algoadvance

1. Max Chunks To Make Sorted

You are given an integer array arr of length n that represents a permutation of the integers in the range [0, n-1]. We split arr into some number of chunks (i.e., partitions), and individually sort each chunk. After concatenating them, the result should equal the sorted array.

Return the maximum number of chunks we can make to achieve this.

Clarifying Questions

1. Range Guarantees: Is it guaranteed that the elements of arr cover the full range from [0, n-1] without any missing or duplicate numbers?
   • Yes, it’s given that arr is a permutation of integers in the range [0, n-1].
2. Constraints on Array Length: Are there any constraints on the length of the array?
   • This should be a typical constraint where n can be reasonably large but not excessively so (e.g., 1 <= n <= 10^5).
3. Return Type: The function should return an integer representing the maximum number of chunks.
   • Correct.

Strategy

To solve this problem, the main idea is to iterate through the array and keep track of the maximum value seen so far. At any position i, if the maximum value seen so far is equal to i, it indicates that all the numbers from 0 to i are present in some order up to that point. This means we can create a chunk ending at i.

Here’s a step-by-step approach:

1. Initialize a variable to keep track of the maximum value seen so far (max_seen).
2. Initialize a counter for the chunks (chunks).
3. Iterate through the array:
   • Update the max_seen value to be the maximum of its current value and the current array value.
   • If max_seen is equal to the current index i, it means we can split and form a chunk. Increment the chunks counter.
4. Return the total number of chunks.

Time Complexity

This approach involves a single pass through the array, making it O(n) in time complexity.

Code

Here’s how you can implement this:

def maxChunksToSorted(arr):
    max_seen = 0
    chunks = 0

    for i, value in enumerate(arr):
        max_seen = max(max_seen, value)
        if max_seen == i:
            chunks += 1

    return chunks

# Example Usage
arr = [1, 0, 2, 3, 4]
print(maxChunksToSorted(arr))  # Output: 4

Explanation

• For arr = [1, 0, 2, 3, 4]:
  • At index 0: max_seen is 1 (cannot form a chunk).
  • At index 1: max_seen is still 1 and now it equals i = 1, so we can form first chunk ([1, 0]).
  • At index 2: max_seen is 2 (cannot form a chunk).
  • At index 3: max_seen is 3 and equals i, so we can form another chunk ([2, 3]).
  • At index 4: max_seen is 4 and equals i, so we can form another chunk ([4]).

Thus, the maximum number of chunks is 4.

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