Problem Description

In this problem, we are provided with an array of integers arr and three separate integers a, b, and c. Our goal is to determine how many triplets (arr[i], arr[j], arr[k]) meet certain criteria, where i, j, and k are distinct indices into the array with i < j < k.

A triplet is considered "good" if it satisfies the following conditions:

• The absolute difference between arr[i] and arr[j] is less than or equal to a.
• The absolute difference between arr[j] and arr[k] is less than or equal to b.
• The absolute difference between arr[i] and arr[k] is less than or equal to c.

The problem asks us to return the total number of these good triplets.

Intuition

The straightforward way to find all good triplets is to check every possible triplet in the array to see if it meets the criteria. This involves using three nested loops to generate all possible combinations of i, j, and k where 0 <= i < j < k < arr.length.

Within these loops, we will calculate the absolute differences between arr[i], arr[j], and arr[k] and check if they satisfy the conditions related to a, b, and c. If a triplet satisfies all these conditions, we increment a counter.

This method is a brute-force approach since it explores all possible triplets without any optimizations. It ensures that all triplets are checked and counted correctly, albeit at the expense of potentially higher computational time, especially for large arrays.

Solution Approach

The implementation of the solution provided in the reference code uses the brute-force approach, which is a common pattern when dealing with problems that require us to explore all possible combinations of elements to find a particular set that satisfies certain conditions.

Algorithm steps:

1. Initialize a counter ans to keep track of the number of good triplets found.
2. Calculate the length n of the input array arr.
3. Iterate over the array with the first pointer i, which runs from the beginning of the array to the third-to-last element.
4. For each i, iterate with the second pointer j, which runs from one element after i to the second-to-last element.
5. For each j, iterate with the third pointer k, which runs from one element after j to the last element.
6. Inside the innermost loop (where k is iterating), check if the current triplet (arr[i], arr[j], arr[k]) is a good triplet by verifying the three conditions with the given a, b, and c:
   • abs(arr[i] - arr[j]) <= a
   • abs(arr[j] - arr[k]) <= b
   • abs(arr[i] - arr[k]) <= c
7. If all three conditions are satisfied, increment the counter ans.
8. After all iterations are complete, return the value of ans.

This approach does not use any specific data structures or complex algorithms, but rather relies on nested loops to examine each triplet one by one. The only operations involved are arithmetic operations (- and abs) and simple comparisons (<=).

While it is an exhaustive solution, it is not the most efficient in terms of time complexity. The time complexity of this approach is O(n^3), as there are three nested loops each iterating over the array. This solution can be quite slow for larger arrays but is perfectly fine for smaller input sizes where a more sophisticated algorithm might not lead to significant improvements in runtime or when fast and simple programming is required over an optimized, complex solution.

Example Walkthrough

Let's consider a small example to illustrate the solution approach. Suppose we have the following:

• arr = [2, 1, 3]
• a = 1
• b = 1
• c = 1

Now we want to find the number of good triplets that satisfy all the given conditions.

We start by initializing our counter ans to 0. Our array length n is 3.

1. We begin iterating with pointer i starting at 0.
2. For i = 0 (arr[i] = 2), we move on to pointer j.
3. We let j iterate starting from i + 1, which is 1.
4. For j = 1 (arr[j] = 1), we then use pointer k.
5. k starts iterating from j + 1, which is 2.
6. For k = 2 (arr[k] = 3), we now check if (arr[i], arr[j], arr[k]) = (2, 1, 3) is a good triplet.
   • The absolute difference between arr[i] and arr[j] is |2 - 1| = 1, which is less than or equal to a.
   • The absolute difference between arr[j] and arr[k] is |1 - 3| = 2, so this is not less than or equal to b and we can ignore this triplet.

Since there are no other combinations to check, our count ans stays at 0. We conclude that there are no good triplets in the array [2, 1, 3] with the given a, b, and c values.

Applying the same approach in a larger array would involve more iterations, but the steps would be similar: iterate over each potential triplet and check if they satisfy the conditions.

Solution Implementation

Time and Space Complexity

The given Python code implements a function to count the number of good triplets in an array. A triplet (arr[i], arr[j], arr[k]) is considered good if it satisfies all of the following conditions:

• 0 <= i < j < k < arr.length
• |arr[i] - arr[j]| <= a
• |arr[j] - arr[k]| <= b
• |arr[i] - arr[k]| <= c

Where |x - y| denotes the absolute value of the difference between x and y.

Time Complexity

The time complexity of the function is determined by the three nested for-loops. The outermost loop runs n times, where n is the length of the array. The middle loop runs up to n-1 times, and the innermost loop runs up to n-2 times. Therefore, the total number of iterations is governed by the series of descending integers starting from n and decreasing to 1.

The time complexity can then be calculated as the sum of the series Σ(i * (i-1) / 2), for i from 1 to n-2. This simplifies to O(n^3) since the loop counters are independent and the series is a sum of cubic numbers.

Space Complexity

The space complexity of the solution is O(1). Other than the input array, only a fixed number of integer variables (ans, n, i, j, k) are used, and their number does not depend on the input size n.

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