F. Dominant Indices

Codeforces

IDCF1009F

Time4000ms

Memory512MB

Difficulty—

data structuresdsutrees

English · Original

Chinese · Translation

Formal · Original

You are given a rooted undirected tree consisting of $n$ vertices. Vertex $1$ is the root. Let's denote a _depth array_ of vertex $x$ as an infinite sequence $[d_{x, 0}, d_{x, 1}, d_{x, 2}, \dots]$, where $d_{x, i}$ is the number of vertices $y$ such that both conditions hold: * $x$ is an ancestor of $y$; * the simple path from $x$ to $y$ traverses exactly $i$ edges. The _dominant index_ of a _depth array_ of vertex $x$ (or, shortly, the _dominant index_ of vertex $x$) is an index $j$ such that: * for every $k < j$, $d_{x, k} < d_{x, j}$; * for every $k > j$, $d_{x, k} \le d_{x, j}$. For every vertex in the tree calculate its _dominant index_. ## Input The first line contains one integer $n$ ($1 \le n \le 10^6$) — the number of vertices in a tree. Then $n - 1$ lines follow, each containing two integers $x$ and $y$ ($1 \le x, y \le n$, $x \ne y$). This line denotes an edge of the tree. It is guaranteed that these edges form a tree. ## Output Output $n$ numbers. $i$\-th number should be equal to the _dominant index_ of vertex $i$. [samples]

你被给定一棵包含 $n$ 个顶点的有根无向树。顶点 $1$ 为根。定义顶点 $x$ 的 _深度数组_ 为一个无限序列 $[ d_(x, 0), d_(x, 1), d_(x, 2), dots.h ]$，其中 $d_(x, i)$ 表示满足以下两个条件的顶点 $y$ 的数量：顶点 $x$ 的 _深度数组_ 的 _主导索引_（简称顶点 $x$ 的 _主导索引_）是指满足以下条件的索引 $j$：请计算树中每个顶点的 _主导索引_。第一行包含一个整数 $n$（$1 lt.eq n lt.eq 10^6$）——树中顶点的数量。接下来 $n - 1$ 行，每行包含两个整数 $x$ 和 $y$（$1 lt.eq x, y lt.eq n$，$x != y$），表示树中的一条边。保证这些边构成一棵树。请输出 $n$ 个数。第 $i$ 个数应为顶点 $i$ 的 _主导索引_。 ## Input 第一行包含一个整数 $n$（$1 lt.eq n lt.eq 10^6$）——树中顶点的数量。接下来 $n - 1$ 行，每行包含两个整数 $x$ 和 $y$（$1 lt.eq x, y lt.eq n$，$x != y$），表示树中的一条边。保证这些边构成一棵树。 ## Output 请输出 $n$ 个数。第 $i$ 个数应为顶点 $i$ 的 _主导索引_。 [samples]

**Definitions** Let $ T = (V, E) $ be a rooted undirected tree with $ n $ vertices, where $ V = \{1, 2, \dots, n\} $ and vertex $ 1 $ is the root. For each vertex $ x \in V $, define the *depth array* $ D_x = (d_{x,0}, d_{x,1}, d_{x,2}, \dots) $, where: $$ d_{x,i} = \left| \left\{ y \in V \mid \text{dist}(x, y) = i \right\} \right| $$ and $ \text{dist}(x, y) $ denotes the number of edges in the unique path between $ x $ and $ y $ in $ T $. Define the *dominant index* of vertex $ x $ as the smallest index $ j \in \mathbb{N}_0 $ such that: $$ d_{x,j} = \max_{i \in \mathbb{N}_0} d_{x,i} $$ **Constraints** 1. $ 1 \le n \le 10^6 $ 2. The input forms a tree with $ n-1 $ edges. 3. For all $ x \in V $, $ d_{x,i} = 0 $ for all $ i > \text{diameter}(T) $, so the depth array is effectively finite. **Objective** For each vertex $ x \in \{1, 2, \dots, n\} $, compute its dominant index $ j_x $, i.e., the smallest $ j \ge 0 $ such that $ d_{x,j} = \max_{i \ge 0} d_{x,i} $. Output the sequence $ (j_1, j_2, \dots, j_n) $.

Samples

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API Response (JSON)

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      "content": "You are given a rooted undirected tree consisting of $n$ vertices. Vertex $1$ is the root. Let's denote a _depth array_ of vertex $x$ as an infinite sequence $[d_{x, 0}, d_{x, 1}, d_{x, 2}, \\dots]$, ",
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