths Problem

- Mathematics, Computer Science
- WALCOM
- 2014

This work presents a combinatorial algorithm to solve the Single Source Shortest Paths for All Flows (SSSP-AF) problem in O(mn) worst case time, followed by an algorithm to solved the All Pairs Shortest paths for all Fl flows (APSP-af) problemIn \(O(\sqrt{t}n^{(\omega+9)/4})\) time. Expand

Algebraic theory on shortest paths for all flows

- Mathematics, Computer Science
- Theor. Comput. Sci.
- 2019

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

Sharing Information in All Pairs Shortest Path Algorithms

- Computer Science, Mathematics
- CATS
- 2011

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

Variations on the bottleneck paths problem

- Computer Science, Mathematics
- Theor. Comput. Sci.
- 2015

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

- Mathematics, Computer Science
- ISAAC
- 2009

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

- Computer Science, Mathematics
- Theor. Comput. Sci.
- 2014

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

#### References

SHOWING 1-8 OF 8 REFERENCES

Formal Procedures for Connecting Terminals with a Minimum Total Wire Length

- Computer Science
- JACM
- 1957

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

On the shortest spanning subtree of a graph and the traveling salesman problem

- Mathematics
- 1956

7. A. Kurosh, Ringtheoretische Probleme die mit dem Burnsideschen Problem uber periodische Gruppen in Zussammenhang stehen, Bull. Acad. Sei. URSS, Ser. Math. vol. 5 (1941) pp. 233-240. 8. J.… Expand

NETWORK FLOW THEORY

- Computer Science
- 1956

Abstract : The labeling algorithm for the solution of maximal network flow problems and its application to various problems of the transportation type are discussed.

Mathematisch Centrum 2e Boerhaavestraat 49 Amsterdam-O (Received ]1tne 11, 19:WJ Druck tier Univ,,rsitiitstlruckerei II

- Theorie des graphes et ses applications,
- 1958

I heorie des graphes et ses applications, pp. 6g _ 69. paris: Dunod t 95g

- Mattreoatisch Centlum 2e Boerhaavestraat 49 Amsterdam-O (Recefued
- 1959

VVEINBERGER: Formal Procedures for Connecting Terminals with a Minimum Total ,vire Length

- J. Ass. Comp. Mach
- 1957

WETNaERGER : Formal procedures ior Co " nJciiog ferminals with a Minimum Total Wire Length

- J . Ass . Comp . Macb . I heorie des graphes et ses applications Mattreoatisch Centlum 2 e Boerhaavestraat 49 AmsterdamO ( Recefued