Tree Of Undirected Graph Research Articles

A Novel Genetic Algorithm Based on all Spanning Trees of Undirected Graph for Distribution Network Reconfiguration

Network reconfiguration is of theoretical and practical significance to guarantee safe and economical operation of distribution system. In this paper, based on all spanning trees of undirected graph, a novel genetic algorithm for electric distribution network reconfiguration is proposed. Above all, all spanning trees of simplified graph of distribution network are found. Tie branches are obtained with spanning tree subtracted from simplified graph. There is one and only one switch open on each tie branch. Decimal identity number of open switch on each tie branch is taken as the optimization variable. Therefore, length of chromosome is very short. Each spanning tree corresponds to one subpopulation. Gene operations of each subpopulation are implemented with parallel computing method. Individuals of offspring after gene operation automatically meet with radial and connected constraints for distribution network operation. Disadvantages of conventional genetic algorithm for network reconfiguration that a large amount of unfeasible solutions are created after crossover and mutation, which result in very low searching efficiency, are completely overcome. High calculation speed and superior capability of the proposed method are validated by two test cases.

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

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.

On the approximability of some maximum spanning tree problems

We study the approximability of some problems which aim at finding spanning trees in undirected graphs which maximize, rather than minimize, a single objective function representing a form of benefit or usefulness of the tree. We prove that the problem of finding a spanning tree which maximizes the number of paths which connect pairs of vertices and pass through a common arc can be polynomially approximated within 9 8 . It is known that this problem can be solved exactly in polynomial time if the graph is 2-connected; we extend this result to graphs having at most two articulation points. We leave open whether in the general case the problem admits a polynomial time approximation scheme or is MAX-SNP hard and therefore not polynomially approximable within 1 + ε, for any fixed ε > 0, unless P = NP. On the other hand, we show that the problems of finding a spanning tree which has maximum diameter, or maximum height with respect to a specified root, or maximum sum of the distances between all pairs of vertices, or maximum sum of the distances from a specified root to all remaining vertices, are not polynomially approximable within any constant factor, unless P = NP. The same result holds for the problem of finding a lineal spanning tree with maximum height, and this solves a problem which was left open in Fellows et al. (1988).

Algorithms for Enumerating All Spanning Trees of Undirected and Weighted Graphs

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 $O(N \log V + V E)$ 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.

Constant time algorithms for the transitive closure and some related graph problems on processor arrays with reconfigurable bus systems

The transitive closure problem in O(1) time is solved by a new method that is far different from the conventional solution method. On processor arrays with reconfigurable bus systems, two O(1) time algorithms are proposed for computing the transitive closure of an undirected graph. One is designed on a three-dimensional n*n*n processor array with a reconfigurable bus system, and the other is designed on a two-dimensional n/sup 2/*n/sup 2/ processor array with a reconfigurable bus system, where n is the number of vertices in the graph. Using the O(1) time transitive closure algorithms, many other graph problems are solved in O(1) time. These problems include recognizing bipartite graphs and finding connected components, articulation points, biconnected components, bridges, and minimum spanning trees in undirected graphs.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>