Two Algorithms for Computing All Spanning Trees of a Simple, Undirected, and Connected Graph: Once Assuming a Complete Graph
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.
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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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.
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.
3 citations
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.
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.
2 citations
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Book•
01 Jan 1974
1,204 citations
01 Oct 1975
TL;DR: Graph Theory and Its Applications to Problems of Society and its Applications to Algorithms and Computer Science.
Abstract: Introductory Graph Theory with ApplicationsGraph Theory with ApplicationsResearch Topics in Graph Theory and Its ApplicationsChemical Graph TheoryMathematical Foundations and Applications of Graph EntropyGraph Theory with Applications to Engineering and Computer ScienceGraphs Theory and ApplicationsQuantitative Graph TheoryApplied Graph TheoryChemical Graph TheoryA First Course in Graph TheoryGraph TheoryGraph Theory with ApplicationsGraph Theory with ApplicationsSpectra of GraphsFuzzy Graph Theory with Applications to Human TraffickingApplications of Graph TheoryChemical Applications of Graph TheoryRecent Advancements in Graph TheoryA Textbook of Graph TheoryGraph Theory and Its Engineering ApplicationsGraph Theory, Combinatorics, and ApplicationsAdvanced Graph Theory and CombinatoricsTopics in Intersection Graph TheoryGraph Theory with Applications to Engineering and Computer ScienceGraph Theory and Its Applications, Second EditionHandbook of Research on Advanced Applications of Graph Theory in Modern SocietyGraph Theory with Applications to Algorithms and Computer ScienceGraph TheoryGraph Theory with Algorithms and its ApplicationsGraph TheoryGraph Theory with ApplicationsGraph Theory ApplicationsHandbook of Graph TheoryGraph Theory and Its Applications to Problems of SocietyBasic Graph Theory with ApplicationsTen
809 citations
"Two Algorithms for Computing All Sp..." refers background in this paper
...Sir Arthur Cayley proposed a formula which states, if Tn is the number of trees of a complete graph of n vertices, then Tn = nn−2 [22]–[24]....
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TL;DR: An algorithm for finding all spanning trees (arborescences) of a directed graph is presented that uses backtracking and a method for detecting bridges based on depth-first search.
Abstract: An algorithm for finding all spanning trees (arborescences) of a directed graph is presented. It uses backtracking and a method for detecting bridges based on depth-first search. The time required is $O(V + E + EN)$ and the space is $O(V + E)$, where V, E, and N represent the number of vertices, edges, and spanning trees, respectively. If the graph is undirected, the time decreases to $O(V + E + VN)$, which is optimal to within a constant factor. The previously best-known algorithm for undirected graphs requires time $O(V + E + EN)$.
193 citations
"Two Algorithms for Computing All Sp..." refers methods in this paper
...[3] H. N. Gabow and E. W. Myers, ‘‘Finding all spanning trees of directed and undirected graphs,’’ SIAM J. Comput., vol. 7, no. 3, pp. 280–287, 1978....
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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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TL;DR: Algorithms for enumeration of spanning trees in undirected graphs, with and without weights, are presented, based on swapping edges in a fundamental cycle to construct a computation tree.
Abstract: In this paper, we present algorithms for enumeration of spanning trees in undirected graphs, with and without weights.
The algorithms use a search tree technique to construct a computation tree. The computation tree can be used to output all spanning trees by outputting only relative changes between spanning trees rather than the entire spanning trees themselves. Both the construction of the computation tree and the listing of the trees is shown to require $O(N+V+E)$ operations for the case of undirected graphs without weights. The basic algorithm is based on swapping edges in a fundamental cycle. For the case of weighted graphs (undirected), we show that the nodes of the computation tree of spanning trees can be sorted in increasing order of weight, in $O(N\log V+VE)$ time. The spanning trees themselves can be listed in $O(NV)$ time. Here $N$, $V$, and $E$ refer respectively to the number of spanning trees, vertices, and edges of the graph.
123 citations
"Two Algorithms for Computing All Sp..." refers methods in this paper
...[13] S. Kapoor and H. Ramesh, ‘‘Algorithms for enumerating all spanning trees of undirected and weighted graphs,’’ SIAM J. Comput., vol. 24, no. 2, pp. 247–265, 1995....
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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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TL;DR: The Shioura and Tamura algorithm is optimal in the sense of both time and space complexities because it decreases the space complexity from O(VE) to O(V + E) while preserving the time complexity.
Abstract: Let G be an undirected graph with V vertices and E edges. Many algorithms have been developed for enumerating all spanning trees in G. Most of the early algorithms use a technique called "backtracking." Recently, several algorithms using a different technique have been proposed by Kapoor and Ramesh (1992), Matsui (1993), and Shioura and Tamura (1993). They find a new spanning tree by exchanging one edge of a current one. This technique has the merit of enabling us to compress the whole output of all spanning trees by outputting only relative changes of edges. Kapoor and Ramesh first proposed an O(N + V + E)-time algorithm by adopting such a "compact" output, where N is the number of spanning trees. Another algorithm with the same time complexity was constructed by Shioura and Tamura. These are optimal in the sense of time complexity but not in terms of space complexity because they take O(VE) space. We refine Shioura and Tamura's algorithm and decrease the space complexity from O(VE) to O(V + E) while preserving the time complexity. Therefore, our algorithm is optimal in the sense of both time and space complexities.
111 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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