Two Algorithms for Computing All Spanning Trees of a Simple, Undirected, and Connected Graph: Once Assuming a Complete Graph

doi:10.1109/ACCESS.2018.2872737

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Two Algorithms for Computing All Spanning Trees of a Simple, Undirected, and Connected Graph: Once Assuming a Complete Graph

Journal Article•DOI•

Two Algorithms for Computing All Spanning Trees of a Simple, Undirected, and Connected Graph: Once Assuming a Complete Graph

Maumita Chakraborty¹, Sumon Chowdhury², Rajat Kumar Pal³•Institutions (3)

Information Technology Institute¹, Tata Consultancy Services², University of Calcutta³

01 Jan 2018-IEEE Access (Institute of Electrical and Electronics Engineers (IEEE))-Vol. 6, pp 56290-56300

TL;DR: This paper proposes altogether different and new approaches for the computation of all possible spanning trees of a simple, undirected, and connected graph and proposes to have novelties and limitations of its own.

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Abstract: In this paper, we have proposed altogether different and new approaches for the computation of all possible spanning trees of a simple, undirected, and connected graph. Our proposed algorithms have the capability to solve the major bottlenecks in this area, namely, generation of duplicate trees and circuit checking. In the first algorithm, the given graph has been converted to its corresponding weighted complete graph, which proposes to have novelties and limitations of its own. In addition, we have also proposed another related algorithm, and as a result, we have been able to come up with new ideas in this research domain of graph theory.

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Journal Article•DOI•

Divide-and-conquer based all spanning tree generation algorithm of a simple connected graph

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Dimitrios Gounopoulos¹•Institutions (1)

University of Engineering & Management¹

01 Jan 2022-Theoretical computer science

TL;DR: In this paper , an algorithm based on a new technique, namely divide-and-conquer, has been proposed for all spanning tree generation of a simple connected graph, which is a well-approached problem in graph theory.

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Abstract: All spanning tree generation of a simple connected graph is a well-approached problem in graph theory. In this paper, an algorithm based on a new technique, namely divide-and-conquer, has been proposed. The performance of the proposed algorithm has also been benchmarked against several existing algorithms, including one algorithm implemented in parallel, in this domain. The basis of comparison is on the number of circuits generated and CPU time taken by each of the algorithms compared, using a set of randomly generated graph instances on a common platform.

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3 citations

Book Chapter•DOI•

Algorithm to Generate All Spanning Tree Structures of a Complete Graph

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Suvrima Datta¹, Sumit Chakraborty², Maumita Chakraborty³, Rajat Kumar Pal¹•Institutions (3)

University of Calcutta¹, Ramsaday College², Information Technology Institute³

01 Jan 2021

TL;DR: In this article, the authors proposed an algorithm to generate all possible structures of spanning trees of an undirected complete graph of n vertices, where the process starts with a star-tree (T) of the given complete graph and then replaces the edges of T one by one to generate different possible structures like chain, branch, etc.

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Abstract: The objective of this paper is to propose an algorithm to generate all possible structures of spanning trees of an undirected complete graph of n vertices. The process starts with a star-tree (T) of the given complete graph and then replacing the edges of T one by one to generate different possible structures like chain, branch, etc. These spanning tree structures repeat themselves as we move from lower to higher values of n. The authors have attempted to find out some generalized expressions for different structures of spanning trees for a complete graph of order n.

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2 citations

References

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An Algorithm for Finding All the Spanning Trees in Undirected Graphs

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Tomomi Matsui

01 Jan 1998

TL;DR: This paper proposes an algorithm for nding all the spanning trees in undirected graphs that requires O(n +m + n) time and O( n + m) space, and is optimal for outputting all the spans trees explicitly.

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Abstract: In this paper, we propose an algorithm for nding all the spanning trees in undirected graphs. The algorithm requires O(n +m + n) time and O(n + m) space, where the given graph has n vertices, m edges and spanning trees. For outputting all the spanning trees explicitly, this algorithm is optimal.

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16 citations

"Two Algorithms for Computing All Sp..." refers methods in this paper

...[14] T. Matsui, ‘‘An algorithm for finding all the spanning trees in undirected graphs,’’ Dept. Math....
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...Some algorithms following this method of tree generation are those of Cherkasskii [11], Hakimi [12], Kapoor and Ramesh [13], Matsui [14], [15], Mayeda and Seshu [16], Shioura and Tamura [17], Shioura et al....
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...[15] T. Matsui, ‘‘A flexible algorithm for generating all the spanning trees in undirected graphs,’’ Algorithmica, vol. 18, no. 4, pp. 530–543, 1997....
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...Some algorithms following this method of tree generation are those of Cherkasskii [11], Hakimi [12], Kapoor and Ramesh [13], Matsui [14], [15], Mayeda and Seshu [16], Shioura and Tamura [17], Shioura et al. [18], and many others. iii....
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...Some of the earlier standard algorithms (namely, algorithms by Shioura and Tamura [17], Matsui [14], Mayeda and Seshu [16], Hakimi [12], Char [2], and Winter [21]) for generating all possible spanning trees of a given graph have also been implemented on the same environment with the same set of instances for a comparative study....
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Journal Article•DOI•

Algorithm 354: generator of spanning trees [H]

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M. Douglas McIlroy¹•Institutions (1)

Bell Labs¹

01 Sep 1969-Communications of The ACM

TL;DR: This procedure finds all trees that span a nondirected graph on n nodes that are partitioned into two classes, and formalizes the effect of partitioning with respect to nodes i and j.

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Abstract: /* This procedure finds all trees that span a nondirected graph on n nodes. The essential step of this procedure partitions the set T(G) of trees which span graph G into two classes. Trees of one class contain a branch connecting a selected pair of nodes, i and j ; trees of the other class exclude such branches. To formalize the effect of partitioning with respect to nodes i and j , we let A~' be the \"attachment se t\" of branches between them, and G~i be the graph derived from G by combining i and j into a single node. Then

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12 citations

"Two Algorithms for Computing All Sp..." refers methods in this paper

...[4] M. D. McIlroy, ‘‘Algorithm 354: Generator of spanning trees,’’ Commun....
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...Some of the well-known algorithms falling under this category are those proposed by authors like Berger [1], Char [2], Gabow and Myers [3], McIlroy [4], Naskar et al....
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...Some of the well-known algorithms falling under this category are those proposed by authors like Berger [1], Char [2], Gabow and Myers [3], McIlroy [4], Naskar et al. [5]–[7], 56290 2169-3536 2018 IEEE....
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Journal Article•DOI•

The Enumeration of Trees without Duplication

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I. Berger

01 Dec 1967-IEEE Transactions on Circuit Theory

9 citations

"Two Algorithms for Computing All Sp..." refers methods in this paper

...These three techniques are tree testing [1]–[10] elementary tree transformation [11]–[18], and successive reduction [19]–[21] methods....
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...REFERENCES [1] I. Berger, ‘‘The enumeration of trees without duplication,’’ IEEE Trans....
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...Some of the well-known algorithms falling under this category are those proposed by authors like Berger [1], Char [2], Gabow and Myers [3], McIlroy [4], Naskar et al....
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...Some of the well-known algorithms falling under this category are those proposed by authors like Berger [1], Char [2], Gabow and Myers [3], McIlroy [4], Naskar et al. [5]–[7], 56290 2169-3536 2018 IEEE....
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Book Chapter•DOI•

Generation of All Spanning Trees in the Limelight

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Saptarshi Naskar¹, Krishnendu Basuli, Samar Sen Sarma²•Institutions (2)

Sarsuna College¹, University of Calcutta²

02 Jan 2012

TL;DR: The present approach avoids the major bottleneck of any tree generation algorithm with a simple but efficient procedure and at the same time ensures that a large number of non-tree subgraphs are not generated at all.

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Abstract: Many problems in science and engineering [1, 3, 8, 10] can be formulated in terms of graphs. There are problems where spanning trees are necessary to be computed from the given graphs. Connected subgraph with all the n vertices of the graph G(V,E), where |V|=n, having exactly of n(1 edges called the spanning tree of the given graph. The major bottleneck of any tree generation algorithm is the prohibitively large cost of testing whether a newly born tree is twin of a previously generated one and also there is a problem that without checking for circuit generated subgraph is tree or non-tree. This problem increases the time complexity of the existing algorithms. The present approach avoids this problem with a simple but efficient procedure and at the same time ensures that a large number of non-tree subgraphs are not generated at all.

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6 citations

Journal Article•DOI•

Generation of All Spanning Trees

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Saptarshi Naskar¹, Krishnendu Basuli¹, Samar Sen Sarma¹•Institutions (1)

University of Calcutta¹

12 Jul 2009-Social Science Research Network

TL;DR: The method here is qualitatively and quantitatively better than existing methods and minimum number of duplicate tree comparison and no circuit testing at all for its realization.

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Abstract: This paper deals with all spanning tree generation of a simple, symmetric and connected graph. Since, number of spanning trees of a graph is asymptotically exponential it is our endeavor to generate, all trees in reasonable amount of time and space[10]. The method here is qualitatively and quantitatively better than existing methods. The reason behind the claim is minimum number of duplicate tree comparison and no circuit testing at all for its realization[10,4,5,7,8,9,12,13]. We are hopeful that betterment of the algorithm lies in the target of no duplicate tree generation.

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5 citations