feat: add solutions to lc problems: No.0962~0964

solution/0900-0999/0962.Maximum Width Ramp/README.md

@@ -60,13 +60,13 @@ tags:
 
 ### 方法一：单调栈
 
-根据题意，我们可以发现，所有可能的 $nums[i]$ 所构成的子序列一定是单调递减的。为什么呢？我们不妨用反证法证明一下。
+根据题意，我们可以发现，所有可能的 $\textit{nums}[i]$ 所构成的子序列一定是单调递减的。为什么呢？我们不妨用反证法证明一下。
 
-假设存在 $i_1<i_2$，并且 $nums[i_1]<=nums[i_2]$，那么实际上 $nums[i_2]$一定不可能是一个候选值，因为 $nums[i_1]$ 更靠左，会是一个更优的值。因此 $nums[i]$ 所构成的子序列一定单调递减，并且 $i$ 一定是从 0 开始。
+假设存在 $i_1<i_2$，并且 $\textit{nums}[i_1]\leq\textit{nums}[i_2]$，那么实际上 $\textit{nums}[i_2]$ 一定不可能是一个候选值，因为 $\textit{nums}[i_1]$ 更靠左，会是一个更优的值。因此 $\textit{nums}[i]$ 所构成的子序列一定单调递减，并且 $i$ 一定是从 0 开始。
 
-我们用一个从栈底到栈顶单调递减的栈 $stk$ 来存储所有可能的 $nums[i]$，然后我们从右边界开始遍历 $j$，若遇到 $nums[stk.top()]<=nums[j]$，说明此时构成一个坡，循环弹出栈顶元素，更新 ans。
+我们用一个从栈底到栈顶单调递减的栈 $\textit{stk}$ 来存储所有可能的 $\textit{nums}[i]$，然后我们从右边界开始遍历 $j$，若遇到 $\textit{nums}[\textit{stk.top()}]\leq\textit{nums}[j]$，说明此时构成一个坡，循环弹出栈顶元素，更新 $\textit{ans}$。
 
-时间复杂度 $O(n)$，其中 $n$ 表示 $nums$ 的长度。
+时间复杂度 $O(n)$，空间复杂度 $O(n)$，其中 $n$ 表示 $\textit{nums}$ 的长度。
 
 <!-- tabs:start -->
 
@@ -215,46 +215,4 @@ function maxWidthRamp(nums) {
 
 <!-- solution:end -->
 
-<!-- solution:start -->
-
-### 方法二：排序
-
-<!-- tabs:start -->
-
-#### TypeScript
-
-```ts
-function maxWidthRamp(nums: number[]): number {
-    const idx = nums.map((x, i) => [x, i]).sort(([a], [b]) => a - b);
-    let [ans, j] = [0, nums.length];
-
-    for (const [_, i] of idx) {
-        ans = Math.max(ans, i - j);
-        j = Math.min(j, i);
-    }
-
-    return ans;
-}
-```
-
-#### JavaScript
-
-```js
-function maxWidthRamp(nums) {
-    const idx = nums.map((x, i) => [x, i]).sort(([a], [b]) => a - b);
-    let [ans, j] = [0, nums.length];
-
-    for (const [_, i] of idx) {
-        ans = Math.max(ans, i - j);
-        j = Math.min(j, i);
-    }
-
-    return ans;
-}
-```
-
-<!-- tabs:end -->
-
-<!-- solution:end -->
-
 <!-- problem:end -->

solution/0900-0999/0962.Maximum Width Ramp/README_EN.md

@@ -54,7 +54,15 @@ tags:
 
 <!-- solution:start -->
 
-### Solution 1: Monotonic stack
+### Solution 1: Monotonic Stack
+
+According to the problem, we can find that the subsequence formed by all possible $\textit{nums}[i]$ must be monotonically decreasing. Why is that? Let's prove it by contradiction.
+
+Suppose there exist $i_1<i_2$ and $\textit{nums}[i_1]\leq\textit{nums}[i_2]$, then actually $\textit{nums}[i_2]$ cannot be a candidate value, because $\textit{nums}[i_1]$ is more to the left and would be a better value. Therefore, the subsequence formed by $\textit{nums}[i]$ must be monotonically decreasing, and $i$ must start from 0.
+
+We use a monotonically decreasing stack $\textit{stk}$ (from bottom to top) to store all possible $\textit{nums}[i]$. Then we traverse $j$ starting from the right boundary. If we encounter $\textit{nums}[\textit{stk.top()}]\leq\textit{nums}[j]$, it means a ramp is formed at this point. We cyclically pop the top elements from the stack and update $\textit{ans}$.
+
+The time complexity is $O(n)$ and the space complexity is $O(n)$, where $n$ represents the length of $\textit{nums}$.
 
 <!-- tabs:start -->
 
@@ -203,46 +211,4 @@ function maxWidthRamp(nums) {
 
 <!-- solution:end -->
 
-<!-- solution:start -->
-
-### Solution 2: Sorting
-
-<!-- tabs:start -->
-
-#### TypeScript
-
-```ts
-function maxWidthRamp(nums: number[]): number {
-    const idx = nums.map((x, i) => [x, i]).sort(([a], [b]) => a - b);
-    let [ans, j] = [0, nums.length];
-
-    for (const [_, i] of idx) {
-        ans = Math.max(ans, i - j);
-        j = Math.min(j, i);
-    }
-
-    return ans;
-}
-```
-
-#### JavaScript
-
-```js
-function maxWidthRamp(nums) {
-    const idx = nums.map((x, i) => [x, i]).sort(([a], [b]) => a - b);
-    let [ans, j] = [0, nums.length];
-
-    for (const [_, i] of idx) {
-        ans = Math.max(ans, i - j);
-        j = Math.min(j, i);
-    }
-
-    return ans;
-}
-```
-
-<!-- tabs:end -->
-
-<!-- solution:end -->
-
 <!-- problem:end -->

solution/0900-0999/0963.Minimum Area Rectangle II/README.md

@@ -74,7 +74,7 @@ tags:
 
 最后，如果找到满足条件的矩形，返回其中面积的最小值。否则，返回 $0$。
 
-时间复杂度 $O(n^3)$，空间复杂度 $O(n)$。其中 $n$ 为数组 $points$ 的长度。
+时间复杂度 $O(n^3)$，空间复杂度 $O(n)$。其中 $n$ 为数组 $\textit{points}$ 的长度。
 
 <!-- tabs:start -->
 

solution/0900-0999/0963.Minimum Area Rectangle II/README_EN.md

@@ -66,7 +66,13 @@ tags:
 
 <!-- solution:start -->
 
-### Solution 1
+### Solution 1: Hash Table + Enumeration
+
+We use a hash table to store all the points, then enumerate three points $p_1 = (x_1, y_1)$, $p_2 = (x_2, y_2)$, $p_3 = (x_3, y_3)$, where $p_2$ and $p_3$ are the two endpoints of the diagonal of the rectangle. If the line formed by $p_1$ and $p_2$ and the line formed by $p_1$ and $p_3$ are perpendicular, and the fourth point $(x_4, y_4)=(x_2 - x_1 + x_3, y_2 - y_1 + y_3)$ exists in the hash table, then we have found a rectangle. At this point, we can calculate the area of the rectangle and update the answer.
+
+Finally, if a rectangle that satisfies the conditions is found, return the minimum area among them. Otherwise, return $0$.
+
+The time complexity is $O(n^3)$ and the space complexity is $O(n)$, where $n$ is the length of the array $\textit{points}$.
 
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solution/0900-0999/0964.Least Operators to Express Number/README_EN.md

@@ -73,7 +73,22 @@ The expression contains 3 operations.
 
 <!-- solution:start -->
 
-### Solution 1
+### Solution 1: Memoization Search
+
+We define a function $dfs(v)$, which represents the minimum number of operators needed to compose the number $v$ using $x$. Then the answer is $dfs(target)$.
+
+The execution logic of function $dfs(v)$ is as follows:
+
+If $x \geq v$, then we can obtain $v$ by adding $v$ instances of $x / x$, with the number of operators being $v \times 2 - 1$; or we can obtain $v$ by subtracting $(x - v)$ instances of $x / x$ from $x$, with the number of operators being $(x - v) \times 2$. We take the minimum of the two.
+
+Otherwise, we enumerate $x^k$ starting from $k=2$ to find the first $k$ where $x^k \geq v$:
+
+- If $x^k - v \geq v$ at this point, then we can only first obtain $x^{k-1}$, then recursively calculate $dfs(v - x^{k-1})$. The number of operators in this case is $k - 1 + dfs(v - x^{k-1})$;
+- If $x^k - v < v$ at this point, then we can obtain $v$ in the above manner, with the number of operators being $k - 1 + dfs(v - x^{k-1})$; or we can first obtain $x^k$, then recursively calculate $dfs(x^k - v)$, with the number of operators being $k + dfs(x^k - v)$. We take the minimum of the two.
+
+To avoid redundant calculations, we implement the $dfs$ function using memoization search.
+
+The time complexity is $O(\log_{x}{target})$ and the space complexity is $O(\log_{x}{target})$.
 
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