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In computer science, Prim's algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex , where the total weight of all the edges in the tree is minimized.
Minimum degree spanning tree; Minimum k-cut; Minimum k-spanning tree; Minor testing (checking whether an input graph contains an input graph as a minor); the same holds with topological minors; Steiner tree, or Minimum spanning tree for a subset of the vertices of a graph. [2] (The minimum spanning tree for an entire graph is solvable in ...
A planar graph and its minimum spanning tree. Each edge is labeled with its weight, which here is roughly proportional to its length. A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. [1]
If is edge-unweighted every spanning tree possesses the same number of edges and thus the same weight. In the edge-weighted case, the spanning tree, the sum of the weights of the edges of which is lowest among all spanning trees of , is called a minimum spanning tree (MST). It is not necessarily unique.
Example of rectilinear minimum spanning tree from random points. In graph theory, the rectilinear minimum spanning tree (RMST) of a set of n points in the plane (or more generally, in ) is a minimum spanning tree of that set, where the weight of the edge between each pair of points is the rectilinear distance between those two points.
Examples of such greedy algorithms are Kruskal's algorithm and Prim's algorithm for finding minimum spanning trees and the algorithm for finding optimum Huffman trees. Greedy algorithms appear in the network routing as well. Using greedy routing, a message is forwarded to the neighbouring node which is "closest" to the destination.
M. Haque, Md. R. Uddin, and Md. A. Kashem (2007) found a linear time algorithm that can find the minimum degree spanning tree of series-parallel graphs with small degrees. [2] G. Yao, D. Zhu, H. Li, and S. Ma (2008) found a polynomial time algorithm that can find the minimum degree spanning tree of directed acyclic graphs. [3]
A minimum-cost spanning-forest game (MCSF game) is a generalization of an MCST game, in which multiple supply-nodes are allowed. In general, the core of an MCSF game may be empty. [1] However, if the underlying network is a tree, the core is always non-empty, and core points can be computed in strongly-polynomial time. [9]