**π‘ Problem Formulation:** In Python, given an array, how can one count the number of index pairs where the elements at those indices are the same? Assume we’re provided with an array like `[3, 6, 2, 6, 3]`

, and we want to output the count of index pairs having the same value. For instance, the pair of indices (0, 4) should be counted since both contain the value 3.

## Method 1: Brute Force Approach

This method iterates through each possible pair in the array and counts when both elements are identical. This is the simplest method but also the least efficient with a time complexity of O(n^2), where n is the number of elements in the array.

Here’s an example:

def count_identical_pairs(arr): count = 0 n = len(arr) for i in range(n): for j in range(i+1, n): if arr[i] == arr[j]: count += 1 return count print(count_identical_pairs([3, 6, 2, 6, 3]))

The output of this code snippet:

2

This snippet defines a function `count_identical_pairs()`

that takes an array as input and returns the number of index pairs with identical corresponding values. It uses a nested loop to compare all unique pairs of indices in the array.

## Method 2: Dictionary Track of Elements

A more efficient method, with a time complexity of O(n), uses a dictionary to track the occurrences of elements. For each element, the index pairs can be calculated using combinations, which are then summed up for the final count.

Here’s an example:

from collections import Counter def count_identical_pairs_efficient(arr): count = 0 element_count = Counter(arr) for key in element_count: count += element_count[key] * (element_count[key] - 1) // 2 return count print(count_identical_pairs_efficient([3, 6, 2, 6, 3]))

The output of this code snippet:

2

We use `collections.Counter`

to count the frequency of each element in the array. The function then calculates the number of index pairs for each element using the formula for combinations and sums these up for the final result.

## Method 3: Using itertools.combinations

The itertools.combinations function can be used to generate all possible index pairs and count the ones meeting the criteria directly. This method is readable but has the same efficiency as the brute force approach, O(n^2).

Here’s an example:

from itertools import combinations def count_identical_pairs_itertools(arr): return sum(1 for i, j in combinations(range(len(arr)), 2) if arr[i] == arr[j]) print(count_identical_pairs_itertools([3, 6, 2, 6, 3]))

The output of this code snippet:

2

The function uses `combinations()`

from the `itertools`

module to generate index pairs and counts those with equal elements using a generator expression.

## Method 4: Use Numpy for Large Datasets

For large datasets, leveraging Numpy’s efficient array operations can greatly improve performance. This method requires knowledge of Numpy but provides a very fast solution compared to pure Python methods.

Here’s an example:

import numpy as np def count_identical_pairs_numpy(arr): arr = np.array(arr) return sum(np.sum(arr == arr[i]) - 1 for i in range(len(arr))) // 2 print(count_identical_pairs_numpy([3, 6, 2, 6, 3]))

The output of this code snippet:

2

This function converts the list to a Numpy array and uses array broadcasting to compare all elements. The count of true comparisons for each element is summed and then corrected for double-counting.

## Bonus One-Liner Method 5: List Comprehension and set

A concise one-liner method combines list comprehension and set to identify unique pairs and then count them. This method is quick and easy to write but not very efficient for larger arrays due to its O(n^2) complexity.

Here’s an example:

arr = [3, 6, 2, 6, 3] identical_pairs_count = sum(1 for i in range(len(arr)) for j in range(i+1, len(arr)) if arr[i] == arr[j]) print(identical_pairs_count)

The output of this code snippet:

2

This one-liner defines a variable `identical_pairs_count`

with a nested list comprehension that counts the index pairs with identical elements using a condition inside the comprehension.

## Summary/Discussion

**Method 1:**Brute Force Approach. Simple to understand and implement. Inefficient for large arrays due to its O(n^2) time complexity.**Method 2:**Dictionary Track of Elements. Efficient use of data structure to lower the time complexity to O(n), making it the preferred method for most cases.**Method 3:**Using itertools.combinations. Offers clean and readable code but has the same downside as the brute force method in terms of performance.**Method 4:**Use Numpy for Large Datasets. Ideal for large datasets where performance is crucial, though it requires Numpy knowledge and installation.**Method 5:**Bonus One-Liner. It provides a quick way to solve the problem in a single line but suffers from inefficiency like the brute force and itertools methods.