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Formatted question description: https://leetcode.ca/all/1799.html
1799. Maximize Score After N Operations
Level
Hard
Description
You are given nums, an array of positive integers of size 2 * n. You must perform n operations on this array.
In the i-th operation (1-indexed), you will:
- Choose two elements,
xandy. - Receive a score of
i * gcd(x, y). - Remove
xandyfromnums.
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 <= 7nums.length == 2 * n1 <= nums[i] <= 10^6
Solution
In this problem, the 2 * n numbers in nums need to be divided into n groups, where each group contains 2 numbers. Find all possible combinations, and for each combination, calculate the score accordingly. Finally, return the maximum score.
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class Solution { int maxScore = 0; Map<Long, Integer> map = new HashMap<Long, Integer>(); public int maxScore(int[] nums) { int length = nums.length; int[] groups = new int[length]; groups[0] = 1; for (int i = 1; i < length; i++) { groups[i] = 1; backtrack(nums, groups, 1, 2, nums.length / 2); groups[i] = 0; } return maxScore; } public void backtrack(int[] nums, int[] groups, int startIndex, int curr, int n) { int length = nums.length; if (curr > n) { int[][] groupsNums = new int[length][2]; for (int i = 0; i < length; i++) { groupsNums[i][0] = groups[i]; groupsNums[i][1] = nums[i]; } Arrays.sort(groupsNums, new Comparator<int[]>() { public int compare(int[] groupNum1, int[] groupNum2) { return groupNum1[0] - groupNum2[0]; } }); int[] gcds = new int[n]; for (int i = 0; i < length; i += 2) { int gcd = gcd(groupsNums[i][1], groupsNums[i + 1][1]); gcds[i / 2] = gcd; } Arrays.sort(gcds); int score = 0; for (int i = 0; i < n; i++) score += (i + 1) * gcds[i]; maxScore = Math.max(maxScore, score); return; } for (int i = startIndex; i < length; i++) { if (groups[i] == 0) { groups[i] = curr; for (int j = i + 1; j < length; j++) { if (groups[j] == 0) { groups[j] = curr; backtrack(nums, groups, i + 1, curr + 1, n); groups[j] = 0; } } groups[i] = 0; } } } public int gcd(int num1, int num2) { return num2 == 0 ? num1 : gcd(num2, num1 % num2); } } ############ 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: def maxScore(self, nums: List[int]) -> int: m = len(nums) 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]) f = [0] * (1 << m) 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] -
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]; } }; -
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 max(a, b int) int { if a > b { return a } return b } 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; }