[Java] Graph union find algorithm

For an undirected graph, we can compute the disjointed sets that represent all connected nodes in the subgraph where each node resides.

For each set, we elect a representative, all nodes reachable in a set will have the same representative. The resulting view will be a tree where the representative sits at the root and all connected nodes are its children.

Example applications include: quickly verify whether 2 nodes in a graph have a path to each other (they must belong to same set) or calculating the minimum number of edges to add to a graph to make it fully connected (or the opposite).

It is based on 2 operations:

find(Vertex x)

which will return for a given node, the representative of its subset. We recurse up the tree where this node resides until the representative is found. We optimize the operation for future searches by including path compression, where once a representative is found, all nodes along the same path are updated to track it. This makes it so that the find operation runs in O(inverse Ackermann(V)), which is considered O(1) but more realistically is O(log(log(...(V))) or how many times we need to apply log(x) to its result starting with V until the output is less than 1. It is an extremely slowly increasing sequence.

union(Vertex x, Vertex y)

which will connect the subtree where node x resides to the subtree of node y, unless they are already part of the same subtree. To improve efficiency we track in O(V) extra space the rank of each subtree (its depth) and when merging two subtrees, we connect the one with minimum depth to the other, so the overall height of the resulting tree is kept as flat as possible.

It uses O(V) extra space, to track for each node who is the representative of its subset and O(V) to track the rank of the subtree rooted at each node.

It runs in O(V Ackermann(V)) time since we run 2 find operation for each edge (pair of nodes) we unite and use the union by rank with path compression method.

You can check my implementation of unionFind on my Gist along with some tests in UnionFindJTests.