## Pages

We are given n nodes and a (possibly huge) number of edges. We are also given the cost of building a library in a city (i.e. a node) and a road (i.e. an edge). Based on these data we want to minimize the cost of creating a forest of graphs from the given nodes and edges, with the requirement that each graph should have a library on one of its nodes. This is a HackerRank problem on Graph Theory algorithms, and I am about to describe my python solution to it.

If a library is cheaper than a road, the solution is immediate. Build a library on every node.
```def solution(n, library, road, edges):
return n * library

# ...
```
Otherwise, we want to create a minimum spanning forest, so to minimize the number of roads, keeping track of the number of edges used and how many graphs are actually generated. I found natural using an adapted form of the Kruskal MST (Minimum Spanning Tree) algorithm, that looks very close to our needs.

Kruskal needs a union-find to work, and this ADT is not commonly available out of the box. So, I first implemented a python UnionFind class, see previous post for details.
Then, while working on this problem, I made a slight change to it. My algorithm was simpler and faster if its union() method returned False when nothing was actually done in it, and True only if it led to a join in two existing graph.

Using such refactored UnionFind.union(), I wrote this piece of code based on Kruskal algorithm:
```uf = UnionFind(n)

for edge in edges:
if uf.union(edge[0] - 1, edge[1] - 1):  # 2
if uf.count == 1:  # 5
break
```
1. The union-find object keeps track of the numbers of disjointed graphs in the forest, but not of edges. This extra variable does.
2. I need to convert the edges from 1-based to 0-based convention before use them. If the two nodes get connected by this call to union(), I have some extra work to do.
4. An edge has been used by union(), keep track of it.
5. If union() connected all the nodes in a single graph, there is no use in going on looping.

Now it is just a matter of adding the cost for roads and libraries to get the result.
```return road_count * road + uf.count * library
```

Complete python code for problem, union-find, and test case on GitHub.