Showing papers on "Prim's algorithm published in 2006"

Minimum Spanning Tree Based Clustering Algorithms

Abstract: The minimum spanning tree clustering algorithm is known to be capable of detecting clusters with irregular boundaries. In this paper, we propose two minimum spanning tree based clustering algorithms. The first algorithm produces a k-partition of a set of points for any given k. The algorithm constructs a minimum spanning tree of the point set and removes edges that satisfy a predefined criterion. The process is repeated until k clusters are produced. The second algorithm partitions a point set into a group of clusters by maximizing the overall standard deviation reduction, without a given k value. We present our experimental results comparing our proposed algorithms to k-means and EM. We also apply our algorithms to image color clustering and compare our algorithms to the standard minimum spanning tree clustering algorithm.

An ant-based algorithm for finding degree-constrained minimum spanning tree

Abstract: A spanning tree of a graph such that each vertex in the tree has degree at most d is called a degree-constrained spanning tree. The problem of finding the degree-constrained spanning tree of minimum cost in an edge weighted graphis well known to be NP-hard. In this paper we give an Ant-Based algorithm for finding low cost degree-constrained spanning trees. Ants are used to identify a set of candidate edges from which a degree-constrained spanning tree can be constructed. Extensive experimental results show that the algorithm performs very well against other algorithms on a set of 572 problem instances.

Power-aware collective tree exploration

Abstract: An n-node tree has to be explored by a group of k mobile robots deployed initially at the root. Robots traverse the edges of the tree until all nodes are visited. We would like to minimize maximal distance traveled by each robot (e.g. to preserve the battery power). First, we assume that a tree is known in advance. For this NP-hard problem we present a 2-approximation. Moreover, we present an optimal algorithm for a case where k is constant. From the 2-approximation algorithm we develop a fast 8-competitive online algorithm, which does not require a previous knowledge of the tree and collects information during the exploration. Furthermore, our online algorithm is distributed and uses only a local communication. We show a lower bound of 1.5 for the competitive ratio of any deterministic online algorithm.

Dynamic Algorithm for Graph Clustering Using Minimum Cut Tree

Abstract: In this paper we introduce a dynamic algorithm for clustering undirected graphs, whose edge property is continuously changing. The algorithm can maintain high-quality clusters efficiently in presence of insertion and deletion (update) of edges. The algorithm is motivated by the minimum-cut tree based partitioning algorithm presented in G. W. Flake et al. (2004) and G. W. Flake (2000). It takes O(k3) time for each update processing, where k is the maximum size of any cluster. This is the worst case time complexity, and in general time taken is much less. To the best of our knowledge, this is the first clustering algorithm, for evolving graphs, providing strong theoretical quality guarantee on clusters

Power-Aware collective tree exploration

Abstract: An n-node tree has to be explored by a group of k mobile robots deployed initially at the root. Robots traverse the edges of the tree until all nodes are visited. We would like to minimize maximal distance traveled by each robot (e.g. to preserve the battery power). First, we assume that a tree is known in advance. For this NP-hard problem we present a 2-approximation. Moreover, we present an optimal algorithm for a case where k is constant. From the 2-approximation algorithm we develop a fast 8-competitive online algorithm, which does not require a previous knowledge of the tree and collects information during the exploration. Furthermore, our online algorithm is distributed and uses only a local communication. We show a lower bound of 1.5 for the competitive ratio of any deterministic online algorithm.

A hybrid heuristic for the minimum weight vertex cover problem

Abstract: Given an undirected graph with weights associated with its vertices, the minimum weight vertex cover problem seeks a subset of vertices with minimum sum of weights such that each edge of the graph has at least one endpoint belonging to the subset. In this paper, we propose a hybrid approach, combining a steady-state genetic algorithm and a greedy heuristic, for the minimum weight vertex cover problem. The genetic algorithm generates vertex cover, which is then reduced to minimal weight vertex cover by the heuristic. We have evaluated the performance of our algorithm on several test problems of varying sizes. Computational results show the effectiveness of our approach in solving the minimum weight vertex cover problem.

Minimum-energy broadcasting in multi-hop wireless networks using a single broadcast tree

Abstract: In this paper we address the minimum-energy broadcast problem in multi-hop wireless networks, so that all broadcast requests initiated by different source nodes take place on the same broadcast tree. Our approach differs from the most commonly used one where the determination of the broadcast tree depends on the source node, thus resulting in different tree construction processes for different source nodes. Using a single broadcast tree simplifies considerably the tree maintenance problem and allows scaling to larger networks. We first show that, using the same broadcast tree, the total power consumed for broadcasting from a given source node is at most twice the total power consumed for broadcasting from any other source node. We next develop a polynomial-time approximation algorithm for the construction of a single broadcast tree. The performance analysis of the algorithm indicates that the total power consumed for broadcasting from any source node is within 2H(n-1) from the optimal, where n is the number of nodes in the network and H(n) is the harmonic function. This approximation ratio is close to the best achievable bound in polynomial time. We also provide a useful relation between the minimum-energy broadcast problem and the minimum spanning tree, which shows that a minimum spanning tree may be a good candidate in sparsely connected networks. The performance of our algorithm is also evaluated numerically with simulations.

FasterDSP: A faster approximation algorithm for directed Steiner tree problem

Abstract: Given a weighted directed graph G = (V, E, c), where c : E → R + is an edge cost function, a subset X of vertices (terminals), and a root vertex ν r , the directed Steiner tree problem (DSP) asks for a minimum-cost tree which spans the paths from root vertex ν r to each terminal. Charikar et al.'s algorithm is well-known for this problem. It achieves an approximation guarantee of l(l - 1)k † in O(n 1 k 21 ) time for any fixed level l > 1, where l is the level of the tree produced by the algorithm, n is the number of vertices, |V|, and k is the number of terminals, |X|. However, it requires a great amount of computing power, and there are some problems in the proof of the approximation guarantee of the algorithm. This paper provides a faster approximation algorithm improving Charikar et al.'s DSP algorithm with a better time complexity, O(n'k' + n 2 k + nm), where m is the number of edges, and an amended √8k - S In k factor for the 2-level Steiner tree, where δ = √6 - 2 = 0.4494.

Enumerate and expand: improved algorithms for connected vertex cover and tree cover

Abstract: We present a new method of solving graph problems related to VERTEX COVER by enumerating and expanding appropriate sets of nodes. As an application, we obtain dramatically improved runtime bounds for two variants of the VERTEX COVER problem: In the case of CONNECTED VERTEX COVER, we take the upper bound from O*(6k) to O*(3.2361k) without large hidden factors. For TREE COVER, we show exactly the same complexity, improving vastly over the previous bound of O*((2k)k). In the process, faster algorithms for solving subclasses of the Steiner tree problem on graphs are investigated.

An effective genetic algorithm for the minimum-label spanning tree problem

Abstract: Given a connected, undirected graph G with labeled edges, the minimum label spanning tree problem seeks a spanning tree on G to whose edges are attached the smallest possible number of labels. A greedy heuristic for this NP-hard problem greedily chooses labels so as to reduce the number of components in the subgraphs they induce as quickly as possible. A genetic algorithm for the problem encodes candidate solutions as per mutations of the labels; an initial segment of such a chromosome lists the labels that appear on the edges in the chromosome's tree. Three versions of the GA apply generic or heuristic crossover and mutation operators and a local search step. In tests on 27 randomly-generated instances of the minimum-label spanning tree problem, versions of the GA that apply generic operators, with and without the local search step, perform less well than the greedy heuristic, but a version that applies the local search step and operators tailored to the problem returns solutions that require on average 10 fewer labels than the heuristic's.

How to trim an MST: a 2-approximation algorithm for minimum cost tree cover

Abstract: The minimum cost tree cover problem is to compute a minimum cost tree T in a given connected graph G with costs on the edges, such that the vertices of T form a vertex cover for G The problem is supposed to arise in applications of vertex cover and edge dominating set when connectivity is additionally required in solutions Whereas a linear-time 2-approximation algorithm for the unweighted case has been known for quite a while, the best approximation ratio known for the weighted case is 3 Moreover, the known 3-approximation algorithm for such case is far from practical in its efficiency In this paper we present a fast, purely combinatorial 2-approximation algorithm for the minimum cost tree cover problem It constructs a good approximate solution by trimming some leaves within a minimum spanning tree (MST), and to determine which leaves to trim, it uses both of the primal-dual schema and the local ratio technique in an interlaced fashion

The geometric generalized minimum spanning tree problem with grid clustering

Abstract: This paper is concerned with a special case of the generalized minimum spanning tree problem. The problem is defined on an undirected graph, where the vertex set is partitioned into clusters, and non-negative costs are associated with the edges. The problem is to find a tree of minimum cost containing at least one vertex in each cluster. We consider a geometric case of the problem where the graph is complete, all vertices are situated in the plane, and Euclidean distance defines the edge cost. We prove that the problem is strongly NP-hard even in the case of a special structure of the clustering called grid clustering. We construct an exact exponential time dynamic programming algorithm and, based on this dynamic programming algorithm, we develop a polynomial time approximation scheme for the problem with grid clustering.

A fully dynamic algorithm for the recognition of P 4 -sparse graphs

Abstract: We consider the dynamic recognition problem for the class of P4-sparse graphs: the objective is to handle edge/vertex additions and deletions, to recognize if each such modification yields a P4-sparse graph, and if yes, to update a representation of the graph. Our approach relies on maintaining the modular decomposition tree of the graph, which we use for solving the recognition problem. We establish conditions for each modification to yield a P4-sparse graph and obtain a fully dynamic recognition algorithm which handles edge modifications in O(1) time and vertex modifications in O(d) time for a vertex of degree d. Thus, our algorithm implies an optimal edges-only dynamic algorithm and a new optimal incremental algorithm for P4-sparse graphs. Moreover, by maintaining the children of each node of the modular decomposition tree in a binomial heap, we can handle vertex deletions in O(log n) time, at the expense of needing O(log n) time for each edge modification and O(d log n) time for the addition of a vertex adjacent to d vertices.

Finding minimum spanning trees more efficiently for tile-based phase unwrapping

Abstract: The tile-based phase unwrapping method employs an algorithm for finding the minimum spanning tree (MST) in each tile. We first examine the properties of a tile's representation from a graph theory viewpoint, observing that it is possible to make use of a more efficient class of MST algorithms. We then describe a novel linear time algorithm which reduces the size of the MST problem by half at the least, and solves it completely at best. We also show how this algorithm can be applied to a tile using a sliding window technique. Finally, we show how the reduction algorithm can be combined with any other standard MST algorithm to achieve a more efficient hybrid, using Prim's algorithm for empirical comparison and noting that the reduction algorithm takes only 0.1% of the time taken by the overall hybrid.

Models and hybrid algorithms for inverse minimum spanning tree problem with stochastic edge weights

Abstract: An inverse minimum spanning tree problem is to make the least modification on the edge weights such that a predetermined spanning tree is a minimum spanning tree with respect to the new edge weights. In this paper, the inverse minimum spanning tree problem with stochastic edge weights is investigated. The concept of -minimum spanning tree is initiated, and subsequently an -minimum spanning tree model and a probability maximization model are presented to formulate the problem according to different decision criteria. In order to solve the two stochastic models, hybrid genetic algorithms and hybrid simulated annealing algorithms are designed and illustrated by some computational experiments.

Fast method for analyzing IC wiring possibility

Abstract: The present invention belongs to IC CAD field It is characterized by that one step operation includes the following steps: computer initialization, creating GRG diagram, reading-in miring resource and circuit network table; using Prim algorithm to separate multi-ended wire net so as to obtain minimum generating tree under the Manhattan distance; pre-estimating wiring probability of every side; selecting wiring route and quickly wiring

Deterministic Energy Conserving Algorithms for Wireless Sensor Networks

Abstract: In this paper we study the problem of maintaining energy efficient topology from the perspective of computational geometry. The paper presented two algorithms that keep the distance small enough among sensors to maintain energy efficient topology. In the design of wireless sensor network, we are given n sensors in the Euclidean plane. The problem is how to increase the network lifetime by choosing the locations of relay sensors such that the distances between sensors are at most some delta i.e., the range of the sensors. The algorithm efficiently combines the classical approach of Steiner tree for 3- and 4-terminals with the concepts based on the Prim's and Kruskal's algorithms. The quantitative analysis shows that after the application of the algorithm the network consumes ((1/2)n + O(n3/2)) less energy

Np-hardness proof and an approximation algorithm for the minimum vertex ranking spanning tree problem

Abstract: The minimum vertex ranking spanning tree problem (MVRST) is to find a spanning tree of G whose vertex ranking is minimum. In this paper, we show that MVRST is NP-hard. To prove this, we polynomially reduce the 3-dimensional matching problem to MVRST. Moreover, we present a (⌈Ds/2⌉ + 1)/(⌊log2(Ds + 1)⌋ + 1)-approximation algorithm for MVRST where Ds is the minimum diameter of spanning trees of G.

Tight bounds for connecting sites across barriers

Abstract: Given m points (sites) and n obstacles (barriers) in the plane, we address the problem of finding a straight-line minimum cost spanning tree on the sites, where the cost is proportional to the number of intersections (crossings) between tree edges and barriers. If the barriers are infinite lines then there is a spanning tree where every barrier is crossed by O(√m) tree edges (connectors), and this bound is asymptotically optimal (spanning tree with low stabbing number). Asano et al. showed that if the barriers are pairwise disjoint line segments, then there is a spanning tree such that every barrier crosses at most 4 tree edges and so the total cost is at most 4n. Constructions with 3 crossings per barrier and 2n total cost provide a lower bound.We obtain tight bounds on the minimum cost spanning tree in the most exciting special case where the barriers are interior disjoint line segments that form a convex subdivision and there is a point in every cell. In particular, we show that there is a spanning tree such that every barrier is crossed by at most 2 tree edges, and there is a spanning tree of total cost 5n/3. Both bounds are tight.

A fully dynamic algorithm for the recognition of P4-sparse graphs

Abstract: We consider the dynamic recognition problem for the class of P 4 -sparse graphs: the objective is to handle edge/vertex additions and deletions, to recognize if each such modification yields a P 4 -sparse graph, and if yes, to update a representation of the graph. Our approach relies on maintaining the modular decomposition tree of the graph, which we use for solving the recognition problem. We establish conditions for each modification to yield a P 4 -sparse graph and obtain a fully dynamic recognition algorithm which handles edge modifications in O(1) time and vertex modifications in O(d) time for a vertex of degree d. Thus, our algorithm implies an optimal edges-only dynamic algorithm and a new optimal incremental algorithm for P 4 -sparse graphs. Moreover, by maintaining the children of each node of the modular decomposition tree in a binomial heap, we can handle vertex deletions in O(logn) time, at the expense of needing O(logn) time for each edge modification and O(d log n) time for the addition of a vertex adjacent to d vertices.

A tight lower bound for a special case of quadratic 0-1 programming

Abstract: We consider a still NP-complete partial case of the unconstrained binary quadratic optimization problem that can be described in terms of an undirected graph with red edges having negative weights and green edges having positive weights The maximum vertex degree of the graph is three It can be assumed wlog that every vertex is incident to a red and a green edge We are looking for a vertex cover with respect to the red edges which covers a subset of green edges of total weight as small as possible We prove that for all connected such graphs except a subclass of special graphs having exactly five green edges it is possible to find a vertex cover with respect to the red edges for which the total weight of uncovered green edges is at least 1/4 fraction of the total weight of all green edges

Real-time minimum vertex cover for two-terminal series-parallel graphs

Abstract: In this paper we show how the tree contraction method can be applied to compute the cardinality of the minimum vertex cover of a two-terminal series-parallel graph. We then construct a real-time paradigm for this problem and show that in the new computational environment, a parallel algorithm is superior to the best possible sequential algorithm, in terms of the accuracy of the solution computed. Specifically, there are cases in which the solution produced by a parallel algorithm that uses p processors is better than the output of any sequential algorithm for the same problem, by a factor superlinear in p.

A Polynomial Time Algorithm for Obtaining a Minimum Vertex Ranking Spanning Tree in Outerplanar Graphs

Abstract: The minimum vertex ranking spanning tree problem is to find a spanning tree of G whose vertex ranking is minimum. This problem is NP-hard and no polynomial time algorithm for solving it is known for non-trivial classes of graphs other than the class of interval graphs. This paper proposes a polynomial time algorithm for solving the minimum vertex ranking spanning tree problem on outerplanar graphs.

An Algorithm for Finding Minimum Edge-Ranking Spanning Tree of Series-Parallel Graphs

Abstract: 1

Associative parallel algorithm for dynamic update of a minimum spanning tree after addition of a new node to a graph

Abstract: This paper proposes a new associative parallel algorithm for dynamic update of a minimum spanning tree after addition of a new node with all its incident edges to a graph. This algorithm is represented as the InsertVert procedure implemented on a model of an associative parallel system of the SIMD type with vertical processing (a STAR machine). The correctness of the procedure is proved and its time complexity is estimated.

How to Trim an MST : A 2-Approximation Algorithm for Minimum Cost Tree Cover

Abstract: The minimum cost tree cover problem is to compute a minimum cost tree T in a given connected graph G with costs on the edges, such that the vertices of T form a vertex cover for G. The problem is supposed to arise in applications of vertex cover and edge dominating set when connectivity is additionally required in solutions. Whereas a linear-time 2-approximation algorithm for the unweighted case has been known for quite a while, the best approximation ratio known for the weighted case is 3. Moreover, the known 3-approximation algorithm for such case is far from practical in its efficiency. In this paper we present a fast, purely combinatorial 2-approximation algorithm for the minimum cost tree cover problem. It constructs a good approximate solution by trimming some leaves within a minimum spanning tree (MST), and to determine which leaves to trim, it uses both of the primal-dual schema and the local ratio technique in an interlaced fashion.