Finding maximum and minimum in union find algorithm

This post is a problem based on union find algorithm. For the algorithm, please see "Weighted Quick Union With Path Compression - Coursera Algorithms"

Problem: Given an element, find the max/min of it's connected components tree.

Following is my, probably brute force, solution to find max/min. For this, we use extra arrays to store the max and min values. We updated the max and min only for root nodes. For all other nodes, just find the root and it's max/min.

Weighted Quick Union With Path Compression - Coursera Algorithms

Weighted Quick Union can be improved further by path compression. The idea here is to avoid flatten the tree. After computing the root of a node,  we set the id of each examined node to point to that root.

Running Time:
For M union-find operations on N elements, running time  ≤ c ( N + M lg* N ) array accesses. 

lg*N = number of times we have to take log of N to get 1.

It's almost linear and union find doesn't have any linear algorithm.

Following is the program:

Weighted Quick Union Algorithm - Coursera Algorithms

Quick Find and Quick Union were slow in performing union and finding if two elements are connected respectively. However, Quick Union can be improvised by using weighted trees. The idea here is to avoid tall trees of Quick Union and always balance the trees such that smaller trees always link up to the larger trees.

We use array to initialize elements to indices itself. To weigh trees, we use an extra array to track the size of trees at each element.

Any node x's depth increases by one when it's tree T1 is merged into another T2.
As |T2|>=|T1| , the size of tree containing x at least doubles.
As N*lg N = N, the size of tree containing x can double at most lg N times.
So the depth of any node at x is at most lg N.

Running Time:
initialize - N
union(p,q) - lg N⁺ - including the root calculations
connected(p,q) - lg N - proportional to depth of p and q


Following is the program:

Quick Union Algorithm - Coursera Algorithms

Quick Find was very slow with quadratic operations. Quick Union is a lazy union-find algorithm to unite sets and to see if an element is connected to another. We use array to initialize elements to indices itself. Later, we update the elements to point to it's parent.  This way we create trees out of elements.


Array accesses:
initialize - N
union(p,q) - N⁺ including the cost of root calculations
connected(p,q) -N

The trees can get quite tall, making the connected operation quite slow.

Following is the program:

Quick Find Algorithm - Coursera Algorithms

Quick Find is an eager union-find algorithm.

Purpose: To find if two elements are connected and to perform union operation on two elements.

We will consider the elements as simple int array. We initialize array to index elements itself and then change the ids in the array when an union operation is performed.

Array accesses: 2N+1
initialize - N
union(p,q) - N
connected(p,q) - 1

If N unions are performed on N elements, union takes N² array access. Quadratic operations don't scale.

Following is the program: