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1799. Maximize Score After N Operations

Description

You are given nums, an array of positive integers of size 2 * n. You must perform n operations on this array.

In the ith operation (1-indexed), you will:

  • Choose two elements, x and y.
  • Receive a score of i * gcd(x, y).
  • Remove x and y from nums.

Return the maximum score you can receive after performing n operations.

The function gcd(x, y) is the greatest common divisor of x and y.

 

Example 1:

Input: nums = [1,2]
Output: 1
Explanation: The optimal choice of operations is:
(1 * gcd(1, 2)) = 1

Example 2:

Input: nums = [3,4,6,8]
Output: 11
Explanation: The optimal choice of operations is:
(1 * gcd(3, 6)) + (2 * gcd(4, 8)) = 3 + 8 = 11

Example 3:

Input: nums = [1,2,3,4,5,6]
Output: 14
Explanation: The optimal choice of operations is:
(1 * gcd(1, 5)) + (2 * gcd(2, 4)) + (3 * gcd(3, 6)) = 1 + 4 + 9 = 14

 

Constraints:

  • 1 <= n <= 7
  • nums.length == 2 * n
  • 1 <= nums[i] <= 106

Solutions

Solution 1: State Compression + Dynamic Programming

We can preprocess to get the greatest common divisor of any two numbers in the array nums, stored in the two-dimensional array $g$, where $g[i][j]$ represents the greatest common divisor of $nums[i]$ and $nums[j]$.

Then define $f[k]$ to represent the maximum score that can be obtained when the state after the current operation is $k$. Suppose $m$ is the number of elements in the array nums, then there are a total of $2^m$ states, that is, the range of $k$ is $[0, 2^m - 1]$.

Enumerate all states from small to large, for each state $k$, first determine whether the number of $1$s in the binary bits of this state $cnt$ is even, if so, perform the following operations:

Enumerate the positions where the binary bits in $k$ are 1, suppose they are $i$ and $j$, then the elements at positions $i$ and $j$ can perform one operation, and the score that can be obtained at this time is $\frac{cnt}{2} \times g[i][j]$, update the maximum value of $f[k]$.

The final answer is $f[2^m - 1]$.

The time complexity is $O(2^m \times m^2)$, and the space complexity is $O(2^m)$. Here, $m$ is the number of elements in the array nums.

  • class Solution {
    public int maxScore(int[] nums) {
        int m = nums.length;
        int[][] g = new int[m][m];
        for (int i = 0; i < m; ++i) {
            for (int j = i + 1; j < m; ++j) {
                g[i][j] = gcd(nums[i], nums[j]);
            }
        }
        int[] f = new int[1 << m];
        for (int k = 0; k < 1 << m; ++k) {
            int cnt = Integer.bitCount(k);
            if (cnt % 2 == 0) {
                for (int i = 0; i < m; ++i) {
                    if (((k >> i) & 1) == 1) {
                        for (int j = i + 1; j < m; ++j) {
                            if (((k >> j) & 1) == 1) {
                                f[k] = Math.max(
                                    f[k], f[k ^ (1 << i) ^ (1 << j)] + cnt / 2 * g[i][j]);
                            }
                        }
                    }
                }
            }
        }
        return f[(1 << m) - 1];
    }

    private int gcd(int a, int b) {
        return b == 0 ? a : gcd(b, a % b);
    }
}

  • class Solution {
public:
    int maxScore(vector<int>& nums) {
        int m = nums.size();
        int g[m][m];
        for (int i = 0; i < m; ++i) {
            for (int j = i + 1; j < m; ++j) {
                g[i][j] = gcd(nums[i], nums[j]);
            }
        }
        int f[1 << m];
        memset(f, 0, sizeof f);
        for (int k = 0; k < 1 << m; ++k) {
            int cnt = __builtin_popcount(k);
            if (cnt % 2 == 0) {
                for (int i = 0; i < m; ++i) {
                    if (k >> i & 1) {
                        for (int j = i + 1; j < m; ++j) {
                            if (k >> j & 1) {
                                f[k] = max(f[k], f[k ^ (1 << i) ^ (1 << j)] + cnt / 2 * g[i][j]);
                            }
                        }
                    }
                }
            }
        }
        return f[(1 << m) - 1];
    }
};

  • class Solution:
    def maxScore(self, nums: List[int]) -> int:
        m = len(nums)
        f = [0] * (1 << m)
        g = [[0] * m for _ in range(m)]
        for i in range(m):
            for j in range(i + 1, m):
                g[i][j] = gcd(nums[i], nums[j])
        for k in range(1 << m):
            if (cnt := k.bit_count()) % 2 == 0:
                for i in range(m):
                    if k >> i & 1:
                        for j in range(i + 1, m):
                            if k >> j & 1:
                                f[k] = max(
                                    f[k],
                                    f[k ^ (1 << i) ^ (1 << j)] + cnt // 2 * g[i][j],
                                )
        return f[-1]


  • func maxScore(nums []int) int {
	m := len(nums)
	g := [14][14]int{}
	for i := 0; i < m; i++ {
		for j := i + 1; j < m; j++ {
			g[i][j] = gcd(nums[i], nums[j])
		}
	}
	f := make([]int, 1<<m)
	for k := 0; k < 1<<m; k++ {
		cnt := bits.OnesCount(uint(k))
		if cnt%2 == 0 {
			for i := 0; i < m; i++ {
				if k>>i&1 == 1 {
					for j := i + 1; j < m; j++ {
						if k>>j&1 == 1 {
							f[k] = max(f[k], f[k^(1<<i)^(1<<j)]+cnt/2*g[i][j])
						}
					}
				}
			}
		}
	}
	return f[1<<m-1]
}

func gcd(a, b int) int {
	if b == 0 {
		return a
	}
	return gcd(b, a%b)
}

  • function maxScore(nums: number[]): number {
    const m = nums.length;
    const f: number[] = new Array(1 << m).fill(0);
    const g: number[][] = new Array(m).fill(0).map(() => new Array(m).fill(0));
    for (let i = 0; i < m; ++i) {
        for (let j = i + 1; j < m; ++j) {
            g[i][j] = gcd(nums[i], nums[j]);
        }
    }
    for (let k = 0; k < 1 << m; ++k) {
        const cnt = bitCount(k);
        if (cnt % 2 === 0) {
            for (let i = 0; i < m; ++i) {
                if ((k >> i) & 1) {
                    for (let j = i + 1; j < m; ++j) {
                        if ((k >> j) & 1) {
                            const t = f[k ^ (1 << i) ^ (1 << j)] + ~~(cnt / 2) * g[i][j];
                            f[k] = Math.max(f[k], t);
                        }
                    }
                }
            }
        }
    }
    return f[(1 << m) - 1];
}

function gcd(a: number, b: number): number {
    return b ? gcd(b, a % b) : a;
}

function bitCount(i: number): number {
    i = i - ((i >>> 1) & 0x55555555);
    i = (i & 0x33333333) + ((i >>> 2) & 0x33333333);
    i = (i + (i >>> 4)) & 0x0f0f0f0f;
    i = i + (i >>> 8);
    i = i + (i >>> 16);
    return i & 0x3f;
}

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