feat: add solutions to lc problem: No.3205 (doocs#3200)

No.3205.Maximum Array Hopping Score I

Author: yanglbme

Diff for: solution/3200-3299/3205.Maximum Array Hopping Score I/README.md

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+---
+comments: true
+difficulty: 中等
+edit_url: https://github.com/doocs/leetcode/edit/main/solution/3200-3299/3205.Maximum%20Array%20Hopping%20Score%20I/README.md
+---
+
+<!-- problem:start -->
+
+# [3205. Maximum Array Hopping Score I 🔒](https://leetcode.cn/problems/maximum-array-hopping-score-i)
+
+[English Version](/solution/3200-3299/3205.Maximum%20Array%20Hopping%20Score%20I/README_EN.md)
+
+## 题目描述
+
+<!-- description:start -->
+
+<p>Given an array <code>nums</code>, you have to get the <strong>maximum</strong> score starting from index 0 and <strong>hopping</strong> until you reach the last element of the array.</p>
+
+<p>In each <strong>hop</strong>, you can jump from index <code>i</code> to an index <code>j &gt; i</code>, and you get a <strong>score</strong> of <code>(j - i) * nums[j]</code>.</p>
+
+<p>Return the <em>maximum score</em> you can get.</p>
+
+<p>&nbsp;</p>
+<p><strong class="example">Example 1:</strong></p>
+
+<div class="example-block">
+<p><strong>Input:</strong> <span class="example-io">nums = [1,5,8]</span></p>
+
+<p><strong>Output:</strong> <span class="example-io">16</span></p>
+
+<p><strong>Explanation:</strong></p>
+
+<p>There are two possible ways to reach the last element:</p>
+
+<ul>
+	<li><code>0 -&gt; 1 -&gt; 2</code> with a score of&nbsp;<code>(1 - 0) * 5 + (2 - 1) * 8 = 13</code>.</li>
+	<li><code>0 -&gt; 2</code> with a score of&nbsp;<code>(2 - 0) * 8 =&nbsp;16</code>.</li>
+</ul>
+</div>
+
+<p><strong class="example">Example 2:</strong></p>
+
+<div class="example-block">
+<p><strong>Input:</strong> <span class="example-io">nums = [4,5,2,8,9,1,3]</span></p>
+
+<p><strong>Output:</strong> <span class="example-io">42</span></p>
+
+<p><strong>Explanation:</strong></p>
+
+<p>We can do the hopping <code>0 -&gt; 4 -&gt; 6</code> with a score of&nbsp;<code>(4 - 0) * 9 + (6 - 4) * 3 = 42</code>.</p>
+</div>
+
+<p>&nbsp;</p>
+<p><strong>Constraints:</strong></p>
+
+<ul>
+	<li><code>2 &lt;= nums.length &lt;= 10<sup>3</sup></code></li>
+	<li><code>1 &lt;= nums[i] &lt;= 10<sup>5</sup></code></li>
+</ul>
+
+<!-- description:end -->
+
+## 解法
+
+<!-- solution:start -->
+
+### 方法一：记忆化搜索
+
+我们设计一个函数 $\text{dfs}(i)$，表示从下标 $i$ 出发，能够获得的最大分数。那么答案就是 $\text{dfs}(0)$。
+
+函数 $\text{dfs}(i)$ 的执行过程如下：
+
+我们枚举下一个跳跃的位置 $j$，那么从下标 $i$ 出发，能够获得的分数就是 $(j - i) \times \text{nums}[j]$，加上从下标 $j$ 出发，能够获得的最大分数，总分数就是 $(j - i) \times \text{nums}[j] + \text{dfs}(j)$。我们枚举所有的 $j$，取分数的最大值即可。
+
+为了避免重复计算，我们使用记忆化搜索的方法，将已经计算过的 $\text{dfs}(i)$ 的值保存起来，下次直接返回即可。
+
+时间复杂度 $O(n^2)$，空间复杂度 $O(n)$。其中 $n$ 是数组的长度。
+
+<!-- tabs:start -->
+
+#### Python3
+
+```python
+class Solution:
+    def maxScore(self, nums: List[int]) -> int:
+        @cache
+        def dfs(i: int) -> int:
+            return max(
+                [(j - i) * nums[j] + dfs(j) for j in range(i + 1, len(nums))] or [0]
+            )
+
+        return dfs(0)
+```
+
+#### Java
+
+```java
+class Solution {
+    private Integer[] f;
+    private int[] nums;
+    private int n;
+
+    public int maxScore(int[] nums) {
+        n = nums.length;
+        f = new Integer[n];
+        this.nums = nums;
+        return dfs(0);
+    }
+
+    private int dfs(int i) {
+        if (f[i] != null) {
+            return f[i];
+        }
+        f[i] = 0;
+        for (int j = i + 1; j < n; ++j) {
+            f[i] = Math.max(f[i], (j - i) * nums[j] + dfs(j));
+        }
+        return f[i];
+    }
+}
+```
+
+#### C++
+
+```cpp
+class Solution {
+public:
+    int maxScore(vector<int>& nums) {
+        int n = nums.size();
+        vector<int> f(n);
+        auto dfs = [&](auto&& dfs, int i) -> int {
+            if (f[i]) {
+                return f[i];
+            }
+            for (int j = i + 1; j < n; ++j) {
+                f[i] = max(f[i], (j - i) * nums[j] + dfs(dfs, j));
+            }
+            return f[i];
+        };
+        return dfs(dfs, 0);
+    }
+};
+```
+
+#### Go
+
+```go
+func maxScore(nums []int) int {
+	n := len(nums)
+	f := make([]int, n)
+	var dfs func(int) int
+	dfs = func(i int) int {
+		if f[i] > 0 {
+			return f[i]
+		}
+		for j := i + 1; j < n; j++ {
+			f[i] = max(f[i], (j-i)*nums[j]+dfs(j))
+		}
+		return f[i]
+	}
+	return dfs(0)
+}
+```
+
+#### TypeScript
+
+```ts
+function maxScore(nums: number[]): number {
+    const n = nums.length;
+    const f: number[] = Array(n).fill(0);
+    const dfs = (i: number): number => {
+        if (f[i]) {
+            return f[i];
+        }
+        for (let j = i + 1; j < n; ++j) {
+            f[i] = Math.max(f[i], (j - i) * nums[j] + dfs(j));
+        }
+        return f[i];
+    };
+    return dfs(0);
+}
+```
+
+<!-- tabs:end -->
+
+<!-- solution:end -->
+
+<!-- solution:start -->
+
+### 方法二：动态规划
+
+我们可以将方法一中的记忆化搜索转换为动态规划。
+
+定义 $f[j]$ 表示从下标 $0$ 出发，到下标 $j$ 结束，能够获得的最大分数。那么答案就是 $f[n - 1]$。
+
+状态转移方程为：
+
+$$
+f[j] = \max_{0 \leq i < j} \{ f[i] + (j - i) \times \text{nums}[j] \}
+$$
+
+时间复杂度 $O(n^2)$，空间复杂度 $O(n)$。其中 $n$ 是数组的长度。
+
+<!-- tabs:start -->
+
+#### Python3
+
+```python
+class Solution:
+    def maxScore(self, nums: List[int]) -> int:
+        n = len(nums)
+        f = [0] * n
+        for j in range(1, n):
+            for i in range(j):
+                f[j] = max(f[j], f[i] + (j - i) * nums[j])
+        return f[n - 1]
+```
+
+#### Java
+
+```java
+class Solution {
+    public int maxScore(int[] nums) {
+        int n = nums.length;
+        int[] f = new int[n];
+        for (int j = 1; j < n; ++j) {
+            for (int i = 0; i < j; ++i) {
+                f[j] = Math.max(f[j], f[i] + (j - i) * nums[j]);
+            }
+        }
+        return f[n - 1];
+    }
+}
+```
+
+#### C++
+
+```cpp
+class Solution {
+public:
+    int maxScore(vector<int>& nums) {
+        int n = nums.size();
+        vector<int> f(n);
+        for (int j = 1; j < n; ++j) {
+            for (int i = 0; i < j; ++i) {
+                f[j] = max(f[j], f[i] + (j - i) * nums[j]);
+            }
+        }
+        return f[n - 1];
+    }
+};
+```
+
+#### Go
+
+```go
+func maxScore(nums []int) int {
+	n := len(nums)
+	f := make([]int, n)
+	for j := 1; j < n; j++ {
+		for i := 0; i < j; i++ {
+			f[j] = max(f[j], f[i]+(j-i)*nums[j])
+		}
+	}
+	return f[n-1]
+}
+```
+
+#### TypeScript
+
+```ts
+function maxScore(nums: number[]): number {
+    const n = nums.length;
+    const f: number[] = Array(n).fill(0);
+    for (let j = 1; j < n; ++j) {
+        for (let i = 0; i < j; ++i) {
+            f[j] = Math.max(f[j], f[i] + (j - i) * nums[j]);
+        }
+    }
+    return f[n - 1];
+}
+```
+
+<!-- tabs:end -->
+
+<!-- solution:end -->
+
+<!-- problem:end -->