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.
A minimum spanning tree is a special kind of tree that minimizes the lengths (or “weights”) of the edges of the tree. An example is a cable company wanting to lay line to multiple neighborhoods; by minimizing the amount of cable laid, the cable company will save money.
As you can probably imagine, larger graphs have more nodes and many more possibilities for subgraphs. The number of subgraphs can quickly reach into the millions, or billions, making it very difficult (and sometimes impossible) to find the minimum spanning tree. Additionally, the lengths usually have different weights; one 5m long edge might be given a weight of 5, another of the same length might be given a weight of 7.
One simple definition is that a tree is a connected graph with no cycles, where a cycle let’s you go from a node to itself without repeating an edge.
A few popular algorithms for finding this minimum distance include: 1.Kruskal’s algorithm, 2.Prim’s algorithm
Kruskal’s algorithm
Kruskal’s algorithm is a minimum-spanning-tree algorithm which finds an edge of the least possible weight that connects any two trees in the forest.It is a greedy algorithm in graph theory as it finds a minimum spanning tree for a connected weighted graph adding increasing cost arcs at each step. 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. If the graph is not connected, then it finds a minimum spanning forest (a minimum spanning tree for each connected component).
Total edges=no.of vertices -1
Kruskal’s algorithm says that always select minimum cost edge but if it is forming a cycle don’t include that edge.
Let G=(V,E)be the given graph.Initially our MST contains only vertices of given graph with no edges. In other words, initially MST has V connected components with each vertex acting as one connected component.The goal is to add minimum weight edges to our MST such that we are left with single connected component that comprises all the vertices of graph.
Prim’s algorithm
Prim’s (also known as Jarník’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. The algorithm operates by building this tree one vertex at a time, from an arbitrary starting vertex, at each step adding the cheapest possible connection from the tree to another vertex.
In Prim’s Algorithm we grow the spanning tree from a starting position. Unlike an edge in Kruskal’s, we add vertex to the growing spanning tree in Prim’s.
In Prim’s Algorithm, we will start with an arbitrary node (it doesn’t matter which one) and mark it. In each iteration we will mark a new vertex that is adjacent to the one that we have already marked. As a greedy algorithm, Prim’s algorithm will select the cheapest edge and mark the vertex.
Prim’s vs Kruskal’s algorithm
APPLICATIONS :
- Cluster Analysis.
- Real-time face tracking and verification (i.e. locating human faces in a video stream).
- Protocols in computer science to avoid network cycles.
- Entropy based image registration.
- Max bottleneck paths.
- Dithering (adding white noise to a digital recording in order to reduce distortion).