Find minimum in K Dimensional Tree

We strongly recommend to refer below post as a prerequisite of this. K Dimensional Tree | Set 1 (Search and Insert) In this post find minimum is discussed. The operation is to find minimum in the given dimension. This is especially needed in delete operation. For example, consider below KD Tree, if given dimension is x, then output should be 5 and if given dimensions is y, then output should be 12.
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In KD tree, points are divided dimension by dimension. For example, root divides keys by dimension 0, level next to root divides by dimension 1, next level by dimension 2 if k is more than 2 (else by dimension 0), and so on. 

To find minimum we traverse nodes starting from root. If dimension of current level is same as given dimension, then required minimum lies on left side if there is left child. This is same as Binary Search Tree Minimum. Above is simple, what to do when current level's dimension is different. When dimension of current level is different, minimum may be either in left subtree or right subtree or current node may also be minimum. So we take minimum of three and return. This is different from Binary Search tree. 

Below is C++ implementation of find minimum operation. 

Output:

Minimum of 0'th dimension is 5
Minimum of 1'th dimension is 12

Time complexity- The time complexity of inserting a point into a k-dimensional tree is O(k log n), where n is the number of points already in the tree. This is because at each level of the tree, we compare the point to be inserted with the current node and recursively move down the left or right subtree based on the comparison. We stop when we reach a null node, and create a new node for the point to be inserted.

The time complexity of finding the minimum value in a given dimension of the k-dimensional tree is O(log n), as we start at the root of the tree and recursively move down the left or right subtree based on the comparison of the given dimension of the point with the current node. We stop when we reach a leaf node or a node with a null subtree.

Space complexity - The space complexity of the k-dimensional tree is O(n), where n is the number of points in the tree. This is because we store a Node structure for each point in the tree, which contains the k-dimensional point and pointers to its left and right subtrees.

Source: https://www.cs.umd.edu/class/spring2008/cmsc420/L19.kd-trees.pdf