Euclidean Minimum Spanning Tree Problem Research Articles

Component-based 2-/3-dimensional nearest neighbor search based on Elias method to GPU parallel 2D/3D Euclidean Minimum Spanning Tree Problem

We present improved data parallel approaches working on graphics processing unit (GPU) compute unified device architecture (CUDA) platform to build hierarchical Euclidean minimum spanning forest or tree (EMSF/EMST) for applications whose input only contains N points with arbitrary data distribution in 2D/3D Euclidean space. Characteristic of the proposed parallel algorithms follows “data parallelism, decentralized control and O(1) local memory occupied by each GPU thread”. This research has to solve GPU parallelism of component-based nearest neighbor search (component-based NNS), tree traversal, and other graph operations like union-find. For exact NNS, instead of using classical K-d tree search or brute-force computing method, we propose a K-d search method working based on dividing the Euclidean K-dimensional space into congruent and non-overlapping square/cubic cells where size of points in each cell is bounded. For component-based NNS, with the uniqueness property based on 2D/3D square/cubic space partition, we propose dynamic and static pruning techniques to prune unnecessary neighbor cells’ search. For tree traversal, instead of using breadth-first-search, this paper proposes CUDA kernels working with a distributed dynamic link list for selecting a local spanning tree’s shortest outgoing edge since size of local EMSTs in EMSF cannot be predicted. Source code is provided online and experimental comparisons are conducted on both 2D and 3D benchmarks with up to 107 points to build final EMST. Results show that applying K-d search with static pruning technique and the proposed operators totally working in parallel on GPU, our current implementation runs faster than our previous work and current optimal sequential dual-tree mlpack EMST library.

Clustering, Randomness, and Regularity: Spatial Distributions and Human Performance on the Traveling Salesperson Problem and Minimum Spanning Tree Problem

We investigated human performance on the Euclidean Traveling Salesperson Problem (TSP) and Euclidean Minimum Spanning Tree Problem (MST-P) in regards to a factor that has previously received little attention within the literature: the spatial distributions of TSP and MST-P stimuli. First, we describe a method for quantifying the relative degree of clustering, randomness or regularity within point distributions. We then review evidence suggesting this factor might influence human performance on the two problem types. Following this we report an experiment in which the participants were asked to solve TSP and MST-P test stimuli that had been generated to be either highly clustered, random, or highly regular. The results indicate that for both the TSP and MST-P the participants tended to produce better quality solutions when the stimuli were highly clustered compared to random, and similarly, better quality solutions for random compared to highly regular stimuli. It is suggested that these results provide support for the ideas that human solvers attend to salient clusters of nodes when solving these problems, and that a similar process (or series of processes) may underlie human performance on these two tasks.

Lower bounds for testing Euclidean Minimum Spanning Trees

The Euclidean Minimum Spanning Tree problem is to decide whether a given graph G = ( P , E ) on a set of points in the two-dimensional plane is a minimum spanning tree with respect to the Euclidean distance. Czumaj et al. [A. Czumaj, C. Sohler, M. Ziegler, Testing Euclidean Minimum Spanning Trees in the plane, Unpublished, Part II of ESA 2000 paper, downloaded from http://web.njit.edu/~czumaj/] gave a 1-sided-error non-adaptive property-tester for this task of query complexity O ˜ ( n ) . We show that every non-adaptive (not necessarily 1-sided-error) property-tester for this task has a query complexity of Ω ( n ) , implying that the test in [A. Czumaj, C. Sohler, M. Ziegler, Testing Euclidean Minimum Spanning Trees in the plane, Unpublished, Part II of ESA 2000 paper, downloaded from http://web.njit.edu/~czumaj/] is of asymptotically optimal complexity. We further prove that every adaptive property-tester has query complexity of Ω ( n 1 / 3 ) . Those lower bounds hold even when the input graph is promised to be a bounded degree tree.

On the Euclidean Minimum Spanning Tree Problem

Given a weighted graph G(V;E), a minimum spanning tree for G can be obtained in linear time using a randomized algorithm or nearly linear time using a deterministic algorithm. Given n points in the plane, we can construct a graph with these points as nodes and an edge between every pair of nodes. The weight on any edge is the Euclidean distance between the two points. Finding a minimum spanning tree for this graph is known as the Euclidean minimum spanning tree problem (EMSTP). The minimum spanning tree algorithms alluded to before will run in time O(n2) (or nearly O(n2)) on this graph. In this note we point out that it is possible to devise simple algorithms for EMSTP in k- dimensions (for any constant k) whose expected run time is O(n), under the assumption that the points are uniformly distributed in the space of interest.CR Categories: F2.2 Nonnumerical Algorithms and Problems; G.3 Probabilistic Algorithms

Euclidean minimum spanning trees and bichromatic closest pairs

We present an algorithm to compute a Euclidean minimum spanning tree of a given setS ofN points inEd in timeO(Fd(N,N) logdN), whereFd(n,m) is the time required to compute a bichromatic closest pair amongn red andm green points inEd. IfFd(N,N)=Ω(N1+?), for some fixed ?>0, then the running time improves toO(Fd(N,N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closest pair in expected timeO((nm logn logm)2/3+m log2n+n log2m) inE3, which yields anO(N4/3 log4/3N) expected time, algorithm for computing a Euclidean minimum spanning tree ofN points inE3. Ind?4 dimensions we obtain expected timeO((nm)1?1/([d/2]+1)+?+m logn+n logm) for the bichromatic closest pair problem andO(N2?2/([d/2]+1)?) for the Euclidean minimum spanning tree problem, for any positive ?.