A Flexible Algorithm for Generating All the Spanning Trees in Undirected Graphs

Abstract

In this paper we propose an algorithm for generating all the spanning trees in undirected graphs. The algorithm requires O(n + m + τ n) time where the given graph has n vertices, m edges, and τ spanning trees. For outputting all the spanning trees explicitly, this time complexity is optimal.Our algorithm follows a special rooted tree structure on the skeleton graph of the spanning tree polytope. The rule by which the rooted tree structure is traversed is irrelevant to the time complexity. In this sense, our algorithm is flexible.If we employ the depth-first search rule, we can save the memory requirement to O(n + m). A breadth-first implementation requires as much as O(m + τ n) space, but when a parallel computer is available, this might have an advantage. When a given graph is weighted, the best-first search rule provides a ranking algorithm for the minimum spanning tree problem. The ranking algorithm requires O(n + m + τ n) time and O(m + τ n) space when we have a minimum spanning tree. Introduction.In this paper we propose an algorithm which generates all the spanning trees explicitly in O(n + m + τ n) time where the given graph has n vertices, m edges, and τ spanning trees. For outputting all the spanning trees explicitly, the time complexity is optimal.Our algorithm generates all the spanning trees by following one-dimensional faces of the spanning tree polytope (the convex hull of the characteristic vectors of spanning trees). More precisely, our algorithm traverses a tree structure (a rooted spanning tree) on the skeleton graph of the spanning tree polytope. From the above, our algorithm can be considered as a specialization of the reverse search method proposed by Avis and Fukuda [4], which is a general scheme for solving enumeration problems.Our algorithm has an advantage in that the tree search rule by which we traverse the tree structure on the skeleton graph is irrelevant to the time complexity. Thus, various algorithms with O(n + m + τ n) time complexity can be easily constructed by using different tree search rules. In this sense, our algorithm is flexible. For example, employing the depth-first search rule we obtain an algorithm whose space complexity is O(n + m). A breadth-first implementation requires as much as O(m +τ n) space, but when a parallel computer is available, this might have an advantage. The two tree search rules can even be combined to construct an algorithm which runs parallel but requires a fixed amount of memory storage. When a weighted graph is given, our algorithm generates all the spanning trees in nondecreasing order of weight by employing best-first search rule. The

Algorithms for generating all possible spanning trees of a simple undirected connected graph: an extensive review

“…However, some logic has to be applied to avoid duplicate tree generation. This method of generating spanning trees has been studied by several researchers [6,10,11,13,17,21]. These algorithms have been discussed in the following sections.…”

“…The algorithm was developed in 1993 by Matsui [11]. The algorithm starts with creating an initial spanning tree T by doing Breadth First Traversal or Depth First Traversal of the given input graph G. After that, edges are renumbered to create a linear ordering of edge-set such that branches are numbered as e 1 , e 2 , e 3 , …, e n − 1 and chords are numbered as e n , e n + 1 , e n + 2 , …, e m .…”

“…So, there are many different viable approaches, unconquered, to analyze the existing algorithms No. of trees generated Hakimi [6] Kapoor and Ramesh [10] Matsui [11] Mayeda and Seshu [13] Naskar et al…”

“…Various efficient algorithms for generating all spanning trees of a graph have been proposed by several researchers [1,3,5,6,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. The algorithms [3, 6, 10, 11, 13, 15-17, 19-21, 25] have been reviewed, explained, and compared in Sect.…”

Listing all the minimum spanning trees in an undirected graph

“…P : List all the MSTs in graph G. Related to this problem are the algorithms to list all the spanning trees [5,9,11], do the same in non-decreasing order of cost [3,13] and find K-shortest spanning trees in a graph [2,8]. Indeed, P may be solved by finding all the spanning trees, or more preferably by applying a K-shortest 3176 T. Yamada et al spanning tree algorithm with sufficiently large K and truncating its execution as soon as we have a spanning tree of weight larger than z .…”