The Application of Union-Find Sets in Kruskal Algorithm

Abstract

It is the key of Krusal Algorithm to select edges and judge loop. Though realized path-compression in operation of searching tree of Union-Find Sets Data Structure, and optimized merging-operation which the shallower tree has been merged into the deeper tree, the operation of selecting edges and judging loop in Krusal Algorithm is optimized and the time complexity of algorithm is reduced. Compared with the common way, the programming operation time of the new way has been reduced to some 1/10. Keywords-Kruskal algorithm; union-find sets; path-compression; minimum cost spanning tree 1 INTRODUCTION Kruskal algorithm is used to solve the minimum cost spanning tree (MCST)of one net. The basic idea of this algorithm is choose the edges based on weight from the small to the large in the net of N vertices. if the selected edge does not form a loop with other selected edges, it will be as an edge of spanning tree; if selected edge form a spanning tree loop with other selected, it will be discarded. In accordance with this process continues, the generated tree must be MCST[1][2]until the number of identified edges is N-1.The key to the algorithm is: how to determine whether the selected edge has formed a loop with other edges which have been identified. So the data structure to complete the operation determines the algorithm's time complexity.Union-find sets(UFS) is a tree structure, which is used to deal with the issue of merging and inquiring of disjoint sets.UFS is often expressed as a forest in the using. In this paper, through using UFS to determine whether the selected edge has formed a loop with other identified edges ,we have obtained a good result in the algorithm 's time complexity. UNION-FIND SETSUFS is a data structure which derives from the handling of the equivalence relation and equivalence class. A. Equivalence relation and equivalence classEquivalence relation[3][4] is a common relationship that broadly existed in the real world, and many problems can be transformed into an ad hoc basis by the equivalence relation according to dividing a collection of objects into equivalence class[3][4]in the real world. Throughout this paper ,let S be a collection with N elements, and R be the equivalence relation to S ,let <x,y> be a dual sequence in R, where x and y are the elements of S. Based on the equivalent relation,the collection is divided into equivalence classes in the general algorithm as follows: Algorithm 1: step 1:Initialize： for each e i ∈S do begin construct the respective single-element set S i ; end; step 2:Divide S's elements into equivalence classes based on the dual sequence <x,y> in R. Suppose x∈S x ，y∈S y ： for each <x,y>∈R do begin if(S x ≠ S y ) then begin

Distributed multi-sensor multi-view fusion based on generalized covariance intersection

“…This usually happens when there are targets at close distance. In order to ensure that each GC is associated with a single subset, we resort to the union find set (UFS) algorithm [37,38]. After application of UFS, a new decomposition is obtained for the GCs of the two sensors…”

“…Proof. Combining (28) with (38), we can derive the bound on v1,2 − v 1,2 1 . For convenience, the following shorthand nota-tion will be used (63)…”

“…We implement the proposed fusion algorithm based on the GM-PHD filter with adaptive birth model. In the implementation, the union find set (UFS) [37,38] algorithm is adopted to ensure that the GC subsets generated during clastering are disjoint, and Optimal SubPattern Assignment (OSPA) metric [39] is used to compute the similarity between different subsets. In addition, Murty's algorithm [41] is utilized to obtain the association pairs effectively.…”

New Strategies and Extensions in Kruskal’s Algorithm in Multicast Routing

“…Guttoski et al (2007) declared that comparing Kruskal's and Prim's algorithms, Kruskal's algorithm presents the most adequate way to build the final solution in the query optimization process, because when the most profitable joins are performed first, the joins that will be formed after are going to have their cost reduced, mainly joins between tables with high number of tuples. Pan et al (2009) used a tree structure called Union-find sets (UFS) used to deal with the issue of merging and inquiring of disjoint sets. UFS is used in Kruskal's algorithm to determine whether the selected edge has formed a loop with other identified edges.…”