Showing papers on "Prim's algorithm published in 2010"

An improved LP-based approximation for steiner tree

Abstract: The Steiner tree problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to the current best 1.55 [Robins,Zelikovsky-SIDMA'05]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than 2 [Vazirani,Rajagopalan-SODA'99]. In this paper we improve the approximation factor for Steiner tree, developing an LP-based approximation algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directed-component cut relaxation for the k-restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together with a minimum-cost terminal spanning tree in the remaining graph. Our algorithm delivers a solution of cost at most ln(4) times the cost of an optimal k-restricted Steiner tree. This directly implies a ln(4)+e

Independent Spanning Trees on Multidimensional Torus Networks

Abstract: Two spanning trees rooted at vertex r in a graph G are called independent spanning trees (ISTs) if for each vertex v in G, vner, the paths from vertex v to vertex r in these two trees are internally distinct. If the connectivity of G is k, the IST problem is to construct k ISTs rooted at each vertex. The IST problem has found applications in fault-tolerant broadcasting, but it is still open for general graphs with connectivity greater than four. In this paper, we shall propose a very simple algorithm for solving the IST problem on multidimensional torus networks. In our algorithm, every vertex can determine its parent for a specific independent spanning tree only depending on its own label. Thus, our algorithm can also be implemented in parallel systems or distributed systems very easily.

Greedy heuristics for the bounded diameter minimum spanning tree problem

Abstract: Given a connected, weighted, undirected graph G and a bound D, the bounded diameter minimum spanning tree problem seeks a spanning tree on G of minimum weight among the trees in which no path between two vertices contains more than D edges. In Prim's algorithm, the diameter of the growing spanning tree can always be known, so it is a good starting point from which to develop greedy heuristics for the bounded diameter problem. Abdalla, Deo, and Gupta described such an algorithm. It imitates Prim's algorithm but avoids edges whose inclusion in the spanning tree would violate the diameter bound. Running the algorithm from one start vertex requires time that is O(n3).A modification of this approach uses the start vertex as the center of the spanning tree (if D is even) or as one of the two center vertices (if D is odd). This yields a simpler algorithm whose time is O(n2). A further modification chooses each next vertex at random rather than greedily, though it still connects each vertex to the growing tree with the lowest-weight feasible edge. On Euclidean problem instances with small diameter bounds, the randomized heuristic is superior to the two fully greedy algorithms, though its advantage fades as the diameter bound grows. On instances whose edge weights have been chosen at random, the fully greedy algorithms outperform the randomized heuristic.

Metaheuristic approaches for the quadratic minimum spanning tree problem

Abstract: Given an undirected graph with costs associated both with its edges and unordered pairs of edges, the quadratic minimum spanning tree problem asks to find a spanning tree that minimizes the sum of costs of all edges and pairs of edges in the tree. We present multistart simulated annealing, hybrid genetic and iterated tabu search algorithms for solving this problem. We report on computational experiments that compare these algorithms on random graphs of size up to 50 vertices. The results indicate that the iterated tabu search algorithm is superior to the other two approaches in terms of both solution quality and computation time.

FPT algorithms for connected feedback vertex set

Abstract: We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G=(V,E) and an integer k, decide whether there exists F⊆V, |F|≤k, such that G[V∖F] is a forest and G[F] is connected. We show that Connected Feedback Vertex Set can be solved in time O(2O(k)nO(1)) on general graphs and in time $O(2^{O(\sqrt{k}\log k)}n^{O(1)})$ on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses, as a subroutine, a parameterized algorithm for Group Steiner Tree, a well studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree of independent interest and believe that it could be useful for obtaining parameterized algorithms for other connectivity problems.

Fast self-stabilizing minimum spanning tree construction: using compact nearest common ancestor labeling scheme

Abstract: We present a novel self-stabilizing algorithm for minimum spanning tree (MST) construction. The space complexity of our solution is O(log2 n) bits and it converges in O(n2) rounds. Thus, this algorithm improves the convergence time of all previously known self-stabilizing 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(log2 n) bits.

Islanding Algorithm for Distributed Generators Based on Improved Prim Algorithm

Abstract: Multi-consumer island operation can be used as an important operation mode to enhance power supply reliability of distribution network containing distributed generation(DG) while the power consumers are supplied by DG.According to the feature of distribution network that it possesses ring structure and operates in dendroid mode,a distributed islanding method based on Prim algorithm is proposed to change the islanding problem into minimum spanning tree to obtain connected graph.The connected graph is searched by improved Prim algorithm to determine effective range of island.The proposed islanding method can adapt to the ring structure of distribution network,and can ensure the contineous power supply to important consumers and maximize range of the island and is favorable to the fast swithing from island operation mode to network-connected operation mode after the fault recovery.Case analysis on typical islanding show that the proposed algorithm can dynamically generate rational islanding scheme after fault occurred in distribution network..

Pointed binary encompassing trees: Simple and optimal

Abstract: For n disjoint line segments in the plane we construct in optimal O(nlogn) time and linear space an encompassing tree of maximum degree three such that at every vertex v all edges of the tree that are incident to v lie in a halfplane bounded by the line through the input segment which v is an endpoint of. In particular, this tree is pointed since every vertex has an incident angle greater than @p. Such a pointed binary tree can be augmented to a minimum pseudo-triangulation. It follows that every set of disjoint line segments in the plane has a constrained minimum pseudo-triangulation whose maximum vertex degree is bounded by a constant.

On Sorting, Heaps, and Minimum Spanning Trees

Abstract: Let A be a set of size m. Obtaining the first k≤m elements of A in ascending order can be done in optimal O(m+klog k) time. We present Incremental Quicksort (IQS), an algorithm (online on k) which incrementally gives the next smallest element of the set, so that the first k elements are obtained in optimal expected time for any k. Based on IQS, we present the Quickheap (QH), a simple and efficient priority queue for main and secondary memory. Quickheaps are comparable with classical binary heaps in simplicity, yet are more cache-friendly. This makes them an excellent alternative for a secondary memory implementation. We show that the expected amortized CPU cost per operation over a Quickheap of m elements is O(log m), and this translates into O((1/B)log (m/M)) I/O cost with main memory size M and block size B, in a cache-oblivious fashion. As a direct application, we use our techniques to implement classical Minimum Spanning Tree (MST) algorithms. We use IQS to implement Kruskal’s MST algorithm and QHs to implement Prim’s. Experimental results show that IQS, QHs, external QHs, and our Kruskal’s and Prim’s MST variants are competitive, and in many cases better in practice than current state-of-the-art alternative (and much more sophisticated) implementations.

Geometric minimum spanning trees with GEOFILTERKRUSKAL

Abstract: Let P be a set of points in ℝd. We propose GeoFilterKruskal, an algorithm that computes the minimum spanning tree of P using well separated pair decomposition in combination with a simple modification of Kruskal’s algorithm. When P is sampled from uniform random distribution, we show that our algorithm takes one parallel sort plus a linear number of additional steps, with high probability, to compute the minimum spanning tree. Experiments show that our algorithm works better in practice for most data distributions compared to the current state of the art [31]. Our algorithm is easy to parallelize and to our knowledge, is currently the best practical algorithm on multi-core machines for d>2.

A neighborhood density estimation clustering algorithm based on minimum spanning tree

Abstract: In this paper a clustering algorithm based on the minimum spanning tree (MST) with neighborhood density difference estimation is proposed. Neighborhood are defined by patterns connected with the edges in the MST of a given dataset. In terms of the difference between patterns and their neighbor density, boundary patterns and corresponding boundary edges are detected. Then boundary edges are cut, and as a result the dataset is split into defined number clusters. For reducing time complexity of detecting boundary patterns, an rough and a refined boundary candidates estimation approach are employed, respectively. The experiments are performed on synthetic and real data. The clustering results demonstrate the proposed algorithm can deal with not well separated, shape-diverse clusters.

An Effective Algorithm for Minimum Weighted Vertex Cover Problem

Abstract: The Minimum Weighted Vertex Cover (MWVC) prob- lem is a classic graph optimization NP - complete problem. Given an undirected graph G = (V, E) and weighting function defined on the vertex set, the minimum weighted vertex cover problem is to find a vertex set S V whose total weight is minimum subject to every edge of G has at least one end point in S. In this paper an effective algorithm, called Support Ratio Algorithm (SRA), is designed to find the minimum weighted vertex cover of a graph. Computational experiments are designed and conducted to study the performance of our proposed algorithm. Extensive simulation results show that the SRA can yield better solutions than other existing algorithms found in the literature for solving the minimum vertex cover problem.

Board Game Supporting Learning Prim's Algorithm and Dijkstra's Algorithm

Abstract: The concept of minimum spanning tree algorithms in data structure is difficult for students to learn and to imagine without practice. Usually, learners need to diagram the spanning trees with pen to realize how the minimum spanning tree algorithm works. In this paper, the authors introduce a competitive board game to motivate students to learn the concept of minimum spanning tree algorithms. They discuss the reasons why it is beneficial to combine graph theories and board game for the Dijkstra and Prim minimum spanning tree theories. In the experimental results, this paper demonstrates the board game and examines the learning feedback for the mentioned two graph theories. Advantages summarizing the benefits of combining the graph theories with board game are discussed.

Minimum spanning tree on spatio-temporal networks

Abstract: Given a spatio-temporal network whose edge properties vary with time, a time-sub-interval minimum spanning tree (TSMST) is a collection of minimum spanning trees where each tree is associated with one or more time intervals; during these time intervals, the total cost of this spanning tree is the least among all spanning trees. The TSMST problem aims to identify a collection of distinct minimum spanning trees and their respective time-sub-intervals. This is an important problem in spatio-temporal application domains such as wireless sensor networks (e.g., energy-efficient routing). As the ranking of candidate spanning trees is non-stationary over a given time interval, computing TSMST is challenging. Existing methods such as dynamic graph algorithms and kinetic data structures assume separable edge weight functions. In contrast, we propose novel algorithms to find TSMST for large networks by accounting for both separable and non-separable piecewise linear edge weight functions. The algorithms are based on the ordering of edges in edge-order-intervals and intersection points of edge weight functions.

An approximate solution method based on tabu search for k-minimum spanning tree problems

Abstract: This paper considers a new tabu search-based approximate solution algorithm for k-minimum spanning tree problems. One of the features of the proposed algorithm is that it efficiently obtains local optimal solutions without applying minimum spanning tree algorithms. Numerical experimental results show that the proposed method provides a good performance especially for dense graphs in terms of solution accuracy over existing algorithms.

Research on the routing algorithm based greedy multicast algorithm

Abstract: Because all the nodes are multi-cast, the minimum spanning tree is the best, therefore, it is expected by the niche tree dynamic greed multicast routing algorithms produce more of the performance with a reasonable level. As for the greedy algorithm and the text of the characters tree dynamic greedy algorithm are made detailed emulation, Simulations result indicates that the results of the algorithm in multicast node are with greater density of the advantages and it shows that the invalid in other cases were acceptable, and has complicated the low quality.

Broadcasting in heterogeneous tree networks

Abstract: We consider the broadcasting problem in heterogeneous tree networks. A heterogeneous tree network is represented by a weighted tree T = (V,E) such that the weight of each edge denotes the communication time between the two end vertices. The broadcasting problem is to find a broadcast center such that the maximum communication time from the broadcast center to all vertices is minimized. In this paper, we propose a linear time algorithm for the broadcasting problem in a heterogeneous tree network following the postal model. As a byproduct of the algorithm, we can compute in linear time the broadcasting time of any vertex in the tree, i.e., the maximum time required to transmit messages from the vertex to every other vertex in the tree. Furthermore, an optimal sequence by which the broadcast center broadcasts its messages to all vertices in T can also be determined in linear time.

On optimal design and expansion of electrical power distribution systems

Abstract: Optimal planning of large-scale distribution networks is a multiobjective combinatorial optimization problem with many complexities. This paper proposes the application of improved genetic algorithm (GA) for the optimal design of large-scale distribution systems in order to provide optimal sizing and locating of the high voltage (HV) substations and medium voltage (MV) feeders routing, using their corresponding fixed and variable costs associated with operational and optimization constraints. The novel approach presented in the paper, solves hard satisfactory optimization problems with different constraints in large-scale distribution networks. This paper presents a new concept based on MST in graph theory and GA for optimal locating of the HV substations and MV feeders routing in a real-size distribution network. Minimum spanning tree solved with Prim's algorithm is employed to generate a set of feasible population. In the present article, to reduce computational burden and avoid huge search space leading to infeasible solutions, special coding method is generated for GA operators to solve optimal feeders routing. The proposed coding method guarantees the validity of the solution during the progress of the GA toward the global optimal solution. The developed GA-based software is tested in a real-size large-scale distribution system and the well-satisfactory results are presented.

Specializations and generalizations of the stackelberg minimum spanning tree game

Abstract: The Stackelberg Minimum Spanning Tree (StackMST) game is a network pricing (bilevel) optimization problem. The game is played by two players on a graph G = (V,E), whose edges are partitioned into two sets: a set R of red edges (inducing a spanning tree of G) with a fixed non-negative real cost, and a set B of blue edges which are instead priced by a leader. This is done with the final intent of maximizing a revenue that will be returned for their purchase by a follower, whose goal in turn is to select a minimum spanning tree of G. StackMST is known to be APX-hard already when the number of distinct red costs is 2, as well as min{k,1 + lnβ, 1 + lnρ}-approximable, where k is the number of distinct red costs, β is the number of blue edges selected by the follower in an optimal pricing, and ρ is the maximum ratio between red costs. In this paper we analyze some meaningful specializations and generalizations of StackMST, which shed some more light on the computational complexity of the game. More precisely, we first show that if G is complete, then the following holds: (i) if there are only 2 distinct red costs, then the problem can be solved optimally (this contrasts with the corresponding APX-hardness of the general problem); (ii) otherwise, the problem can be approximated within 7/4 + e, for any e > 0. Afterwards, we define a natural extension of StackMST, namely that in which blue edges have a non-negative activation cost associated, and the leader has a global activation budget that must not be exceeded, and, after showing that the very same approximation ratio as that of the original game can be achieved, we prove that if the spanning tree induced by the red edges has radius h (in terms of number of edges), then the problem admits a (2h + e)-approximation algorithm.

Simulation Research of Communication Networks Based on Prim Algorithm

Abstract: In order to improve the efficiency of the communication networks, we used the Kruskal algorithm and the Prim algorithm through algorithm comparison and analysis methods of data structure. A dynamic framework for the communication network installed system is built. Moreover, according to the actual framework of the communication network specific issues, the module chooses the Minimum Cost Spanning Tree Prim algorithm ultimately. The assumptions process and outcomes simulation have proper analysis and certification by C language. Experimental results show that the Minimum Cost Spanning Tree of Prim algorithm on the installed system for communication networks dynamic planning process have the advantage of speed and effectively reduce the waste of resources, which not only can ensure efficiency but also can effectively improve communication networks installed cost.

The Research of Wireless Sensor Networks Optimization Algorithm Based on the Energy Control

Abstract: Wireless Sensor Networks(WSN) is a hot spot of the research of wireless networks currently, the key of achieving efficient transmission business is to control node energy and improve the network lifetime i Chen You-rong n wireless sensor networks. The paper first uses Boolean sensing model based on Poisson point process to identify the function of the rate of coverage and the node density in unit area, and then calculates the total number of nodes in the region, next uses the greedy strategy of the Prim algorithm to find a spanning tree with the maximum weight, and constructs a approximate solution for the minimum connected dominating set. In order to control the commotions of the nodes, make the nodes in spanning tree to work, and other nodes are in sleep state. At last, further analyses the relationship between the number of nodes in connected dominating and the coverage radius.

On symbolic OBDD-based algorithms for the minimum spanning tree problem

Abstract: The minimum spanning tree problem is one of the most fundamental algorithmic graph problems and OBDDs are a very common dynamic data structure for Boolean functions. Since in some applications graphs become larger and larger, a research branch has emerged which is concerned with the design and analysis of so-called symbolic algorithms for classical graph problems on OBDD-represented graph instances. Here, a symbolic minimum spanning tree algorithm using O(log3 |V|) functional operations is presented, where V is the set of vertices of the input graph. Furthermore, answering an open problem posed by Sawitzki (2006) it is shown that every symbolic OBDD-based algorithm for the minimum spanning tree problem needs exponential space (with respect to the OBDD size of the input graph). This result even holds for planar input graphs.

Research and improvement of Kruskal algorithm

Abstract: In the application of Kruskal algorithm to get the minimum spanning tree,the time of selecting edge at least was n-1.When the number of side m and vertex n satisfy the relationship m≤2n-2,Kruskal algorithm can be improved.When the number of selecting edge time was at most n-1,the improved algorithm was used to get the solution.The idea of the improved algorithm is deleting the edge which has the maximum weight and does not influence diagram connectivity when deleted it,till there are number of n-1 edges.The time of improved algorithm is reduced by theory.

Evaluation of Interconnected Power Systems Splitting

Abstract: Controlled splitting of an interconnected power system is the last defense line against wide-area blackout. As a special protection scheme, the methodology of system splitting is a comprehensive decision-making problem. This article introduces a novel approach for separation of the integrated power systems into several stable islands. The proposed method combines both the dynamic and static characteristics of interconnected power networks and determines the proper splitting schemes. The presented algorithm searches for proper islanding strategies using the Krylov projection method and a new optimization algorithm to find the proper splitting points such that the total load shedding is minimized. The method reduces the huge initial search space of islanding strategies by considering dynamic characteristics of integrated power systems. A spanning-tree-based Prim algorithm is used to find all possible combination of the islands. The speed of the proposed algorithm is remarkably high, and it can be u...