On Single-Objective Sub-Graph-Based Mutation for Solving the Bi-Objective Minimum Spanning Tree Problem.

In a complete graphwith independent uniform(or exponential) edge weights, letbe the minimum‐weight spanning tree (MST), andthe MST after deleting the edges of all previous trees. We show that each tree's weightconverges in probability to a constant, with, and we conjecture that. The problem is distinct from Frieze and Johansson's minimum combined weightofkedge‐disjoint spanning trees; indeed,. With an edge of weightw“arriving” at time, Kruskal's algorithm defines forests, initially empty and eventually equal to, each edge added to the first possible. Using tools of inhomogeneous random graphs we obtain structural results including that the fraction of vertices in the largest component ofconverges to some. We conjecture that the functionstend to time translations of a single function.

We present a new bicriteria approximation algorithm for the degree-bounded minimum-cost spanning tree (MST) problem: Given an undirected graph with nonnegative edge weights and a degree bound B, find a spanning tree of maximum node-degree B and minimum total edge-cost. Our algorithm outputs a tree of maximum degree at most a constant times B and total edge-cost at most a constant times that of a minimum-cost degree-B-bounded spanning tree. While our new algorithm is based on ideas from Lagrangian relaxation, as is our previous work [SIAM J. Comput., 31 (2002), pp. 1783--1793], it does not rely on computing a solution to a linear program. Instead, it uses a repeated application of Kruskal's MST algorithm interleaved with a combinatorial update of approximate Lagrangian node-multipliers maintained by the algorithm. These updates cause subsequent repetitions of the spanning tree algorithm to run for longer and longer times, leading to overall progress and a proof of the performance guarantee. A second useful feature of our algorithm is that it can handle nonuniform degree bounds on the nodes: Given distinct bounds Bv for every node $v \in V$, the output tree has degree at most O(Bv + log|V|) for every $v \in V$. As before, the cost of the tree is at most a constant times that of a minimum-cost tree obeying all degree bounds.

In this paper, we study undirected multiple graphs of any natural multiplicity $$k > 1$$ . There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of $$k$$ linked edges, which connect 2 or $$k + 1$$ vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges, and it can be the common ending vertex to $$k$$ linked edges of a multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of any other multi-edge. Special attention is paid to the class of divisible multiple graphs. The main peculiarity of them is a possibility to divide the graph into $$k$$ parts, which are adjusted on the linked edges and which have no common edges. Each part is an ordinary graph. The definition of a multiple tree is stated and the basic properties of such trees are studied. Unlike ordinary trees, the number of edges in a multiple tree is not fixed. In the article, the estimation of the minimum and maximum number of edges in the divisible tree is stated and proved. Next, the definitions of the spanning tree and the complete spanning tree of a multiple graph are given. The criterion of completeness of the spanning tree is proved for divisible graphs. It is also proved that a complete spanning tree exists in any divisible graph. If the multiple graph is weighted, the minimum spanning tree problem and the minimum complete spanning tree problem can be set. In the article, we suggest a heuristic algorithm for the minimum complete spanning tree problem for a divisible graph.

The addition of new houses in the Kota Mas Cimahi housing complex has triggered an increase in the need for electricity distribution installations. To optimize the electricity distribution network, in this case minimizing the installation of electrical cables, an application is needed that can optimize the installation of electrical cables, namely the Kruskal algorithm which is an algorithm for finding the minimum spanning tree value on a weighted graph. In this study, two methods will be used, namely the manual method and the POM-QM for Windows software to help solve the minimum spanning tree problem. This study's goal is to identify the Kota Mas Cimahi housing complex's energy distribution network model's graph structure and determine a minimum spanning tree solution with POM-QM for Windows software and the Kruskal technique done by hand. In order to gather data for this study, PLN was the primary and secondary source Cimahi UP3. The distribution data for the electricity cable network is arranged in the form of a network image. By representing the electricity distribution network in the form of a graph, the minimum spanning tree is then obtained, and the solution to the MST (Minimum Spanning Tree) problem uses the Kruskal algorithm and TORA software. The results of the study showed that the minimum spanning tree cable length which was originally 1424.61 meters, after using the Kruskal algorithm manually became 1258.05 meters and also by using the POM-QM for Windows software obtained a minimum result of 1258.05 meters, so it can be concluded that there is a cable length efficiency of 11.69%.

In a previous article, using underlying graph theoretical properties, Gouveia and Magnanti (2003) described several network flow‐based formulations for diameter‐constrained tree problems. Their computational results showed that, even with several enhancements, models for situations when the tree diameter D is odd proved to be more difficult to solve than those when D is even. In this article we provide an alternative modeling approach for the situation when D is odd. The approach views the diameter‐constrained minimum spanning tree as being composed of a variant of a directed spanning tree (from an artificial root node) together with two constrained paths, a shortest and a longest path, from the root node to any node in the tree. We also show how to view the feasible set of the linear programming relaxation of the new formulation as the intersection of two integer polyhedra, a so‐called triangle‐tree polyhedron and a constrained path polyhedron. This characterization improves upon a model of Gouveia and Magnanti (2003) whose linear programming relaxation feasible set is the intersection of three rather than two integer polyhedra. The linear programming gaps for the tightened model are very small, typically less than 0.5%, and are usually one third to one tenth of the gaps of the best previous model described in Gouveia and Magnanti (2003). Moreover, using the new model, we have been able to solve large Euclidean problem instances that are not solvable by the previous approaches. © 2004 Wiley Periodicals, Inc.

This paper studies the minimum spanning tree problem on a graph with uncertain edge weights, which are formulated as uncertain variables. The concept of ideal uncertain minimum spanning tree (ideal UMST) is initiated by extending the definition of the uncertain [Formula: see text]-minimum spanning tree to reect the overall properties of the α-minimum spanning tree weights at any confidence level [Formula: see text]. On the basis of this new concept, the definition of uncertain distribution-minimum spanning tree is proposed in three ways. Particularly, by considering the tail value at risk from the perspective of risk management, the notion of uncertain [Formula: see text]-distribution-minimum spanning tree ([Formula: see text]-distribution-UMST) is suggested. It is shown that the [Formula: see text]-distribution-UMST is just the uncertain expected minimum spanning tree when [Formula: see text] = 0. For any [Formula: see text], this problem can be effectively solved via the proposed deterministic graph transformation-based approach with the aid of the [Formula: see text]-distribution-path optimality condition. Furthermore, the proposed definitions and solutions are illustrated by some numerical examples.

Motivated by the increasing need for fast distributed processing of large-scale graphs such as the Web graph and various social networks, we study a number of fundamental graph problems in the message-passing model, where we have k machines that jointly perform computation on an arbitrary n-node (typically, n > k) input graph. The graph is assumed to be randomly partitioned among the k ≥ 2 machines (a common implementation in many real world systems). The communication is point-to-point, and the goal is to minimize the time complexity i.e., the number of communication rounds, of solving various fundamental graph problems.We present lower bounds that quantify the fundamental time limitations of distributively solving graph problems. We first show a lower bound of Ω(n/k) rounds for computing a spanning tree (ST) of the input graph. This result also implies the same bound for other fundamental problems such as computing a minimum spanning tree (MST), breadth-first tree (BFS), and shortest paths tree (SPT). We also show an Ω(n/k2) lower bound for connectivity ST verification and other related problems. Our lower bounds develop and use new bounds in random-partition communication complexity.To complement our lower bounds, we also give algorithms for various fundamental graph problems, e.g., PageRank, MST, connectivity, ST verification, shortest paths, cuts, spanners, covering problems, densest subgraph, subgraph isomorphism, finding triangles, etc. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved in O(n/k) time (the notation O hides polylog(n) factors and an additive polylog(n) term); this shows that one can achieve almost linear (in k) speedup, whereas for shortest paths, we present algorithms that run in O(n/Ok) time (for (1 + e)-factor approximation) and in O(n/k) time (for O(log n)-factor approximation) respectively.Our results are a step towards understanding the complexity of distributively solving large-scale graph problems.

A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in $\R^d$. Tightness of the distribution, as $\delta \to 0$, is established for the following two-dimensional examples: the uniformly random spanning tree on $\delta \Z^2$, the minimal spanning tree on $\delta \Z^2$ (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in $\R^2$ with density $\delta^{-2}$. In each case, sample trees are proven to have the following properties, with probability one with respect to any of the limiting measures: i) there is a single route to infinity (as was known for $\delta > 0$), ii) the tree branches are given by curves which are regular in the sense of H\"older continuity, iii) the branches are also rough, in the sense that their Hausdorff dimension exceeds one, iv) there is a random dense subset of $\R^2$, of dimension strictly between one and two, on the complement of which (and only there) the spanning subtrees are unique with continuous dependence on the endpoints, v) branching occurs at countably many points in $\R^2$, and vi) the branching numbers are uniformly bounded. The results include tightness for the loop erased random walk (LERW) in two dimensions. The proofs proceed through the derivation of scale-invariant power bounds on the probabilities of repeated crossings of annuli.

A method to design resilient protocols for spanning tree problems is proposed. The method consists of a generic resilient protocol that can be used for any spanning tree problem. A resilient protocol for a particular spanning tree problem can be obtained by adjusting various run-time parameters of the generic protocol. The generic protocol is designed by using the concepts of replacement sets that contain the links that can be used to replace a link that fails and replacement values that measure the change in the spanning tree when a link in a replacement set is used to recreate a spanning tree. The approach is as general as the common approach (for the case of spanning trees) because the generic protocol can solve a new spanning tree problem without an individual design. The protocols obtained with this method are efficient and easy to implement and verify because they are all derived from the generic protocol. >

Degree-bounded minimum spanning trees

The minimum spanning tree (MST), a graph constructed from a distribution of points, draws lines between pairs of points so that all points are linked in a single skeletal structure that contains no loops and has minimal total edge length. The MST has been used in a broad range of scientific fields such as particle physics (to distinguish classes of events in collider collisions), in astronomy (to detect mass segregation in star clusters) and cosmology (to search for filaments in the cosmic web). Its success in these fields has been driven by its sensitivity to the spatial distribution of points and the patterns within. MiSTree, a public Python package, allows a user to construct the MST in a variety of coordinates systems, including Celestial coordinates used in astronomy. The package enables the MST to be constructed quickly by initially using a k-nearest neighbour graph (kNN, rather than a matrix of pairwise distances) which is then fed to Kruskal's algorithm to construct the MST. MiSTree enables a user to measure the statistics of the MST and provides classes for binning the MST statistics (into histograms) and plotting the distributions. Applying the MST will enable the inclusion of high-order statistics information from the cosmic web which can provide additional information to improve cosmological parameter constraints. This information has not been fully exploited due to the computational cost of calculating N-point statistics. MiSTree was designed to be used in cosmology but could be used in any field which requires extracting non-Gaussian information from point distributions. The source code for MiSTree is available on GitHub at https://github.com/knaidoo29/mistree

Rectilinear Steiner spanning trees are used in several phases of the VLSI physical design process when collections of terminals must be connected to a common electrical point. The authors present a new linear time heuristic algorithm for generating these rectilinear Steiner trees. They begin with minimal spanning trees and transform those configurations which are never found in the shortest Steiner trees. This decreases the length of the spanning trees. In addition to its speed, the algorithm is straightforward and can be implemented efficiently. Empirical results demonstrate that reductions of over 13% in length are possible, while an average of over 8% reduction is achieved. This compares very favorably to much more complex algorithms. The algorithm is recommended for applications where many spanning trees are needed in a very short time. >

The Minimum Conflict Weighted Spanning Tree Problem is a variant of the Minimum Spanning Tree Problem in which, given a list of conflicting edges modelled as a conflict graph, we want to find a weighted spanning tree with the minimum number of conflicts as main objective function and minimize the total weight of spanning trees as secondary objective function. The problem is proved to be NP-Hard in its general form and finds applications in several real-case scenarios such as the modelling of road networks in which some movements are prohibited. We propose a genetic algorithm designed to minimize the number of conflict edge pairs and the total weight of the spanning tree. We tested our approach on benchmark instances, the results of our GA showed that we outperform the other approaches proposed in the literature.

Kruskal’s Algorithm is an algorithm used to find the minimum spanning tree in graphical connectivity that provides the option to continue processing the least-weighted margins. In the Kruskal algorithm, ordering the weight of the ribs makes it easy to find the shortest path. This algorithm is independent in nature which will facilitate and improve path creation. Based on the results of the application system trials that have been carried out in testing and comparisons between the Kruskal algorithm and the Dijkstra algorithm, the following conclusions can be drawn: that a strength that is the existence of weight sorting will facilitate the search for the shortest path. And considering the characteristics of Kruskal’s independent algorithm, it will facilitate and improve the formation of the path. The weakness of the Kruskal algorithm is that if the number of nodes is very large, it will be slower than Dijkstra’s algorithm because it has to sort thousands of vertices first, then form a path.

Minimum Spanning Trees (MSTs) are ubiquitous in optimization problems in networks. Even though fast algorithms exist to solve the MST problem, real world applications are usually subject to constraints that do not let us apply such methods directly. In these cases we confront a version of the MST called the “Weighted Spanning Tree” (WST) in which we look for a spanning tree in a graph that satisfies other side constraints and is of minimum cost. In this paper we implement this constraint using a lower bound and learning to accelerate the search and thus reduce the solving time. We show that having this propagator is tremendously beneficial for solvers and we show the benefits of learning.