feat: add solutions to lc problem: No.0699 (doocs#3328)

No.0699.Falling Squares

Author: yanglbme

solution/0600-0699/0699.Falling Squares/README.md

@@ -72,12 +72,14 @@ tags:
 
 ### 方法一：线段树
 
-线段树将整个区间分割为多个不连续的子区间，子区间的数量不超过 $log(width)$。更新某个元素的值，只需要更新 $log(width)$ 个区间，并且这些区间都包含在一个包含该元素的大区间内。区间修改时，需要使用**懒标记**保证效率。
+根据题目描述，我们需要维护一个区间集合，支持区间的修改和查询操作。这种情况下，我们可以使用线段树来解决。
+
+线段树将整个区间分割为多个不连续的子区间，子区间的数量不超过 $\log(width)$，其中 $width$ 是区间的长度。更新某个元素的值，只需要更新 $\log(width)$ 个区间，并且这些区间都包含在一个包含该元素的大区间内。区间修改时，需要使用**懒标记**保证效率。
 
 -   线段树的每个节点代表一个区间；
--   线段树具有唯一的根节点，代表的区间是整个统计范围，如 $[1, N]$；
+-   线段树具有唯一的根节点，代表的区间是整个统计范围，如 $[1, n]$；
 -   线段树的每个叶子节点代表一个长度为 1 的元区间 $[x, x]$；
--   对于每个内部节点 $[l, r]$，它的左儿子是 $[l, mid]$，右儿子是 $[mid + 1, r]$, 其中 $mid = ⌊(l + r) / 2⌋$ (即向下取整)。
+-   对于每个内部节点 $[l, r]$，它的左儿子是 $[l, mid]$，右儿子是 $[mid + 1, r]$, 其中 $\textit{mid} = \frac{l + r}{2}$；
 
 对于本题，线段树节点维护的信息有：
 
@@ -86,6 +88,8 @@ tags:
 
 另外，由于数轴范围很大，达到 $10^8$，因此我们采用动态开点。
 
+时间复杂度方面，每次查询和修改的时间复杂度为 $O(\log n)$，总时间复杂度为 $O(n \log n)$。空间复杂度为 $O(n)$。
+
 <!-- tabs:start -->
 
 #### Python3
@@ -388,7 +392,7 @@ func newNode(l, r int) *node {
 	return &node{
 		l:   l,
 		r:   r,
-		mid: int(uint(l+r) >> 1),
+		mid: (l + r) >> 1,
 	}
 }
 
@@ -474,6 +478,111 @@ func fallingSquares(positions [][]int) []int {
 }
 ```
 
+#### TypeScript
+
+```ts
+class Node {
+    left: Node | null = null;
+    right: Node | null = null;
+    l: number;
+    r: number;
+    mid: number;
+    v: number = 0;
+    add: number = 0;
+
+    constructor(l: number, r: number) {
+        this.l = l;
+        this.r = r;
+        this.mid = (l + r) >> 1;
+    }
+}
+
+class SegmentTree {
+    private root: Node = new Node(1, 1e9);
+
+    public modify(l: number, r: number, v: number): void {
+        this.modifyNode(l, r, v, this.root);
+    }
+
+    private modifyNode(l: number, r: number, v: number, node: Node): void {
+        if (l > r) {
+            return;
+        }
+        if (node.l >= l && node.r <= r) {
+            node.v = v;
+            node.add = v;
+            return;
+        }
+        this.pushdown(node);
+        if (l <= node.mid) {
+            this.modifyNode(l, r, v, node.left!);
+        }
+        if (r > node.mid) {
+            this.modifyNode(l, r, v, node.right!);
+        }
+        this.pushup(node);
+    }
+
+    public query(l: number, r: number): number {
+        return this.queryNode(l, r, this.root);
+    }
+
+    private queryNode(l: number, r: number, node: Node): number {
+        if (l > r) {
+            return 0;
+        }
+        if (node.l >= l && node.r <= r) {
+            return node.v;
+        }
+        this.pushdown(node);
+        let v = 0;
+        if (l <= node.mid) {
+            v = Math.max(v, this.queryNode(l, r, node.left!));
+        }
+        if (r > node.mid) {
+            v = Math.max(v, this.queryNode(l, r, node.right!));
+        }
+        return v;
+    }
+
+    private pushup(node: Node): void {
+        node.v = Math.max(node.left!.v, node.right!.v);
+    }
+
+    private pushdown(node: Node): void {
+        if (node.left == null) {
+            node.left = new Node(node.l, node.mid);
+        }
+        if (node.right == null) {
+            node.right = new Node(node.mid + 1, node.r);
+        }
+        if (node.add != 0) {
+            let left = node.left,
+                right = node.right;
+            left!.add = node.add;
+            right!.add = node.add;
+            left!.v = node.add;
+            right!.v = node.add;
+            node.add = 0;
+        }
+    }
+}
+
+function fallingSquares(positions: number[][]): number[] {
+    const ans: number[] = [];
+    const tree = new SegmentTree();
+    let mx = 0;
+    for (const [l, w] of positions) {
+        const r = l + w - 1;
+        const h = tree.query(l, r) + w;
+        mx = Math.max(mx, h);
+        ans.push(mx);
+        tree.modify(l, r, h);
+    }
+    return ans;
+}
+```
+
 <!-- tabs:end -->
 
 <!-- solution:end -->

solution/0600-0699/0699.Falling Squares/README_EN.md

@@ -68,7 +68,25 @@ Note that square 2 only brushes the right side of square 1, which does not count
 
 <!-- solution:start -->
 
-### Solution 1
+### Solution 1: Segment Tree
+
+According to the problem description, we need to maintain a set of intervals that support modification and query operations. In this case, we can use a segment tree to solve the problem.
+
+A segment tree divides the entire interval into multiple non-contiguous sub-intervals, with the number of sub-intervals not exceeding $\log(width)$, where $width$ is the length of the interval. To update the value of an element, we only need to update $\log(width)$ intervals, and these intervals are all contained within a larger interval that includes the element. When modifying intervals, we need to use **lazy propagation** to ensure efficiency.
+
+-   Each node of the segment tree represents an interval;
+-   The segment tree has a unique root node representing the entire statistical range, such as $[1, n]$;
+-   Each leaf node of the segment tree represents a primitive interval of length 1, $[x, x]$;
+-   For each internal node $[l, r]$, its left child is $[l, \textit{mid}]$, and its right child is $[\textit{mid} + 1, r]$, where $\textit{mid} = \frac{l + r}{2}$;
+
+For this problem, the information maintained by the segment tree nodes includes:
+
+1. The maximum height $v$ of the blocks in the interval
+2. Lazy propagation marker $add$
+
+Additionally, since the range of the number line is very large, up to $10^8$, we use dynamic node creation.
+
+In terms of time complexity, each query and modification has a time complexity of $O(\log n)$, and the total time complexity is $O(n \log n)$. The space complexity is $O(n)$.
 
 <!-- tabs:start -->
 
@@ -372,7 +390,7 @@ func newNode(l, r int) *node {
 	return &node{
 		l:   l,
 		r:   r,
-		mid: int(uint(l+r) >> 1),
+		mid: (l + r) >> 1,
 	}
 }
 
@@ -458,6 +476,111 @@ func fallingSquares(positions [][]int) []int {
 }
 ```
 
+#### TypeScript
+
+```ts
+class Node {
+    left: Node | null = null;
+    right: Node | null = null;
+    l: number;
+    r: number;
+    mid: number;
+    v: number = 0;
+    add: number = 0;
+
+    constructor(l: number, r: number) {
+        this.l = l;
+        this.r = r;
+        this.mid = (l + r) >> 1;
+    }
+}
+
+class SegmentTree {
+    private root: Node = new Node(1, 1e9);
+
+    public modify(l: number, r: number, v: number): void {
+        this.modifyNode(l, r, v, this.root);
+    }
+
+    private modifyNode(l: number, r: number, v: number, node: Node): void {
+        if (l > r) {
+            return;
+        }
+        if (node.l >= l && node.r <= r) {
+            node.v = v;
+            node.add = v;
+            return;
+        }
+        this.pushdown(node);
+        if (l <= node.mid) {
+            this.modifyNode(l, r, v, node.left!);
+        }
+        if (r > node.mid) {
+            this.modifyNode(l, r, v, node.right!);
+        }
+        this.pushup(node);
+    }
+
+    public query(l: number, r: number): number {
+        return this.queryNode(l, r, this.root);
+    }
+
+    private queryNode(l: number, r: number, node: Node): number {
+        if (l > r) {
+            return 0;
+        }
+        if (node.l >= l && node.r <= r) {
+            return node.v;
+        }
+        this.pushdown(node);
+        let v = 0;
+        if (l <= node.mid) {
+            v = Math.max(v, this.queryNode(l, r, node.left!));
+        }
+        if (r > node.mid) {
+            v = Math.max(v, this.queryNode(l, r, node.right!));
+        }
+        return v;
+    }
+
+    private pushup(node: Node): void {
+        node.v = Math.max(node.left!.v, node.right!.v);
+    }
+
+    private pushdown(node: Node): void {
+        if (node.left == null) {
+            node.left = new Node(node.l, node.mid);
+        }
+        if (node.right == null) {
+            node.right = new Node(node.mid + 1, node.r);
+        }
+        if (node.add != 0) {
+            let left = node.left,
+                right = node.right;
+            left!.add = node.add;
+            right!.add = node.add;
+            left!.v = node.add;
+            right!.v = node.add;
+            node.add = 0;
+        }
+    }
+}
+
+function fallingSquares(positions: number[][]): number[] {
+    const ans: number[] = [];
+    const tree = new SegmentTree();
+    let mx = 0;
+    for (const [l, w] of positions) {
+        const r = l + w - 1;
+        const h = tree.query(l, r) + w;
+        mx = Math.max(mx, h);
+        ans.push(mx);
+        tree.modify(l, r, h);
+    }
+    return ans;
+}
+```
+
 <!-- tabs:end -->
 
 <!-- solution:end -->

solution/0600-0699/0699.Falling Squares/Solution.go

@@ -9,7 +9,7 @@ func newNode(l, r int) *node {
 	return &node{
 		l:   l,
 		r:   r,
-		mid: int(uint(l+r) >> 1),
+		mid: (l + r) >> 1,
 	}
 }