Problem Description

In this problem, we are tasked with constructing an array nums with the following constraints:

• The length of the array nums is equal to the given integer n.
• Each element in the array is a positive integer.
• The absolute difference between any two consecutive elements is at most 1.
• The sum of all elements in nums does not exceed a given integer maxSum.
• Among all possible nums arrays that satisfy the above conditions, we want to maximize the value of nums[index].

Our objective is to find out what that maximized nums[index] is, given the parameters n (the length of the array), index (the specific position in the array we want to maximize), and maxSum (the maximum allowed sum of all elements in the array).

Intuition

The intuition behind the solution is to leverage binary search to efficiently find the maximum possible value of nums[index].

1. We know that nums[index] must be a positive integer and that the sum of all elements in the array must not exceed maxSum. This means that nums[index] has an upper bound given by maxSum.

2. The idea is to perform a binary search, starting with the lowest possible value for nums[index] (which is 1) and the maximum possible value, which would be maxSum (assuming all other values in the array are 1).

3. For each possible value of nums[index] we test in our binary search, we calculate the sum of elements that would be required to form a valid array if nums[index] were that value. To do this, the sum function is used, which calculates the sum of elements in a portion of the array that slopes upwards or downwards by 1 with each step away from nums[index].

4. If the calculated sum is less than or equal to maxSum while maintaining the constraints of the problem, it means we can potentially increase nums[index]. On the other hand, if the sum exceeds maxSum, then nums[index] must be lower.

By using this method, when we eventually narrow down to a single value through binary search, we find the maximum value of nums[index] that can exist within an array satisfying all the described constraints.

Learn more about Greedy and Binary Search patterns.

Solution Approach

The solution provided uses a binary search algorithm to find the maximum value of nums[index]. The binary search algorithm is a classic approach to efficiently search for an element in a sorted array by repeatedly dividing the search interval in half.

The following steps are taken in this implementation:

1. Initialize Search Range: The search for the maximum value of nums[index] begins by setting the left bound to 1, which is the smallest possible value for any element in the array, and right bound to maxSum, the highest possible value for the nums[index] (assuming all other elements are at the minimum value of 1).

2. Binary Search Loop: A while loop runs as long as the left bound is less than right. The loop calculates the mid value as the average of left and right, setting up the next guess for nums[index].

3. Calculate Required Sum: In each iteration, the program calculates the required sum for the array if nums[index] were equal to mid. This is done using a custom sum function, which accounts for the sum of the pyramid-like sequence that forms when values decrease by 1 on each side of the index.

   • sum(x, cnt) Function: This function calculates the sum of the first cnt terms of an arithmetic series that starts at x and decreases by 1 each term until it reaches 1 or runs out of terms. If x is greater than cnt, the sum is the sum of cnt terms starting at x and subtracting down to (x - cnt + 1). If x is less than or equal to cnt, then the sum includes all numbers down to 1, and the remaining terms are 1s. The formula is based on the sum of the first n natural numbers n(n + 1)/2 and adjusted for the start being x instead of 1.

   The function calculates two sums:

   • The sum for the left side from nums[index] to the start of the array.
   • The sum for the right side from nums[index] to the end of the array.

4. Update Search Bounds: Depending on whether the sum of the sequence with nums[index] equal to mid exceeds maxSum or not, we adjust the binary search range accordingly:

   • If the total sum does not exceed maxSum, it is safe to move the left bound up to mid because a larger or equal nums[index] is viable.
   • If the total sum exceeds maxSum, the right bound is set to mid - 1 because we need a smaller nums[index] to reduce the total sum.

5. Determine the Maximum Value: After exit from the loop, the maximum possible value for nums[index] is found, which is pointed by left. At this point, left is the largest value that did not violate the sum constraint. Since the constraints ensure that the sequence is increasing then decreasing around nums[index] and that the maximum sum does not exceed maxSum, left indeed maximizes nums[index].

By the end of this process, the solution has efficiently zeroed in on the largest possible value for nums[index] in compliance with all the problem's constraints using binary search.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach using the mentioned constraints.

Suppose we have the following inputs:

• n = 5 (length of the array)
• index = 2 (position in the array we want to maximize)
• maxSum = 10 (maximum allowed sum of all elements in the array)

Step by Step Process:

1. Initialize Search Range:

   • We start with left = 1 and right = maxSum = 10.

2. Binary Search Loop:

   • Begin with left = 1 and right = 10.
   • Calculate mid = (left + right) / 2, let's assume integer division, so with left = 1 and right = 10, mid = 5.

3. Calculate Required Sum:

   • We calculate the sum for nums[index] = mid.
   • Using the sum function, we calculate the sum for the left part (from start to index) and the right part (from index to end).
   • For the left part, we need 2 values (elements at index 0 and 1), and for the right part, we also need 2 values (elements at index 3 and 4).
   • sumLeft = sum(mid, 3) = sum(5, 3) = 5 + 4 + 1 (as the third term would be 0) = 10.
   • sumRight = sum(mid, 2) = sum(5, 2) = 5 + 4 = 9.
   • Total sum = sumLeft + sumRight - mid (we subtract mid because it's counted in both the left and right sums) = 10 + 9 - 5 = 14.
   • This sum exceeds maxSum; therefore, we need to reduce mid.

4. Update Search Bounds:

   • Since 14 exceeds maxSum, we set right = mid - 1 = 4.

5. Loop Continuation:

   • Adjust mid with the new bounds, left = 1 and right = 4.
   • New mid = (1 + 4) / 2 = 2.
   • Calculate new sums with mid = 2.
   • sumLeft = sum(2, 3) = 2 + 1 + 1 = 4.
   • sumRight = sum(2, 2) = 2 + 1 = 3.
   • Total sum = sumLeft + sumRight - mid = 4 + 3 - 2 = 5.
   • This sum fits within maxSum, so we try to increase mid by moving left up.

6. Update Search Bounds Again:

   • As 5 is less than maxSum, we now set left = mid = 2.
   • Now left = 2 and right = 4.

7. Finishing the Search:

   • Continue the binary search until left and right meet.
   • Suppose in the next iteration mid = 3 does not exceed maxSum but mid = 4 does, we will stop with left at 3.

8. Determination:

   • The binary search concludes when left equals right, which is the value just before the sum exceeded maxSum.
   • We find left to be 3, so nums[index] = 3 is the largest possible value that does not violate the constraints.

This example illustrates the solution approach, showing how a binary search systematically narrows down the maximum value for nums[index] while adhering to the problem's constraints. By calculating sums that would form a valid array configuration for each guess and adjusting our bounds accordingly, we efficiently pinpoint the solution.

Solution Implementation

Time and Space Complexity

The time complexity of the provided code is O(log(maxSum)). The binary search algorithm runs between 1 and maxSum, which determines the number of iterations needed to find the solution. In each iteration, the sum function is called twice, each call of which is O(1) because the operations involve simple arithmetic and a conditional check, and thus don't depend on the size of n or maxSum.

The space complexity of the code is O(1). There are only a fixed number of variables used (left, right, mid, and within the sum function), and no extra space that scales with the input size is required. Therefore, the amount of memory used is constant.

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