+ problem 1799

Author: fruit-in

Problemset/1799-Maximize Score After N Operations/README.md

@@ -0,0 +1,76 @@
+# 1799. Maximize Score After N Operations
+You are given `nums`, an array of positive integers of size `2 * n`. You must perform `n` operations on this array.
+
+In the <code>i<sup>th</sup></code> 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:
+<pre>
+<strong>Input:</strong> nums = [1,2]
+<strong>Output:</strong> 1
+<strong>Explanation:</strong> The optimal choice of operations is:
+(1 * gcd(1, 2)) = 1
+</pre>
+
+#### Example 2:
+<pre>
+<strong>Input:</strong> nums = [3,4,6,8]
+<strong>Output:</strong> 11
+<strong>Explanation:</strong> The optimal choice of operations is:
+(1 * gcd(3, 6)) + (2 * gcd(4, 8)) = 3 + 8 = 11
+</pre>
+
+#### Example 3:
+<pre>
+<strong>Input:</strong> nums = [1,2,3,4,5,6]
+<strong>Output:</strong> 14
+<strong>Explanation:</strong> The optimal choice of operations is:
+(1 * gcd(1, 5)) + (2 * gcd(2, 4)) + (3 * gcd(3, 6)) = 1 + 4 + 9 = 14
+</pre>
+
+#### Constraints:
+* `1 <= n <= 7`
+* `nums.length == 2 * n`
+* <code>1 <= nums[i] <= 10<sup>6</sup></code>
+
+## Solutions (Python)
+
+### 1. Solution
+```Python
+import math
+from functools import cache
+
+
+class Solution:
+    def maxScore(self, nums: List[int]) -> int:
+        @cache
+        def gcd(x: int, y: int) -> int:
+            return math.gcd(x, y)
+
+        n = len(nums) // 2
+        pairmask = [[] for _ in range(n + 1)]
+        dp = [0] * (1 << len(nums))
+
+        for num in range(len(dp)):
+            if bin(num).count('1') % 2 == 0:
+                pairmask[bin(num).count('1') // 2].append(num)
+
+        for i in range(1, n + 1):
+            for j in range(len(nums)):
+                x = nums[j]
+                for k in range(j + 1, len(nums)):
+                    y = nums[k]
+                    for prevmask in pairmask[i - 1]:
+                        mask = (1 << j) | (1 << k)
+                        if prevmask & mask == 0:
+                            dp[prevmask | mask] = max(
+                                dp[prevmask | mask], dp[prevmask] + i * gcd(x, y))
+
+        return dp[-1]
+```

Problemset/1799-Maximize Score After N Operations/README_CN.md

@@ -0,0 +1,76 @@
+# 1799. N 次操作后的最大分数和
+给你 `nums` ，它是一个大小为 `2 * n` 的正整数数组。你必须对这个数组执行 `n` 次操作。
+
+在第 `i` 次操作时（操作编号从 **1** 开始），你需要：
+* 选择两个元素 `x` 和 `y` 。
+* 获得分数 `i * gcd(x, y)` 。
+* 将 `x` 和 `y` 从 `nums` 中删除。
+
+请你返回 `n` 次操作后你能获得的分数和最大为多少。
+
+函数 `gcd(x, y)` 是 `x` 和 `y` 的最大公约数。
+
+#### 示例 1:
+<pre>
+<strong>输入:</strong> nums = [1,2]
+<strong>输出:</strong> 1
+<strong>解释:</strong> 最优操作是：
+(1 * gcd(1, 2)) = 1
+</pre>
+
+#### 示例 2:
+<pre>
+<strong>输入:</strong> nums = [3,4,6,8]
+<strong>输出:</strong> 11
+<strong>解释:</strong> 最优操作是：
+(1 * gcd(3, 6)) + (2 * gcd(4, 8)) = 3 + 8 = 11
+</pre>
+
+#### 示例 3:
+<pre>
+<strong>输入:</strong> nums = [1,2,3,4,5,6]
+<strong>输出:</strong> 14
+<strong>解释:</strong> 最优操作是：
+(1 * gcd(1, 5)) + (2 * gcd(2, 4)) + (3 * gcd(3, 6)) = 1 + 4 + 9 = 14
+</pre>
+
+#### 提示:
+* `1 <= n <= 7`
+* `nums.length == 2 * n`
+* <code>1 <= nums[i] <= 10<sup>6</sup></code>
+
+## 题解 (Python)
+
+### 1. 题解
+```Python
+import math
+from functools import cache
+
+
+class Solution:
+    def maxScore(self, nums: List[int]) -> int:
+        @cache
+        def gcd(x: int, y: int) -> int:
+            return math.gcd(x, y)
+
+        n = len(nums) // 2
+        pairmask = [[] for _ in range(n + 1)]
+        dp = [0] * (1 << len(nums))
+
+        for num in range(len(dp)):
+            if bin(num).count('1') % 2 == 0:
+                pairmask[bin(num).count('1') // 2].append(num)
+
+        for i in range(1, n + 1):
+            for j in range(len(nums)):
+                x = nums[j]
+                for k in range(j + 1, len(nums)):
+                    y = nums[k]
+                    for prevmask in pairmask[i - 1]:
+                        mask = (1 << j) | (1 << k)
+                        if prevmask & mask == 0:
+                            dp[prevmask | mask] = max(
+                                dp[prevmask | mask], dp[prevmask] + i * gcd(x, y))
+
+        return dp[-1]
+```

Problemset/1799-Maximize Score After N Operations/Solution.py

@@ -0,0 +1,30 @@
+import math
+from functools import cache
+
+
+class Solution:
+    def maxScore(self, nums: List[int]) -> int:
+        @cache
+        def gcd(x: int, y: int) -> int:
+            return math.gcd(x, y)
+
+        n = len(nums) // 2
+        pairmask = [[] for _ in range(n + 1)]
+        dp = [0] * (1 << len(nums))
+
+        for num in range(len(dp)):
+            if bin(num).count('1') % 2 == 0:
+                pairmask[bin(num).count('1') // 2].append(num)
+
+        for i in range(1, n + 1):
+            for j in range(len(nums)):
+                x = nums[j]
+                for k in range(j + 1, len(nums)):
+                    y = nums[k]
+                    for prevmask in pairmask[i - 1]:
+                        mask = (1 << j) | (1 << k)
+                        if prevmask & mask == 0:
+                            dp[prevmask | mask] = max(
+                                dp[prevmask | mask], dp[prevmask] + i * gcd(x, y))
+
+        return dp[-1]

README.md

@@ -1156,6 +1156,7 @@
 [1796][1796l]|[Second Largest Digit in a String][1796]                                              |![rs]
 [1797][1797l]|[Design Authentication Manager][1797]                                                 |![py]
 [1798][1798l]|[Maximum Number of Consecutive Values You Can Make][1798]                             |![rs]
+[1799][1799l]|[Maximize Score After N Operations][1799]                                             |![py]
 [1800][1800l]|[Maximum Ascending Subarray Sum][1800]                                                |![rb]  ![rs]
 [1801][1801l]|[Number of Orders in the Backlog][1801]                                               |![rs]
 [1802][1802l]|[Maximum Value at a Given Index in a Bounded Array][1802]                             |![rs]
@@ -2793,6 +2794,7 @@
 [1796]:Problemset/1796-Second%20Largest%20Digit%20in%20a%20String/README.md#1796-second-largest-digit-in-a-string
 [1797]:Problemset/1797-Design%20Authentication%20Manager/README.md#1797-design-authentication-manager
 [1798]:Problemset/1798-Maximum%20Number%20of%20Consecutive%20Values%20You%20Can%20Make/README.md#1798-maximum-number-of-consecutive-values-you-can-make
+[1799]:Problemset/1799-Maximize%20Score%20After%20N%20Operations/README.md#1799-maximize-score-after-n-operations
 [1800]:Problemset/1800-Maximum%20Ascending%20Subarray%20Sum/README.md#1800-maximum-ascending-subarray-sum
 [1801]:Problemset/1801-Number%20of%20Orders%20in%20the%20Backlog/README.md#1801-number-of-orders-in-the-backlog
 [1802]:Problemset/1802-Maximum%20Value%20at%20a%20Given%20Index%20in%20a%20Bounded%20Array/README.md#1802-maximum-value-at-a-given-index-in-a-bounded-array
@@ -4424,6 +4426,7 @@
 [1796l]:https://leetcode.com/problems/second-largest-digit-in-a-string/
 [1797l]:https://leetcode.com/problems/design-authentication-manager/
 [1798l]:https://leetcode.com/problems/maximum-number-of-consecutive-values-you-can-make/
+[1799l]:https://leetcode.com/problems/maximize-score-after-n-operations/
 [1800l]:https://leetcode.com/problems/maximum-ascending-subarray-sum/
 [1801l]:https://leetcode.com/problems/number-of-orders-in-the-backlog/
 [1802l]:https://leetcode.com/problems/maximum-value-at-a-given-index-in-a-bounded-array/

README_CN.md

@@ -1156,6 +1156,7 @@
 [1796][1796l]|[字符串中第二大的数字][1796]                              |![rs]
 [1797][1797l]|[设计一个验证系统][1797]                                  |![py]
 [1798][1798l]|[你能构造出连续值的最大数目][1798]                        |![rs]
+[1799][1799l]|[N 次操作后的最大分数和][1799]                            |![py]
 [1800][1800l]|[最大升序子数组和][1800]                                  |![rb]  ![rs]
 [1801][1801l]|[积压订单中的订单总数][1801]                              |![rs]
 [1802][1802l]|[有界数组中指定下标处的最大值][1802]                      |![rs]
@@ -2793,6 +2794,7 @@
 [1796]:Problemset/1796-Second%20Largest%20Digit%20in%20a%20String/README_CN.md#1796-字符串中第二大的数字
 [1797]:Problemset/1797-Design%20Authentication%20Manager/README_CN.md#1797-设计一个验证系统
 [1798]:Problemset/1798-Maximum%20Number%20of%20Consecutive%20Values%20You%20Can%20Make/README_CN.md#1798-你能构造出连续值的最大数目
+[1799]:Problemset/1799-Maximize%20Score%20After%20N%20Operations/README_CN.md#1799-n-次操作后的最大分数和
 [1800]:Problemset/1800-Maximum%20Ascending%20Subarray%20Sum/README_CN.md#1800-最大升序子数组和
 [1801]:Problemset/1801-Number%20of%20Orders%20in%20the%20Backlog/README_CN.md#1801-积压订单中的订单总数
 [1802]:Problemset/1802-Maximum%20Value%20at%20a%20Given%20Index%20in%20a%20Bounded%20Array/README_CN.md#1802-有界数组中指定下标处的最大值
@@ -4424,6 +4426,7 @@
 [1796l]:https://leetcode.cn/problems/second-largest-digit-in-a-string/
 [1797l]:https://leetcode.cn/problems/design-authentication-manager/description/
 [1798l]:https://leetcode.cn/problems/maximum-number-of-consecutive-values-you-can-make/
+[1799l]:https://leetcode.cn/problems/maximize-score-after-n-operations/
 [1800l]:https://leetcode.cn/problems/maximum-ascending-subarray-sum/
 [1801l]:https://leetcode.cn/problems/number-of-orders-in-the-backlog/
 [1802l]:https://leetcode.cn/problems/maximum-value-at-a-given-index-in-a-bounded-array/