Applied Soft Computing Journal 76 (2019) 105–120 Contents lists available at ScienceDirect Applied Soft Computing Journal journal homepage: www.elsevier.com/locate/asoc GPU implementation of Borůvka’s algorithm to Euclidean minimum spanning tree based on Elias method Wen-bao Qiao ∗ , Jean-charles Créput Le2i FRE2005, CNRS, Arts et Métiers, Univ. Bourgogne Franche-Comté, France highlights • An improvement of Bentley spiral search adapted to EMST/EMSF and its GPU parallelism. • GPU two-direction parallel Breadth-first search to collect the shortest outgoing edge. • GPU parallel EMST/EMSF for applications that only have Euclidean points as input. article info Article history: Received 12 December 2017 Received in revised form 1 June 2018 Accepted 24 October 2018 Available online 12 December 2018 Keywords: Euclidean minimum spanning tree EMST GPU parallel EMST Breadth first search GPU parallel BFS Decentralized control Data parallel abstract We present both sequential and data parallel approaches to build hierarchical minimum spanning forest (MSF) or tree (MST) in Euclidean space (EMSF/EMST) for applications whose input N points are uniformly or boundedly distributed in Euclidean space. Each iteration of the sequential approach takes O(N) time complexity through combining Borůvka’s algorithm with an improved component-based neighborhood search algorithm, namely sliced spiral search, which is a newly proposed improvement to Bentley’s spiral search for finding a component graph’s closest outgoing point on the plane. It works based on the uniqueness property in Euclidean space, and allows O(1) time complexity for one search from a query point to find the component’s closest outgoing point at different iterations of Borůka’s algorithm. The data parallel approach includes a newly proposed two-direction breadth-first search (BFS) implementation on graphics processing unit (GPU) platform, which is specialized for selecting a spanning tree’s minimum outgoing weight. This GPU two-direction parallel BFS enables a tree traversal operation to start from any one of its vertex acting as root. These GPU parallel implementations work by assigning N threads with one thread associated to one input point, one thread occupies O(1) local memory and the whole algorithm occupies O(N) global memory. Experiments are conducted on both uniformly distributed data sets and TSPLIB database. We evaluate computation time of the proposed approaches on more than 80 benchmarks with size N growing up to 10 6 points on personal laptop. © 2018 Elsevier B.V. All rights reserved. 1. Introduction Given a general weighted un-directed graph G = (V , E ) with N vertexes and explicit edge list E , a minimum spanning tree (MST) means a subset of E that connects all vertexes without any circles and with minimum total edges’ weight. A minimum spanning forest (MSF) consists of disjoint MSTs on each of the local connected components of that graph G. In graph theory, a connected component (or just component) of an un-directed graph G is a sub-graph of G in which any two vertexes are connected by a finite or infinite sequence of edges. MST/MSF is widely used in machine learning and data mining applications, for example, MSF consisting various small MSTs naturally forms data clusters of an input [1–3], like image segmentation [4,5]; an exact MST ∗ Corresponding author. E-mail address: rapidbao@outlook.com (W.-b. Qiao). also provides upper bound for tour length estimation of traveling salesman problems [6]. Three classical MST algorithms like Borůvka’s [7] (see [8] for translation), Kruskal’s (1956) [9] and Prim’s (1957) [10] algorithms establish the basis of MST implementations. They all work sequen- tially on standard general graph with predefined distance matrix (or edge list). Kruskal’s algorithm works by firstly sorting all edges of the graph in non-decreasing order, and then beginning with each vertex being a component, inserting edges one by one until all points in V have been connected without circles. Prim’s algorithm is a greedy algorithm. It firstly selects a random vertex as a tree, and then repeatedly adds the shortest edge that connects a new vertex to the tree until all the vertexes are included [11]. Borůvka’s algorithm begins with each vertex of the graph G being a component, iteratively finds minimum weighted outgoing edge for each component and adds all such edges to the MSF in one iteration. Borůvka’s algorithm has nature attribute to perform https://doi.org/10.1016/j.asoc.2018.10.046 1568-4946/© 2018 Elsevier B.V. All rights reserved.