research-article

On Finding and Updating Spanning Trees and Shortest Paths

Authors:

P. M. Spira,

A. PanAuthors Info & Claims

SIAM Journal on Computing, Volume 4, Issue 3

Pages 375 - 380

https://doi.org/10.1137/0204032

Published: 01 September 1975 Publication History

Abstract

We consider one origin shortest path and minimum spanning tree computations in weighted graphs. We give a lower bound on the number of analytic functions of the input computed by a tree program which solves either of these problems equal to half the number of worst-case comparisons which well-known algorithms attain. We consider the work necessary to update spanning tree and shortest path solutions when the graph is altered after the computation has terminated. Optimal or near-optimal algorithms are attained for the cases considered. The most notable result is that a spanning tree solution can be updated in $O(n)$ when a new node is added to an n-node graph whose minimum spanning tree is known.

References

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E. W. Dijkstra, A note on two problems in connexion with graphs, Numer. Math., 1 (1959), 269–271

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Michael O. Rabin, Proving simultaneous positivity of linear forms, J. Comput. System Sci., 6 (1972), 639–650

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P. M. Spira, A new algorithm for finding all shortest paths in a graph of positive arcs in average time $O(n\sp{2}{\rm log}\sp{2}n)$, SIAM J. Comput., 2 (1973), 28–32

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Yang BZheng W(2024)A Powerful Local Search Method for Minimum Steiner Tree ProblemWeb and Big Data10.1007/978-981-97-7241-4_4(50-66)Online publication date: 31-Aug-2024
https://dl.acm.org/doi/10.1007/978-981-97-7241-4_4
Chen QLachish OHelmer SBöhlen M(2022)Dynamic spanning trees for connectivity queries on fully-dynamic undirected graphsProceedings of the VLDB Endowment10.14778/3551793.355186815:11(3263-3276)Online publication date: 1-Jul-2022
https://dl.acm.org/doi/10.14778/3551793.3551868
Ming YChen SChen YFu Z(2018)A Fast Vertex-Swap Operator for the Prize-Collecting Steiner Tree ProblemComputational Science – ICCS 201810.1007/978-3-319-93701-4_43(553-560)Online publication date: 11-Jun-2018
https://dl.acm.org/doi/10.1007/978-3-319-93701-4_43
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Index Terms

On Finding and Updating Spanning Trees and Shortest Paths
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
2. Theory of computation
  1. Design and analysis of algorithms
  2. Randomness, geometry and discrete structures

Index terms have been assigned to the content through auto-classification.

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Published In

SIAM Journal on Computing Volume 4, Issue 3

Sep 1975

188 pages

ISSN:0097-5397

DOI:10.1137/smjcat.1975.4.issue-3

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 September 1975

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Yang BZheng W(2024)A Powerful Local Search Method for Minimum Steiner Tree ProblemWeb and Big Data10.1007/978-981-97-7241-4_4(50-66)Online publication date: 31-Aug-2024
https://dl.acm.org/doi/10.1007/978-981-97-7241-4_4
Chen QLachish OHelmer SBöhlen M(2022)Dynamic spanning trees for connectivity queries on fully-dynamic undirected graphsProceedings of the VLDB Endowment10.14778/3551793.355186815:11(3263-3276)Online publication date: 1-Jul-2022
https://dl.acm.org/doi/10.14778/3551793.3551868
Ming YChen SChen YFu Z(2018)A Fast Vertex-Swap Operator for the Prize-Collecting Steiner Tree ProblemComputational Science – ICCS 201810.1007/978-3-319-93701-4_43(553-560)Online publication date: 11-Jun-2018
https://dl.acm.org/doi/10.1007/978-3-319-93701-4_43
Yasmin NMenezes R(2006)Incremental computation of shortest paths in semi-dynamic graphs using software componentsProceedings of the 44th annual ACM Southeast Conference10.1145/1185448.1185470(96-101)Online publication date: 10-Mar-2006
https://dl.acm.org/doi/10.1145/1185448.1185470
Koenig SLikhachev MLiu YFurcy D(2004)Incremental Heuristic Search in AIAI Magazine10.1609/aimag.v25i2.176325:2(99-112)Online publication date: 1-Jun-2004
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Frederickson G(1985)Data Structures for On-Line Updating of Minimum Spanning Trees, with ApplicationsSIAM Journal on Computing10.1137/021405514:4(781-798)Online publication date: 1-Nov-1985
https://dl.acm.org/doi/10.1137/0214055

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Rectilinear Shortest Paths and Minimum Spanning Trees in the Presence of Rectilinear Obstacles

Balancing minimum spanning trees and shortest-path trees

Low-Degree Spanning Trees of Small Weight

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