Problem Description

In this problem, you are given an array nums of integers and two additional integers, indexDifference and valueDifference. Your task is to find any two indices i and j such that the following two conditions are satisfied:

1. The absolute difference between the indices i and j is greater than or equal to indexDifference, that is abs(i - j) >= indexDifference.
2. The absolute difference between the values at indices i and j in the array nums is greater than or equal to valueDifference, that is abs(nums[i] - nums[j]) >= valueDifference.

You need to return an array where the first element is the index i and the second element is the index j. If no such pair of indices exists, you should return [-1, -1].

Note that the indices i and j may be the same, which implies that the constraints do not preclude the possibility of comparing an element with itself, as long as the index difference is non-existent (zero).

Intuition

The intuition behind the solution involves iterating through the array while keeping track of the minimum and maximum values seen so far, separated by at least indexDifference. As we traverse the array from the starting index set by indexDifference, we keep updating the minimum index mi and the maximum index mx when a smaller or larger element appears in the range that satisfies the index difference condition.

During this traversal:

• We want to check if the current maximum value we have seen so far (nums[mx]) minus the current value (nums[i]) is greater than or equal to valueDifference. If this condition is met, we have found a valid pair of indices [mx, i] and return them.
• Similarly, we check if the current value (nums[i]) minus the current minimum value we have seen so far (nums[mi]) is also greater than or equal to valueDifference. If this condition is met, we have found another valid pair of indices [mi, i] and return them.

The reason we can perform these checks is that by maintaining the smallest and largest values with the required index difference, we maximize the chances of finding a valid valueDifference. If no such pair is found throughout the traversal, we return [-1, -1] indicating the absence of such a pair of indices satisfying the given conditions.

Solution Approach

The solution uses a simple linear scan approach, enhanced with clever tracking of minimum and maximum values found within a valid index difference range. Here's the breakdown of the approach step-by-step, using the Solution provided:

1. The function findIndices takes the array nums, indexDifference, and valueDifference as its parameters.
2. Two pointers, mi and mx, are initialized to 0. They will keep track of the indices of the minimum and maximum values found within the range specified by the indexDifference.
3. We use a for loop to iterate over the array starting from the indexDifference up to the length of the array. This ensures that we always have a range of elements where the earliest element (nums[j]) is exactly indexDifference apart from the current element (nums[i]).
4. At each iteration, we calculate j to point to the element which is indexDifference behind the current element i.
5. Next, we update the mi and mx pointers if the value at nums[j] is less than the current minimum or greater than the current maximum, respectively.
6. Immediately after updating mi and mx, we perform the check for the valueDifference condition:
   • If the difference between the current element's value (nums[i]) and the minimum value (nums[mi]) is greater than or equal to valueDifference, we return the indices [mi, i] as a result.
   • Similarly, if the difference between the maximum value (nums[mx]) and the current element's value is greater than or equal to valueDifference, we return the indices [mx, i] as a result.
7. If we do not find any indices that satisfy both conditions by the end of the loop, we return [-1, -1].

Let's detail the functionality with the help of an example: Suppose nums = [5, 3, 4, 2, 1, 3], indexDifference = 3, and valueDifference = 2.

• We initialize mi = mx = 0.
• When i = 3 (indexDifference = 3), j = 0. Now, nums[i] = 2 and nums[j] = 5. mi is unchanged as nums[j] < nums[mi] does not hold true, but nums[j] > nums[mx] holds true so mx is updated to 0.
• Next checks for value difference, nums[i] - nums[mi] < valueDifference and nums[mx] - nums[i] is 5 - 2 = 3 which is greater than valueDifference so the function returns [mx, i] which is [0,3].

The solution is efficient, traversing the list only once (O(n)), with constant space complexity (O(1)), as it doesn't require additional storage proportional to the input size.

Example Walkthrough

Let's walk through an example to illustrate the solution approach using the following inputs: nums = [1, 5, 9, 3, 5], indexDifference = 2, and valueDifference = 4.

1. We initialize pointers mi = mx = 0 to keep track of the indices of the minimum and maximum values found.
2. We iterate over the array starting from i = indexDifference, which is 2 in this case. At i = 2, nums[i] = nums[2] = 9.
3. For each i, we calculate j = i - indexDifference. When i = 2, j = 2 - 2 = 0.
4. Now we compare the current element nums[i] with nums[mi] and nums[mx] to check if we need to update mi or mx. Since nums[mi] = nums[0] = 1 and nums[mx] = nums[0] = 1, and the current element nums[i] = 9 is greater than both, we update mx to 2.
5. We check the valueDifference condition with the updated indices. Here, nums[mx] - nums[i] is 0 since mx = i = 2. However, this check is redundant now as it's comparing the element with itself.
6. We move to the next iteration, with i = 3, and nums[i] = nums[3] = 3, and we calculate j = i - indexDifference = 1. Now, nums[j] = nums[1] = 5. At this point, mi remains 0 because nums[1] > nums[0], but we don't update mx because it currently points to 2 where the value is higher (9).
7. We check valueDifference again for these values. We find that abs(nums[mx] - nums[i]) = abs(9 - 3) = 6, which is greater than valueDifference, so we return the indices [mx, i] which is [2, 3].
8. Since we have found a valid pair that satisfies the conditions, the function terminates and returns [2, 3].

Through this example, we demonstrated how the algorithm scans the list while maintaining the indices of the minimum and maximum values to efficiently find a valid pair [i, j] that satisfies both the index difference and value difference conditions. If no such pair is found by the time the algorithm has iterated through the entire array, it would return [-1, -1].

Solution Implementation

Time and Space Complexity

Time Complexity

The given code iterates over the array nums using a single loop starting from indexDifference up to len(nums). For each iteration, the code performs constant-time operations such as comparisons and assignments. These operations do not depend on the size of the input array, except for the iteration that is linear in terms of the number of elements in nums. Therefore, the time complexity of the code is O(n), where n is the length of the nums array.

Space Complexity

The space complexity of the code is determined by the extra space used aside from the input. The code uses a constant number of extra variables (mi, mx, i, j) that do not scale with the size of the input array. Therefore, the space complexity is O(1), as there is no additional data structure or space-dependent on the input size used within the function.

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