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2826. Sorting Three Groups

Description

You are given an integer array nums. Each element in nums is 1, 2 or 3. In each operation, you can remove an element from nums. Return the minimum number of operations to make nums non-decreasing.

 

Example 1:

Input: nums = [2,1,3,2,1]

Output: 3

Explanation:

One of the optimal solutions is to remove nums[0], nums[2] and nums[3].

Example 2:

Input: nums = [1,3,2,1,3,3]

Output: 2

Explanation:

One of the optimal solutions is to remove nums[1] and nums[2].

Example 3:

Input: nums = [2,2,2,2,3,3]

Output: 0

Explanation:

nums is already non-decreasing.

 

Constraints:

  • 1 <= nums.length <= 100
  • 1 <= nums[i] <= 3

 

Follow-up: Can you come up with an algorithm that runs in O(n) time complexity?

Solutions

Solution 1: Dynamic Programming

We define \(f[i][j]\) as the minimum number of operations to turn the first \(i\) numbers into a beautiful array, and the \(i\)th number is changed to \(j+1\). The answer is \(\min(f[n][0], f[n][1], f[n][2])\).

We can enumerate all cases where the \(i\)th number is changed to \(j+1\), and then take the minimum value. Here, we can use a rolling array to optimize the space complexity.

The time complexity is \(O(n)\), where \(n\) is the length of the array. The space complexity is \(O(1)\).

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class Solution:
    def minimumOperations(self, nums: List[int]) -> int:
        f = [0] * 3
        for x in nums:
            g = [0] * 3
            if x == 1:
                g[0] = f[0]
                g[1] = min(f[:2]) + 1
                g[2] = min(f) + 1
            elif x == 2:
                g[0] = f[0] + 1
                g[1] = min(f[:2])
                g[2] = min(f) + 1
            else:
                g[0] = f[0] + 1
                g[1] = min(f[:2]) + 1
                g[2] = min(f)
            f = g
        return min(f)

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class Solution {
    public int minimumOperations(List<Integer> nums) {
        int[] f = new int[3];
        for (int x : nums) {
            int[] g = new int[3];
            if (x == 1) {
                g[0] = f[0];
                g[1] = Math.min(f[0], f[1]) + 1;
                g[2] = Math.min(f[0], Math.min(f[1], f[2])) + 1;
            } else if (x == 2) {
                g[0] = f[0] + 1;
                g[1] = Math.min(f[0], f[1]);
                g[2] = Math.min(f[0], Math.min(f[1], f[2])) + 1;
            } else {
                g[0] = f[0] + 1;
                g[1] = Math.min(f[0], f[1]) + 1;
                g[2] = Math.min(f[0], Math.min(f[1], f[2]));
            }
            f = g;
        }
        return Math.min(f[0], Math.min(f[1], f[2]));
    }
}

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class Solution {
public:
    int minimumOperations(vector<int>& nums) {
        vector<int> f(3);
        for (int x : nums) {
            vector<int> g(3);
            if (x == 1) {
                g[0] = f[0];
                g[1] = min(f[0], f[1]) + 1;
                g[2] = min({f[0], f[1], f[2]}) + 1;
            } else if (x == 2) {
                g[0] = f[0] + 1;
                g[1] = min(f[0], f[1]);
                g[2] = min(f[0], min(f[1], f[2])) + 1;
            } else {
                g[0] = f[0] + 1;
                g[1] = min(f[0], f[1]) + 1;
                g[2] = min(f[0], min(f[1], f[2]));
            }
            f = move(g);
        }
        return min({f[0], f[1], f[2]});
    }
};

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func minimumOperations(nums []int) int {
    f := make([]int, 3)
    for _, x := range nums {
        g := make([]int, 3)
        if x == 1 {
            g[0] = f[0]
            g[1] = min(f[0], f[1]) + 1
            g[2] = min(f[0], min(f[1], f[2])) + 1
        } else if x == 2 {
            g[0] = f[0] + 1
            g[1] = min(f[0], f[1])
            g[2] = min(f[0], min(f[1], f[2])) + 1
        } else {
            g[0] = f[0] + 1
            g[1] = min(f[0], f[1]) + 1
            g[2] = min(f[0], min(f[1], f[2]))
        }
        f = g
    }
    return min(f[0], min(f[1], f[2]))
}

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function minimumOperations(nums: number[]): number {
    let f: number[] = new Array(3).fill(0);
    for (const x of nums) {
        const g: number[] = new Array(3).fill(0);
        if (x === 1) {
            g[0] = f[0];
            g[1] = Math.min(f[0], f[1]) + 1;
            g[2] = Math.min(f[0], Math.min(f[1], f[2])) + 1;
        } else if (x === 2) {
            g[0] = f[0] + 1;
            g[1] = Math.min(f[0], f[1]);
            g[2] = Math.min(f[0], Math.min(f[1], f[2])) + 1;
        } else {
            g[0] = f[0] + 1;
            g[1] = Math.min(f[0], f[1]) + 1;
            g[2] = Math.min(f[0], Math.min(f[1], f[2]));
        }
        f = g;
    }
    return Math.min(...f);
}

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