Count the smaller items on the right side
In the realm of computer science, one intriguing problem is the 'Count Smaller' problem, where the goal is to construct another array given an unsorted array of distinct integers. This array contains the count of smaller elements on the right side of each element in the array.
Let's delve into some common approaches to solve this problem and compare their efficiency.
Naive Approach
A naive approach using nested loops has a time complexity of O(N) and auxiliary space of O(N). Although simple, it's not the most efficient method due to its high time complexity. For instance, for an array , the output would be .
Merge Sort Approach
An approach using Merge Sort has a time complexity of O(N * log N) and auxiliary space of O(N). This method is more efficient than the naive approach, but it's still not the fastest. For an array , the output would be .
Binary Indexed Tree (Fenwick Tree) or Balanced Binary Search Tree
The fastest programming and data structure method for calculating 'countSmaller[]' is typically a Binary Indexed Tree (Fenwick Tree) or a Balanced Binary Search Tree. These methods are favoured due to their efficient update and query operations with a time complexity of approximately O(n log n).
Self-Balancing Binary Search Tree
Another approach uses a Self-Balancing Binary Search Tree (AVL, Red Black, etc.). This method has a time complexity of O(N log N) and uses space to store the pair values in a different array. When inserting a new key in an AVL tree, the size of the left subtree is added to the count of smaller elements for the key being inserted.
Binary Search Tree with Additional Fields
Another approach uses a BST with 2 extra fields: one to hold the elements on the left side of a node and another to store the frequency of the element. This method also has a time complexity of O(N log N).
Set in C++ STL Approach
A third approach uses a Set in C++ STL to count smaller elements on the right side. This method has a time complexity of O(n) and the space complexity is O(n) due to the use of two lists. contains the count of smaller elements on the right side of each element in the array.
In conclusion, while each approach solves the 'Count Smaller' problem, the Binary Indexed Tree (Fenwick Tree) or Balanced Binary Search Tree methods are generally the fastest and most efficient solutions. However, the choice of method may depend on the specific programming language and the problem constraints.
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