Given a set x = {e₁, e2, ..., en} and a collection of sets S₁, S_2, ..., Sn. Each element e_i has a corresponding set S_i = {e_i}.

You are now given a sequence of commands of type Union(ex , ey ) interspersed with commands Query(ex , ey ). The union operation takes the two sets one containing ex , and the other containing ey (they might be the same set, but not at the beginning) and union it to a single set containing the elements of both. As we go creating sets we query command

asks whether the two elements in its argument NOW (at the time you reached it in the sequence) belong to the same set or a different set.

(a) Given a list of O(n) operations of Union and Query. Design an algorithm that processes

the sequence of commands in O(n log n) time. To process the sequence means to answer all

the queries in the sequence. (Hint: keep a set as a rooted tree with depth that is O(log n).

The Crux is how to union two trees and keep this property.)

(b) Given an undirected weighted graph G = (V, E). Show an implementation of Kruskals Minimum

Spanning Tree(MST) algorithm by using the Union and Query commands. Your implementation

should have O(|E| log |V|) time complexity. (Hint: In Kruskals algorithm, it

builds the MST by processing every edge in G and asks whether two endpoints of an edge

belong to two disjoint trees. If so, the algorithm adds the edge making two disjoint trees

into a single tree. )