💡 Problem Formulation: Given an unsorted array of integers, we want to find the maximum number of “chunks” that can be created such that, when these chunks are individually sorted, the entire array is sorted. For example, in the array [2,0,1,4,3], the maximum number of chunks is 1 because the array needs to be sorted as a whole to be sorted correctly.
Method 1: Brute Force Approach
The brute force approach checks each possible partition of the array to see if sorting each chunk results in a sorted array. It’s the most straightforward technique but can be inefficient for large arrays.
Here’s an example:
def max_chunks_brute_force(arr):
n = len(arr)
count = 0
for i in range(n):
max_in_chunk = max(arr[:i+1])
if max_in_chunk == i:
count += 1
return count
arr = [2,0,1,4,3]
print(max_chunks_brute_force(arr))Output: 1
This code goes through the array and checks if the maximum value within a chunk up to index i is i, indicating that all previous elements are smaller and thus the chunk can be sorted independently.
Method 2: Greedy Approach
The greedy approach tracks the maximum element found so far and the number of chunks. It increments the chunk count each time the current maximum matches the current index, indicating that a sortable chunk was identified.
Here’s an example:
def max_chunks_greedy(arr):
count = max_so_far = 0
for i, num in enumerate(arr):
max_so_far = max(max_so_far, num)
if max_so_far == i:
count += 1
return count
arr = [2,0,1,4,3]
print(max_chunks_greedy(arr))Output: 1
The code iterates through the array, updates the maximum value so far, and increases the chunk counter every time a chunk that can be sorted independently is found.
Method 3: Sorting and Mapping Approach
This method involves sorting a copy of the array and creating a mapping. Then, by comparing the original and sorted arrays while tracking indices, we can identify chunks that can be sorted to match the sorted array.
Here’s an example:
def max_chunks_sorting_mapping(arr):
sorted_arr = sorted(arr)
chunks = count = 0
for i in range(len(arr)):
chunks += arr[i] != sorted_arr[i]
if chunks == 0:
count += 1
return count
arr = [2,0,1,4,3]
print(max_chunks_sorting_mapping(arr))Output: 1
This code compares the original array with the sorted one and counts discrepancies. Whenever the number of discrepancies returns to zero, we know we’ve completed a chunk.
Method 4: Use of Stack
This technique utilizes a stack to keep track of potential chunk boundaries. It’s a more sophisticated method ideal for complex scenarios and benefits from optimized sorting of subarrays.
Here’s an example:
def max_chunks_stack(arr):
stack = []
for num in arr:
if not stack or num >= stack[-1]:
stack.append(num)
else:
cur_max = stack.pop()
while stack and num < stack[-1]:
stack.pop()
stack.append(cur_max)
return len(stack)
arr = [2,0,1,4,3]
print(max_chunks_stack(arr))Output: 1
The stack method continuously compares the current element with the last element in the stack (if any) and manipulates the stack’s length to find the maximum number of sortable chunks.
Bonus One-Liner Method 5: Pythonic Approach
A more pythonic approach to the problem may involve a clever one-liner that combines list comprehension, enumeration, and the all() function to concisely solve the problem.
Here’s an example:
max_chunks_pythonic = lambda arr: sum(all(x <= i for x in arr[:i+1]) for i in range(len(arr))) arr = [2,0,1,4,3] print(max_chunks_pythonic(arr))
Output: 1
The one-liner checks, for each index in the array, if all elements in the chunk are less than or equal to the index itself. It then sums up the boolean results to obtain the total number of chunks.
Summary/Discussion
- Method 1: Brute Force. Simple to implement but not optimal for large arrays due to its time complexity.
- Method 2: Greedy Approach. Efficient and easy to understand. Works well with any array size.
- Method 3: Sorting and Mapping. Involves additional sorting which may increase the time complexity depending on the array size.
- Method 4: Use of Stack. This method is less intuitive but handles complex cases well.
- Method 5: Pythonic Approach. Compact and elegant, it is ideal for those who appreciate Python’s capabilities for conciseness.