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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Journal Article•DOI•

Generation of Trees without Duplications

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W. Mayeda¹, S. Seshu¹•Institutions (1)

University of Illinois at Urbana–Champaign¹

01 Jun 1965-IEEE Transactions on Circuit Theory

TL;DR: A practical procedure satisfying these conditions is presented and it is shown that for practical networks the number of trees exceeds the fast memory of any computer and, hence, slows down search for duplicates.

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Abstract: Recent emphasis on the use of electronic digital computers has focused attention on a number of practical problems which were hitherto treated on an "existence" basis. One problem which arises in computer analysis of electrical networks is the listing of all trees in the network. Any practically useful scheme must find all the trees of the network without generating duplicates (for practical networks the number of trees exceeds the fast memory of any computer and, hence, slows down search for duplicates). Also, the trees must be generated by replacement of one branch at a time in order to keep track of signs (for active network analysis). A practical procedure satisfying these conditions is presented in this paper.

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91 citations

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

...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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...[16] W. Mayeda and S. Seshu, ‘‘Generation of trees without duplications,’’ IEEE Trans....
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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 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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Journal Article•DOI•

A Simple Algorithm for Listing All the Trees of a Graph

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G. Minty¹•Institutions (1)

University of Michigan¹

01 Mar 1965-IEEE Transactions on Circuit Theory

86 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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...Algorithms falling under this category are by Minty [19], Smith [20], Winter [21], and so on....
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...[19] G. Minty, ‘‘A simple algorithm for listing all the trees of a graph,’’ IEEE Trans....
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Journal Article•DOI•

A note on the enumeration and listing of all possible trees in a connected linear graph.

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Horace M. Trent¹•Institutions (1)

United States Naval Research Laboratory¹

01 Oct 1954-Proceedings of the National Academy of Sciences of the United States of America

81 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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Journal Article•DOI•

Generation of Trees, Two-Trees, and Storage of Master Forests

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J. Char

01 Sep 1968-IEEE Transactions on Circuit Theory

TL;DR: While an edge-numbering convention and a criterion for a tree play the key roles in systematic generation of trees, the storage technique makes it possible to obtain all the co-factors and determinants of a node-admittance matrix of any network by merely operating on one single master forest matrix.

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Abstract: A new method of listing all possible trees of any given graph without duplication or redundancy, using simple geometrical properties of the graph, is proposed. The procedure given is suitable for both manual and automatic computation, and any modifications to the given graph can be catered to by suitable interpolation and extrapolation. An alternate method for complete graphs, derived from it, gives the trees arranged in an order most suitable for their storage as master forest matrices and for directly obtaining trees and 2-trees of any given graph through simple modifications to them instead of starting from scratch every time. Some properties of master forest matrices are discussed, which., inter alia, lead to a formula for the number of trees in a sub-graph of a complete graph. While an edge-numbering convention and a criterion for a tree play the key roles in systematic generation of trees, the storage technique makes it possible to obtain all the co-factors and determinants of a node-admittance matrix of any network (within the range of storage) by merely operating on one single master forest matrix.

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42 citations

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

...[2] J. Char, ‘‘Generation of trees, two-trees, and storage of master forests,’’ 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 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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...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•

Efficiently scanning all spanning trees of an undirected graph

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Akiyoshi Shioura, Akihisa Tamura

01 Sep 1995-Journal of The Operations Research Society of Japan

TL;DR: In this article, the problem of enumerating all spanning trees of an undirected graph with V vertices and E edges was considered and a new algorithm was proposed for the problem with time complexity O(N + V + E) and space complexities O(V E).

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Abstract: Let G be an undirected graph with V vertices and E edges. We consider the problem of enumerating all spanning trees of G: In order to explicitly output all spanning trees, the output size is of (NV ), where N is the number of spanning trees. This, however, can be compressed into (N) size. In this paper, we propose a new algorithm for enumerating all spanning trees of G in such compact form. The time and space complexities of our algorithm are O(N + V + E) and O(V E), respectively. The algorithm is optimal in the sense of time complexity.

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39 citations