algoadvance

You are given an array of integers, arr. A subarray is called incrementable if the sum of its elements can be increased by exactly 1 by incrementing exactly one of its elements by 1.

Your task is to return the number of such incrementable subarrays.

Constraints:

Clarifying Questions

To better understand the problem, here are some clarifying questions:

  1. What is considered a valid incrementable subarray?
    • A subarray is valid if you can pick exactly one element from it and increase it by 1, causing the sum of the entire subarray to increase by exactly 1.
  2. Can the subarrays be of any length?
    • Yes, subarrays can be of any length from 1 to the length of arr.
  3. Does the order of elements in the subarray need to be retained from the original array?
    • Yes, the subarray must be contiguous as per its occurrence in the original array.

Strategy

Given the problem requirements and constraints, the approach involves:

  1. Each subarray of any length can potentially be incremented by 1 by simply adding 1 to one of its elements.
  2. To count the incremovable subarrays, observe that for any contiguous subarray of length L, there are L possible ways to increment it. Thus, the count of subarrays of length L is straightforward.
  3. The simplest way to count required subarrays is to exploit the overlapping nature of subarrays:
    • For an array of length n, the total number of subarrays is the sum from 1 to n.
  4. Use the formula for the sum of the first n natural numbers: sum = n * (n + 1) / 2.

Thus, the total number of incrementable subarrays in arr is simply the sum of the lengths of all possible subarrays.

Code

Here is the implementation of the above logic:

def count_incremovable_subarrays(arr):
    n = len(arr)
    return n * (n + 1) // 2

# Example usage
arr = [1, 2, 3, 4]
print(count_incremovable_subarrays(arr))  # Output: 10

Explanation and Time Complexity

  1. Calculation:
    • n: Length of the array.
    • The formula n * (n + 1) // 2 calculates the total number of all incrementable subarrays.
  2. Time Complexity:
    • The time complexity is O(1) since it only involves a few arithmetic operations.
  3. Space Complexity:
    • The space complexity is O(1) as it only uses a small, constant amount of space for storing variables.

This approach efficiently counts the number of incrementable subarrays in linear time, making it effective given the constraint limits.

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