Innovative method to solve the minimum spanning tree problem: The Dhouib-Matrix-MSTP (DM-MSTP)

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 consider the number c(n, m) of connected labeled graphs on n nodes and m edges and the intimately related object, the expected length of the minimal spanning tree of a complete graphs with random edge lengths. We use a very simple recursive procedure for computing the values of c(n, m) for computing the expected length of the minimal spanning tree exactly, under the uniform and the exponential distributions. Our computations are recursive, scale very well with the size of the problem, and we provide the values of the expected minimal length spanning trees for complete graphs Kn with sizes n ≤ 45, extending recent results of Steele [Ste02], and Fill and Steele [FS04]. The main proof technique is based on introducing an artificial root to a graph and subsequently using a very simple inductive argument.

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.

Remmel and Williamson recently defined a class of directed graphs, called filtered digraphs, and described a natural class of bijections between oriented spanning forests of these digraphs and associated classes of functions [12]. Filtered digraphs include many specialized graphs such as complete k-partite graphs. The Remmel-Williamson bijections provide explicit formulas for various multivariate generating functions for the oriented spanning forests which arise in this context. In this paper, we prove another important property of these bijections, namely, that it allows one to construct efficient algorithms for ranking and unranking spanning trees or spanning forests of filtered digraphs G. For example, we show that if G=(V,E) is a filtered digraph and SP(G) is the collection of spanning trees of G, then our algorithm requires O(|V|) operations of sum, difference, product, quotient, and comparison of numbers less than or equal |SP(G)| to rank or unrank spanning trees of G.

Degree-bounded minimum spanning trees

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.

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. >

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 minimum spanning tree problem with non-terminal set

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.

Complexities of some interesting problems on spanning trees

We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline geometric MST problem for a planar point set subject to insertions and deletions; our algorithm for this problem performs O(log2n) work per modification. No previous dynamic geometric MST algorithm was known.

This study’s objective is to assess the performance of routing processing between Shortest Paths Tree (SPT) and Minimum Spanning Tree (MST). In this paper we propose and create software to calculate the efficiency path of routing processing in the network. Extensive simulations are being carried out with two different algorithms specific for Dijkistra’s algorithm and Kruskal’s algorithm. The network loads in the performance plots are characterized by two important parameters (a number of vertices) - 7 nodes and 10 nodes, more vertices could show good performance in the simulation results and showed that the performance of Kruskal’s algorithm is better than Dijkistra’s algorithm when the network loads are increased from 7 nodes to 10 nodes. In this simulation, Because of the total cost of distance matrix for Kruskal is less than Dijkistra when the number of routers or nodes is increased from 7 nodes to 10 nodes. This software namely Network Aids Design (NAD), is useful for calculation about the performance of routing processing in the network.