Examples of using Minimum spanning in English and their translations into Serbian
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And(V, F′) is a minimum spanning tree.
If the graph is connected,the forest has a single component and forms a minimum spanning tree.
Let Y1 be a minimum spanning tree of graph P.
At the termination of the algorithm,the forest forms a minimum spanning forest of the graph.
So T1- f+ e is a minimum spanning tree containing E1+ e and again P holds.
Now all the vertices have been selected and the minimum spanning tree is shown in green.
Let Y1 be a minimum spanning tree of graph P. If Y1=Y then Y is a minimum spanning tree.
This shows Y is a minimum spanning tree.
If each edge has a distinct weight then therewill be only one, unique minimum spanning tree.
Run a graph minimum spanning tree algorithm on this graph to find a minimum spanning tree.
If Y1=Y then Y is a minimum spanning tree.
The set of these minimum spanning trees is called a minimum spanning forest, which contains every vertex in the graph.
Finally, the process finishes with the edge EG of length 9, and the minimum spanning tree is found.
The first algorithm for finding a minimum spanning tree was developed by Czech scientist Otakar Borůvka in 1926.
Seth Pettie and Vijaya Ramachandran have found a provably optimal deterministic comparison-based minimum spanning tree algorithm.
The first algorithm for finding a minimum spanning tree was developed by Czech scientist Otakar Borůvka in 1926(see Borůvka's algorithm).
For uniformly random point sets it is possible to compute minimum spanning trees as quickly as sorting.
If the weights are positive, then a minimum spanning tree is in fact a minimum-cost subgraph connecting all vertices, since subgraphs containing cycles necessarily have more total weight.
This inverse appears in the time complexity of some algorithms, such as the disjoint-set data structure andChazelle's algorithm for minimum spanning trees.
If the graph is not connected,then it finds a minimum spanning forest(a minimum spanning tree for each connected component).
For example, the minimum spanning tree of the graph associated with an instance of the Euclidean TSP is a Euclidean minimum spanning tree, and so can be computed in expected O(n log n) time for n points(considerably less than the number of edges).
Using min heap priority queue in Prim's algorithm to find the minimum spanning tree of a connected and undirected graph, one can achieve a good running time.
The best known(deterministic) minimum spanning tree algorithm by Bernard Chazelle is also based in part on Borůvka's and runs in O(E α(E, V)) time, where α is the inverse of the Ackermann function.
More generally, any edge-weighted undirected graph(not necessarily connected) has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components.
The realization problem for Euclidean minimum spanning trees is stated as follows: Given a tree T=(V,E), find a location D(u) for each vertex u∈ V so that T is a minimum spanning tree of D(u): u∈ V, or determine that no such locations exist.
In higher dimensions, the connectivity determined by the Delaunay triangulation(which, likewise,partitions the convex hull into d-dimensional simplices) contains the minimum spanning tree; however, the triangulation might contain the complete graph.
A data structure constructed from the minimum spanning tree allows the minimax distance between any pair of vertices to be queried in constant time per query, using lowest common ancestor queries in a Cartesian tree.
It is easy to show that tree Y2 is connected, has the same number of edges as tree Y1, and the total weights of its edges is not larger than that of tree Y1,therefore it is also a minimum spanning tree of graph P and it contains edge e and all the edges added before it during the construction of set V.
The root of the Cartesian tree represents the heaviest minimum spanning tree edge, and the children of the root are Cartesian trees recursively constructed from the subtrees of the minimum spanning tree formed by removing the heaviest edge.
Now assume P is true for some non-final edge set F andlet T be a minimum spanning tree that contains F. If the next chosen edge e is also in T, then P is true for F+ e.