Max chunks to make sorted II [Sliding Window]

Time: O(NLogN); Space: O(N); hard

This question is the same as “Max Chunks to Make Sorted” except the integers of the given array are not necessarily distinct, the input array could be up to length 2000, and the elements could be up to 10^8. Given an array arr of integers (not necessarily distinct), we split the array into some number of “chunks” (partitions), and individually sort each chunk. After concatenating them, the result equals the sorted array.

What is the most number of chunks we could have made?

Example 1:

Input: arr = [5,4,3,2,1]

Output: 1

Explanation:

  • Splitting into two or more chunks will not return the required result.

  • For example, splitting into [5, 4], [3, 2, 1] will result in [4, 5, 1, 2, 3], which isn’t sorted.

Example 2:

Input: arr = [2,1,3,4,4]

Output: 4

Explanation:

  • We can split into two chunks, such as [2, 1], [3, 4, 4].

  • However, splitting into [2, 1], [3], [4], [4] is the highest number of chunks possible.

Notes:

  • arr will have length in range [1, 2000].

  • arr[i] will be an integer in range [0, 10^8].

Intuition Let’s try to find the smallest left-most chunk.

Algorithm Notice that if a_1, a_2,…, a_ma is a chunk, and a_1, a_2,…, a_n is a chunk (m < nm<n), then a_m+1, a_m+2, …, a_n is a chunk too. This shows that a greedy approach produces the highest number of chunks.

We know the array arr should end up like expect = sorted(arr). If the count of the first k elements minus the count what those elements should be is zero everywhere, then the first k elements form a valid chunk. We repeatedly perform this process.

We can use a variable nonzero to count the number of letters where the current count is non-zero.

[1]:

import functools

class Solution1(object):
    def maxChunksToSorted(self, arr):
        """
        :type arr: List[int]
        :rtype: int
        """
        def compare(i1, i2):
            return arr[i1] - arr[i2] if arr[i1] != arr[i2] else i1 - i2

        idxs = [i for i in range(len(arr))]
        result, max_i = 0, 0
        for i, v in enumerate(sorted(idxs, key=functools.cmp_to_key(compare))):
            max_i = max(max_i, v)
            if max_i == i:
                result += 1
        return result

[2]:

s = Solution1()
arr = [5,4,3,2,1]
assert s.maxChunksToSorted(arr) == 1
arr = [2,1,3,4,4]
assert s.maxChunksToSorted(arr) == 4

[3]:

import collections

class Solution2(object):
    def maxChunksToSorted(self, arr):
        """
        Sliding Window
        :type arr: List[int]
        :rtype: int
        """
        count = collections.defaultdict(int)
        result = nonzero = 0

        for x, y in zip(arr, sorted(arr)):

            count[x] += 1
            if count[x] == 0: nonzero -= 1
            if count[x] == 1: nonzero += 1

            count[y] -= 1
            if count[y] == -1: nonzero += 1
            if count[y] == 0: nonzero -= 1

            if nonzero == 0:
                result += 1

        return result

[4]:

s = Solution2()
arr = [5,4,3,2,1]
assert s.maxChunksToSorted(arr) == 1
arr = [2,1,3,4,4]
assert s.maxChunksToSorted(arr) == 4