Maintaining a minimum spanning tree under transient node failures (original) (raw)

Nearly linear time minimum spanning tree maintenance for transient node failures

2004

Abstract Given a 2-node connected, real weighted, and undirected graph G=(V,E), with n nodes and m edges, and given a minimum spanning tree (MST) T=(V,E_T) of G, we study the problem of finding, for every node v∈V, a set of replacement edges which can be used for constructing an MST of Gv (ie, the graph G deprived of v and all its incident edges). We show that this problem can be solved on a pointer machine in \calO(m⋅α(m,n)) time and \calO(m) space, where α is the functional inverse of Ackermann's function.

Distributed Minimum Spanning Tree Maintenance for Transient Node Failures

IEEE Transactions on Computers, 2012

In many network applications, the computation takes place on the minimum-cost spanning tree (MST) of the network G; unfortunately, a single link or node failure disconnects the tree. The ALL NODES REPLACEMENT (ANR) problem is the problem of precomputing, for each node u in G, the new MST should u fail. This problem has been extensively investigated for serial and

A Linear Time Algorithm for Finding Minimum Spanning Tree Replacement Edges

ArXiv, 2019

Given an undirected, weighted graph, the minimum spanning tree (MST) is a tree that connects all of the vertices of the graph with minimum sum of edge weights. In real world applications, network designers often seek to quickly find a replacement edge for each edge in the MST. For example, when a traffic accident closes a road in a transportation network, or a line goes down in a communication network, the replacement edge may reconnect the MST at lowest cost. In the paper, we consider the case of finding the lowest cost replacement edge for each edge of the MST. A previous algorithm by Tarjan takes O(malpha(m,n))O(m \alpha(m, n))O(malpha(m,n)) time, where alpha(m,n)\alpha(m, n)alpha(m,n) is the inverse Ackermann's function. Given the MST and sorted non-tree edges, our algorithm is the first that runs in O(m+n)O(m+n)O(m+n) time and O(m+n)O(m+n)O(m+n) space to find all replacement edges. Moreover, it is easy to implement and our experimental study demonstrates fast performance on several types of graphs. Additionally, since the most vital edg...

A New Approach to Find a Minimum Spanning Tree (NAFMST)

While forming reliable communication networks, we must guarantee that, after failure of a node or links, the surviving network still allows communication between all other nodes by choosing alternate path which gives strict requirement on the connectivity of the corresponding graph. A general network design problem which requires the underlying network to be resilient to link failures is known as the edge-connectivity survivable network design problem. In this work we present a new approach to find a Minimum Spanning Tree (NAFMST) used to parallelize efficiently Kruskal's Minimum Spanning Tree algorithm. This algorithm is known for exhibiting inherently sequential Characteristics. More specifically, the strict order by which the algorithm checks the edges of a given graph is the main reason behind the lack of explicit parallelism. Our proposed scheme attempts to overcome the imposed restrictions and improve the performance of the algorithm.

Distributed computation of all node replacements of a minimum spanning tree

In many network applications the computation takes place on the minimum-cost spanning tree (M ST ) of the network G; unfortunately, a single link or node failure disconnects the tree. The All Node Replacements (ANR) problem is the problem of precomputing, for each node u in G, the new M ST should u fail. The simpler problem dealing with single edge failures is called All Edge Replacements (AER). Both problems have been extensively investigated for serial and parallel settings, and efficient solutions have been designed for those environments. The situation is surprisingly different in distributed settings. In fact, for either problem, no distributed solution exists to date which performs better than the brute-force repeated application of a M ST construction protocol. In other words, currently, the solution of either problems uses O(nm + n 2 log n) messages, where n is the number of nodes and m is the number of edges. In this paper we consider the problem of computing all the replacement minimum-cost spanning trees distributively, and we efficiently solve the problem. We design a solution protocol for the ANR problem and we prove that the total amount of data items communicated during the computation (the data complexity) is O(n 2 ). This communication can be achieved transmitting only O(n) long messages between neighbours, if the system so allows; otherwise O(n 2 ) standard messages suffice. Even in systems that do not allow long messages, the proposed protocol constitutes a significant improvement over the individual computation of the replacement trees. Indeed, for dense graphs, our protocol constructs all the n replacement M ST s of the minimum spanning tree T with the same number of messages required just to compute T .

Fully Sequential and Distributed Dynamic Algorithms for Minimum Spanning Trees

arXiv preprint cs/0002005, 2000

In this paper, we present a fully-dynamic distributed algorithm for maintaining a minimum spanning tree on general graphs with positive real edge weights. The goal of a dynamic MST algorithm is to update efficiently the minimum spanning tree after dynamic changes like edge weight changes, rather than having to recompute it from scatch each time. The first part of the paper surveys various algorithms available today both in sequential and distributed environments to solve static MST problem. We also present some of the efficient sequential algorithms for computing dynamic MST like the Frederickson's algorithm and Eppstein's sparsification technique. Lastly we present our new sequential and distributed algorithms for dynamic MST problem. To our knowledge, this is the first of the distributed algorithms for computing dynamic MSTs.

Optimal algorithms for the single and multiple vertex updating problems of a minimum spanning tree

Algorithmica, 1996

The vertex updating problem for a minimum spanning tree (MST) is de ned as follows: Given a graph G = (V; E G ) and an MST T for G, nd a new MST for G to which a new vertex z has been added along with weighted edges that connect z with the vertices of G. We present a set of rules that produce simple optimal parallel algorithms that run in O(lg n) time using n= lg n EREW PRAM processors, where n = jV j. These algorithms employ any valid tree-contraction schedule that can be produced within the stated resource bounds. These rules can also be used to derive simple linear-time sequential algorithms for the same problem. The previously best known parallel result was a rather complicated algorithm that used n processors in the more powerful CREW PRAM model. Furthermore, we show how our solution can be used to solve the multiple vertex updating problem: Update a given MST when k new vertices are introduced simultaneously. This problem is solved in O(lg k lg n) parallel time using k n lg k lg n EREW PRAM processors. This is optimal for graphs having (kn) edges.

Optimal maintenance of a spanning tree

Journal of the ACM, 2008

Those who cannot remember the past are condemned to repeat it." (George Santayana) In this paper, we show that keeping track of history enables significant improvements in the communication complexity of dynamic network protocols. We present a communication optimal maintenance of a spanning tree in a dynamic network. The amortized (on the number of topological changes) message complexity is O(V ), where V is the number of nodes in the network. The message size used by the algorithm is O(log |ID|) where |ID| is the size of the name space of the nodes. Typically, log |ID| = O(log V ).

Optimal algorithms for the vertex updating problem of a minimum spanning tree

Parallel Processing Symposium, 1992

The vertex updating problem for a minimum spanning tree (MST) is de ned as follows: Given a graph G = (V; E G ) and its MST T, update T when a new vertex z is introduced along with weighted edges that connect z with the vertices of G. We present a set of rules that, together with a valid tree-contraction schedule, are used to produce simple optimal parallel algorithms that run in O(lgn) parallel time using n= lgn EREW PRAMs where n = jV j. These rules can also be used to derive simple linear-time sequential algorithms for the same problem. The previously best known parallel result was a rather complicated algorithm that used n processors of the more powerful CREW PRAM model. Furthermore, we show how our solution can be used to solve the multiple vertex updating problem: Update a given MST when k new vertices are introduced simultaneously. This problem is solved in O(lg k lgn) parallel time using k n lg k lgn EREW PRAM processors.

A New Approach to Find a Minimum Spanning Tree (NAFMST) 1 Software Engineer OracleTech Pvt. Ltd

A Simple Distributed Algorithm for the Maintenance of a Spanning Tree

Proceedings VECoS, 2007

This work is devoted to the problem of spanning tree maintenance in the presence of crash failures in a distributed environment using only local knowledge. Using a pre-constructed spanning tree of a k-connected graph, we present a protocol to maintain a ...

Finding the most vital edges with respect to the number of spanning trees

IEEE Transactions on Reliability, 1994

Key Words -Reliability optimization, number of spanning trees, most vital edges, matrix operation Reader Aids -General purpose: Sensitivity analysis of network reliability Special math needed for derivations: Matrix operations, graph Special math needed to use results: Same Results useful to: Reliability analysts, network designers theory, and associated jargon

Fast Self-stabilizing Minimum Spanning Tree Construction

Lecture Notes in Computer Science, 2010

We present a novel self-stabilizing algorithm for minimum spanning tree (MST) construction. The space complexity of our solution is O(log 2 n) bits and it converges in O(n 2) rounds. Thus, this algorithm improves the convergence time of all previously known selfstabilizing asynchronous MST algorithms by a multiplicative factor Θ(n), to the price of increasing the best known space complexity by a factor O(log n). The main ingredient used in our algorithm is the design, for the first time in self-stabilizing settings, of a labeling scheme for computing the nearest common ancestor with only O(log 2 n) bits.

On the PROBABILISTIC MIN SPANNING TREE problem

2010

We study a probabilistic optimization model for min spanning tree, where any vertex v i of the input-graph G(V, E) has some presence probability p i in the final instance G ′ ⊂ G that will effectively be optimized. Suppose that when this "real" instance G ′ becomes known, a spanning tree T , called anticipatory or a priori spanning tree, has already been computed in G and one can run a quick algorithm (quicker than one that recomputes from scratch), called modification strategy, that modifies the anticipatory tree T in order to fit G ′ . The goal is to compute an anticipatory spanning tree of G such that, its modification for any G ′ ⊆ G is optimal for G ′ . This is what we call probabilistic min spanning tree problem. In this paper we study complexity and approximation of probabilistic min spanning tree in complete graphs under two distinct modification strategies leading to different complexity results for the problem. For the first of the strategies developed, we also study two natural subproblems of probabilistic min spanning tree, namely, the probabilistic metric min spanning tree and the probabilistic min spanning tree 1,2 that deal with metric complete graphs and complete graphs with edge-weights either 1, or 2, respectively.

The capacitated minimum spanning tree problem

Tdx, 2013

Along the development of this thesis I've had the pleasure of working with some researchers from whom I learnt a lot. I am specially grateful to Elena Fernández and Mari Albareda for their unconditional support and patience, specially in the most difficult moments. I also want to thank Mauricio Resende for his contributions to do part of this work. Most of this work was developed in the Departament d'Estadística i Investigació Operativa of the Universitat Politècnica de Catalunya. I want to thank the department and all its members for all the facilities and help I found here. Specially to Sonia Navarro and Laura Marí for their help during the thesis deposit. I also want to thank the CONACYT for their financial support. Finally, I want to thank my parents for the gift of life; my brothers and sister for all the time and experiences that we have shared; and all my friends for their support and friendship.

Design of a degree-constrained minimal spanning tree with unreliable links and node outage costs

European Journal of Operational Research, 2004

The degree-constrained minimal spanning tree (DCMST) problem with unreliable links and node outage costs consists of finding links in a network to connect a set of terminal nodes to a central node while minimizing the expected annual expenditure. The number of ports available on each terminal node limits the number of incident links (the degree constraint). Each terminal node in the network has an associated node outage cost, which is the economic cost incurred by the network user whenever that node is disabled due to failure of a link. We formulate this problem as an integerprogramming problem and present a Lagrangian relaxation method which, for each choice of Lagrangian multipliers, provides a lower bound for the optimal objective function value. A subgradient optimization method is used to search for multipliers which yield good lower bounds. A branch exchange heuristic procedure makes modifications to each infeasible solution of the Lagrangian relaxation in order to find good feasible solutions. The quality of these heuristic solutions is estimated using the best obtained lower bounds. Experimental results over a wide range of problem structures show that the branch exchange heuristic method yields verifiably good solutions to this problem.

A self-stabilizing distributed algorithm for minimal spanning tree problem in a symmetric graph

Computers & Mathematics with Applications, 1998

Minimal Spanning Tree (MST) problem in an arbitrary undirected graph is an important problem in graph theory and has extensive applications. Numerous algorithms are available to compute an MST. Our purpose here is to propose a self-stabilizing distributed algorithm for the MST problem and to prove its correctness. The algorithm utilizes an interesting result of [1]. We show the correctness of the proposed algorithm by using a new technique involving induction.