Arrays are one of the most fundamental data structures in computer science. Manipulating and analyzing arrays form the core of many algorithmic challenges. One such intriguing problem is the “Partition Array Into Three Parts With Equal Sum” problem on Leetcode. It tests one’s ability to dissect an array and find patterns within its elements. This article will deep dive into this problem, exploring its nuances and culminating in a Pythonic solution.

### Table of Contents

- Problem Statement
- Preliminary Analysis
- Decoding the Problem
- Solving Strategy
- Python Implementation
- Time and Space Complexity Analysis
- Conclusion

### 1. Problem Statement

Given an array `A`

of integers, the task is to partition the array into three non-empty parts such that all of these parts represent the same sum. The elements in each partition should appear in the order they appear in the original array.

Return `True`

if the array can be partitioned in this way, otherwise return `False`

.

**Example:**

```
Input: A = [0,2,1,-6,6,-7,9,1,2,0,1]
Output: True
Explanation: The array can be split as [0,2,1], [-6,6,-7,9,1] and [2,0,1], each of which sums to 3.
```

### 2. Preliminary Analysis

If an array can be split into three parts with equal sum, the total sum of the entire array should be divisible by 3. If not, we can instantly return `False`

.

### 3. Decoding the Problem

The main challenge here is not just determining the equal sum but also ensuring that the array can be split into three parts respecting the given condition.

### 4. Solving Strategy

**Calculate the Total Sum:**Compute the sum of the entire array. If it’s not divisible by 3, return`False`

.**Determine the Target Sum:**Divide the total sum by 3. This gives the target sum for each of the three parts.**Iterative Splitting:**Traverse the array, trying to form parts with the target sum. Once you form a part, reset the current sum and try to form the next part.**Count the Successful Splits:**If you can successfully form three parts by the end of the array, return`True`

. Else, return`False`

.

### 5. Python Implementation

```
def canThreePartsEqualSum(A):
total_sum = sum(A)
# If total sum is not divisible by 3, return False
if total_sum % 3 != 0:
return False
target_sum = total_sum // 3
num_parts = 0 # To count the number of parts with the target sum
current_sum = 0
for num in A:
current_sum += num
# If we reach the target sum, form a part and reset current sum
if current_sum == target_sum:
num_parts += 1
current_sum = 0
# Return True only if we can split the array into 3 parts
return num_parts >= 3
```

### 6. Time and Space Complexity Analysis

**Time Complexity:**- Calculating the total sum takes O(n) time.
- Traversing the array to form parts also takes O(n) time.
- Thus, the overall time complexity is O(n).

**Space Complexity:**- We are using a constant amount of extra space, irrespective of the size of the input array.
- Thus, the space complexity is O(1).

### 7. Conclusion

The “Partition Array Into Three Parts With Equal Sum” problem presents an opportunity to understand array manipulations and find patterns in sequential data. It’s a classic example that showcases how intuitive reasoning, combined with algorithmic strategies, can lead to an efficient solution.