Offline algorithms for dynamic minimum spanning tree problems

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 (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(log^2 n) work per modification. No previous dynamic geometric MST algorithm was known

An Empirical Study of Locality-Sensitive Hashing to Approximate the Minimum Spanning Tree

The minimum spanning tree is a problem with important applications but for which there are no known efficient algorithms for large data sets. Locality-sensitive hashing has been used to solve the near-neighbor problem and further applications in clustering, which indicates its potential for approximating the minimum spanning tree as well. An algorithm by Sariel Har-Peled, Piotr Indyk, and Rajeev Motwani utilizes locality-sensitive hashing to provide a c-approximation of the minimum spanning tree in O(dn1+1/c log2 n) time. In this thesis, we implement and test this algorithm. We determine that the algorithm is suited to provide a better-than-random approximation of the MST with significant improvements to computational performance over classic minimum spanning tree algorithms on clustered data

A constrained minimum spanning tree problem

In the classical general framework of the minimum spanning tree problem for a weighted graph we consider the case in which a predetermined vertex has a certain fixed degree. In other words, given a weighted graph G, one of its vertices v0 and a positive integer k, we consider the problem of finding the minimum spanning tree of G in which the vertex v0 has degree k, that is the number of edges coming out of v0. We recall that among the various methods for the solution of the unconstrained problem an efficient way to find the minimum spanning tree is based on the simple procedure of choosing one after the other an edge of minimum weight that has not be chosen yet and does not create cycles if added to the previously chosen edges. This technique is known as the \u201cgreedy algorithm\u201d. There are problems for which the greedy algorithm works and problems for which it does not. We prove that for the solution of the one degree constrained minimum spanning tree problem the classical greedy algorithm finds a right solution

Performanace of Improved Minimum Spanning Tree Based on Clustering Technique

Clustering technique is one of the most important and basic tool for data mining. Cluster algorithms have the ability to detect clusters with irregular boundaries, minimum spanning tree-based clustering algorithms have been widely used in practice. In such clustering algorithms, the search for nearest objects in the construction of minimum spanning trees is the main source of computation and the standard solutions take O(N2) time. In this paper, we present a fast minimum spanning tree-inspired clustering algorithm, which, by using an efficient implementation of the cut and the cycle property of the minimum spanning trees, can have much better performance than O(N2)

Minimum Cuts in Near-Linear Time

We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a ``semi-duality'' between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized algorithm that finds a minimum cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n^2 log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n^2 log^3 n). Other applications of the tree-packing approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a space-efficient manner

I/O-optimal algorithms on grid graphs

Given a graph of which the n vertices form a regular two-dimensional grid, and in which each (possibly weighted and/or directed) edge connects a vertex to one of its eight neighbours, the following can be done in O(scan(n)) I/Os, provided M = Omega(B^2): computation of shortest paths with non-negative edge weights from a single source, breadth-first traversal, computation of a minimum spanning tree, topological sorting, time-forward processing (if the input is a plane graph), and an Euler tour (if the input graph is a tree). The minimum-spanning tree algorithm is cache-oblivious. The best previously published algorithms for these problems need Theta(sort(n)) I/Os. Estimates of the actual I/O volume show that the new algorithms may often be very efficient in practice.Comment: 12 pages' extended abstract plus 12 pages' appendix with details, proofs and calculations. Has not been published in and is currently not under review of any conference or journa

A Hybrid Optimized Weighted Minimum Spanning Tree for the Shortest Intrapath Selection in Wireless Sensor Network

Wireless sensor network (WSN) consists of sensor nodes that need energy efficient routing techniques as they have limited battery power, computing, and storage resources. WSN routing protocols should enable reliable multihop communication with energy constraints. Clustering is an effective way to reduce overheads and when this is aided by effective resource allocation, it results in reduced energy consumption. In this work, a novel hybrid evolutionary algorithm called Bee Algorithm-Simulated Annealing Weighted Minimal Spanning Tree (BASA-WMST) routing is proposed in which randomly deployed sensor nodes are split into the best possible number of independent clusters with cluster head and optimal route. The former gathers data from sensors belonging to the cluster, forwarding them to the sink. The shortest intrapath selection for the cluster is selected using Weighted Minimum Spanning Tree (WMST). The proposed algorithm computes the distance-based Minimum Spanning Tree (MST) of the weighted graph for the multihop network. The weights are dynamically changed based on the energy level of each sensor during route selection and optimized using the proposed bee algorithm simulated annealing algorithm