## Cheapest Link and Kruskal's Algorithms

The *Cheapest-Link* and *Kruskal's* are similar algoritms that perform dissimilar tasks on *weighted graphs*. A *weighted graph* is a graph whose edges have been assigned numbers - their weights. Any weighted graph, in particular, a subgraph of a weighted graph, is also assigned weight - the sum of weights of all its edges.

### The Cheapest-Link Algorithm

- On every step mark the cheapest unmarked edge.
- See that the marked edges
- do not form circuits, except when algorithm has stopped,
- do not converge three at a vertex.

- Repeat 1 and 2 while possible.

### Kruskal's Algorithm

- On every step mark the cheapest unmarked edge.
- See that the marked edges do not form circuits.
- Repeat 1 and 2 while possible.

The Cheapest-Link algorithm leads to a Hamilton circuit, if any exists.

Kruskal's algorithm always leads to a *minimum spanning tree* of the given graph. A graph is a *tree* if (1) it's connected and (2) it has no circuits. A spanning tree of a graph is any of its subgraphs that contains all its vertices and is also a tree. A minimum spanning tree has the least weight among all spanning trees of the given graph.

(In the examples below, always click on the cheapest remaining unmarked edge.)

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