POJ1988 Cube Stacking

题目链接

题目

Description

Farmer John and Betsy are playing a game with N (1 <= N <= 30,000)identical cubes labeled 1 through N. They start with N stacks, each containing a single cube. Farmer John asks Betsy to perform P (1<= P <= 100,000) operation. There are two types of operations:
moves and counts.
* In a move operation, Farmer John asks Bessie to move the stack containing cube X on top of the stack containing cube Y.
* In a count operation, Farmer John asks Bessie to count the number of cubes on the stack with cube X that are under the cube X and report that value.

Write a program that can verify the results of the game.

Input

* Line 1: A single integer, P

* Lines 2..P+1: Each of these lines describes a legal operation. Line 2 describes the first operation, etc. Each line begins with a 'M' for a move operation or a 'C' for a count operation. For move operations, the line also contains two integers: X and Y.For count operations, the line also contains a single integer: X.

Note that the value for N does not appear in the input file. No move operation will request a move a stack onto itself.

Output

Print the output from each of the count operations in the same order as the input file.

Sample Input

6
M 1 6
C 1
M 2 4
M 2 6
C 3
C 4

Sample Output

1
0
2

Source

USACO 2004 U S Open

题解

知识点：并查集。

一道典型的带权并查集题。我们用栈底方块作为根节点，权值代表到栈底（根节点）有多少个方块。但还有个问题，两个栈合并时我们并不知道合并到底部的栈有多少个方块，该给被合并的栈的根节点多少权值，于是我们可以用另一个权值数组，或者直接用原来的根节点权值代表这个栈有多少方块，这样合并时候就能直到给根节点赋值多少了。

带权并查集的路径压缩和合并操作都能用向量理解。

首先是路径压缩，已知 \(A\) 到父节点 \(B\) 的权值，即 \(A\) 下面到 \(B\) 有多少方块，以及 \(B\) 到根节点的权值，那么 \(A\) 到根节点的权值就是 \(A\) 到父节点 \(B\) 的权值加上 \(B\) 到根节点的权值，即 \(\vec{AB} + \vec{BR} = \vec{AR}\)。这样递归实现即可，先解决 \(B\) 到根节点的权值问题，再解决 \(A\) 到根节点的权值问题。

其次是合并操作，我们要把栈 \(A\) 放在栈 \(B\) 之上，那么 \(B\) 权就为 \(A\) 方块数加上 \(B\) 的方块数，\(A\) 权就为 \(B\) 的方块数，即 \(\vec{AB} = |B|\) ，\(|B| = |A|+|B|\) 。这里因为根节点的权值与其他节点意义不同，没有体现向量解释的妙qwq。

时间复杂度 \(O(P\log N + N)\)

空间复杂度 \(O(N)\)

代码

#include <bits/stdc++.h>

using namespace std;

int fa[30007], a[30007];

int find(int x) {
    if (fa[x] == x) return x; ///返回根节点
    int pre = fa[x];///保存原父节点
    fa[x] = find(fa[x]);///更新原父节点到根节点的距离，更新父节点为根节点
    if (pre != fa[pre]) a[x] += a[pre];///原父节点不是根节点，则距离为原父节点到根节点距离加自身到原父节点距离
    return fa[x];///返回根节点
}

///根节点栈底元素，权值表示整个栈元素个数
///其他节点表示非栈底元素，权值表示到父节点的距离
///当然也可以把距离和栈大小分开成两个数组，而不是合并在一个权值里，会方便一点
int main() {
    std::ios::sync_with_stdio(0), cin.tie(0), cout.tie(0);
    for (int i = 1;i <= 30000;i++) a[i] = 1, fa[i] = i;
    int p;
    cin >> p;
    while (p--) {
        char op;
        cin >> op;
        if (op == 'M') {
            int x, y;
            cin >> x >> y;

            int rx = find(x);
            int ry = find(y);

            fa[rx] = ry;///x根的父更新为y根
            int t = a[ry];///y权 = 原x权+原y权，x权 = 原y权
            a[ry] += a[rx];
            a[rx] = t;
        }
        else if (op == 'C') {
            int x;
            cin >> x;
            cout << (find(x) == x ? 0 : a[x]) << '\n'; ///距离在查之前要更新的，因为不知道这条路径是否查询过
        }
    }
    return 0;
}