Problem Description

The task requires us to find the number of unique pairs (i, j) from a given integer array nums, where the first element of the pair is at an index less than the second element of the pair (i < j). The condition for these pairs to count is two-fold:

1. The values at these indices must be the same (nums[i] == nums[j]).
2. The product of the indices i * j must be evenly divisible by a given integer k (which means (i * j) % k == 0 should be true).

In other words, for array nums with n integers and the given integer k, we need to count how many distinct pairs of indices have the same value in nums and their index product is a multiple of k.

This is a combinatorial problem that involves both array traversal and elementary number theory.

Intuition

The intuition behind the solution is based on a brute-force approach where we evaluate all possible pairs to see if they satisfy the two conditions mentioned. This means we:

1. Iterate through the array using two loops:
   • An outer loop, where we pick every element i starting from index 0 to n-2
   • An inner loop, where we pick every element j starting from index i+1 to n-1
2. For each pair (i, j) found through this iteration, check if nums[i] and nums[j] are equal.
3. If the numbers at i and j are equal, further check if the product (i * j) is divisible by k.
4. If both conditions hold true, count this pair towards our total count.

The solution uses a single return statement with a sum of a generator expression. This generator expression iterates over all pairs as described and uses a conditional expression to count only those pairs that satisfy the two conditions. Since it loops through half of the possible combinations (since i < j), it ensures that each unique pair is only considered once. This approach is simple and direct but can be inefficient for large arrays since it performs checks for every index pair.

Solution Approach

The implementation of the solution uses a brute-force algorithm with nested loops to iterate over all possible pairs of indices in the given array. There is no use of additional data structures; the solution operates directly on the input array nums. Here is how the solution is implemented:

1. We define a function countPairs that takes in an array nums and an integer k as input.
2. Inside the function, we calculate the length of the array n. This is necessary to know the range for our iterations.
3. We use a nested loop construct to generate all possible pairs:
   • The outer loop runs from 0 to n-1. The variable i in this loop represents the first index of the pair.
   • The inner loop runs from i+1 to n. The variable j represents the second index of the pair.
4. For each pair (i, j), the algorithm performs two checks encapsulated in a conditional statement:
   • First, it checks if nums[i] is equal to nums[j]. This is done by nums[i] == nums[j], ensuring that the values at these indices are the same.
   • Second, it checks if the product of i and j (i * j) is divisible by k. This is achieved through (i * j) % k == 0, confirming that the product is divisible by k without a remainder.
5. The conditional statement is part of a generator expression that evaluates to True for each pair that meets the above conditions and False otherwise.
6. The sum function is then applied directly to this generator expression. In Python, True equates to 1 and False to 0, so summing the generator expression effectively counts the number of True (valid pairs) instances, which is the answer we need to return.

One key reason why this is a brute-force algorithm is because it does not attempt to optimize by using any patterns or early exit conditions. The solution evaluates every index pair before reaching the count. This solution is straightforward and easy to understand, but it has a time complexity of O(n^2), which means that the algorithm's running time increases quadratically with the size of the input array. Such time complexity is not ideal for large datasets.

The algorithm does not use additional space beyond the input and intermediate variables, thus has a space complexity of O(1).

To summarize, the implementation uses simple iteration and basic conditional checks to find and count the pairs that satisfy both conditions. This brute-force approach is naturally chosen for its simplicity when dealing with small to moderately sized arrays.

Example Walkthrough

Let's illustrate the solution approach with a small example.

Suppose we have an array nums = [1, 2, 1, 3, 2] and our given integer k = 2. We are interested in finding pairs (i, j) where nums[i] == nums[j] and (i * j) % k == 0.

Following the steps outlined in the solution approach:

1. We define our function: countPairs(nums, k).

2. We determine the length of nums which is 5. We will need this for our loop ranges.

3. We start our loops:

   • The outer loop will have i iterate from 0 to 4 (the range is 0 to n-1).
   • The inner loop will begin from i+1 and go up to 5. So for i = 0, the inner loop will start at 1.

4. Now, we need to check every pair (i, j):

   • For i = 0, our pairs would be (0, 1), (0, 2), (0, 3), and (0, 4).
   • We see that nums[0] == nums[2] which is 1 == 1, and (0 * 2) % 2 == 0. So (0, 2) is a valid pair.
   • No other pairs with i = 0 satisfy both conditions, so we move to i = 1.

5. With i = 1, we consider pairs (1, 2), (1, 3), and (1, 4).

   • None of these satisfy both conditions. Either nums[i] != nums[j] or (i * j) % 2 != 0.
   • Continue to the next index i = 2.

6. With i = 2, we consider pairs (2, 3) and (2, 4).

   • Again, none of these are valid because the numbers at the indices don't match or their product isn't divisible by 2.
   • Move on to i = 3.

7. At i = 3, the only pair to consider is (3, 4).

   • Since nums[3] != nums[4], this is not a valid pair.

Putting it all together, the only valid pair we have found is (0, 2). Therefore, the function countPairs(nums, k) will return 1.

This example helps to illustrate how the solution iteratively checks each possible pair according to the algorithm's conditions, incrementing the count when both conditions are met. The result is a count of unique index pairs where the array values are the same and their product is divisible by the given k.

Solution Implementation

Time and Space Complexity

Time Complexity

The provided code has a nested loop, where the outer loop runs from 0 to n-1 and the inner loop runs from i+1 to n-1. Here, n is the length of the nums list.

In the worst case, the number of iterations will be the sum of the series from 1 to n-1, which is (n-1)*(n)/2 or O(n^2/2). Since constants are dropped in Big O notation, the time complexity is O(n^2).

Space Complexity

The code doesn't use any additional data structures that grow with the input size. The only variables used are for loop counters and temporary variables for calculations, which occupy constant space. Therefore, the space complexity is O(1).

Learn more about how to find time and space complexity quickly using problem constraints.