The Preliminary Contest for ICPC Asia Shanghai 2019 B. Light bulbs（卡了線段樹空間的思維題）

傳送門：https://nanti.jisuanke.com/t/41399

題目描述

There are $N$ light bulbs indexed from $0$ to $N−1$. Initially, all of them are off.

A $FLIP$ operation switches the state of a contiguous subset of bulbs. $FLIP(L, R)$ means to flip all bulbs xx such that $L \leq x \leq R$. So for example, $FLIP(3, 5)$ means to flip bulbs 3 , 4 and 5, and $FLIP(5,5)$ means to flip bulb 5.

Given the value of $N$ and a sequence of $M$ flips, count the number of light bulbs that will be on at the end state.

輸入格式

The first line of the input gives the number of test cases, T. T test cases follow. Each test case starts with a line containing two integers N and M, the number of light bulbs and the number of operations, respectively. Then, there are M more lines, the ii-th of which contains the two integers $L_i$and $R_i$, indicating that the ii-th operation would like to flip all the bulbs from $L_i$
to $R_i$, inclusive.

$1 \leq T \leq 1000$

$1 \leq N \leq 10^6$

$1 \leq M \leq 1000$

$0 \leq L_i \leq R_i \leq N-1$

輸出格式

For each test case, output one line containing Case #$x: y$, where x is the test case number (starting from 1) and y is the number of light bulbs that will be on at the end state, as described above.

樣例輸入

2
10 2
2 6
4 8
6 3
1 1
2 3
3 4

樣例輸出

Case #1: 4
Case #2: 3

① 線段樹做法(MLE)

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
 
const int maxn = 1e6 + 10;
int t, n, m, x, y;
struct node { int l, r, sum, flip; } tree[maxn << 2];
 
inline const int read()
{
    int x = 0, f = 1; char ch = getchar();
    while (ch < '0' || ch > '9') { if (ch == '-') f = -1; ch = getchar(); }
    while (ch >= '0' && ch <= '9') { x = (x << 3) + (x << 1) + ch - '0'; ch = getchar(); }
    return x * f;
}
 
inline int ls(int id) { return id << 1; }
inline int rs(int id) { return id << 1 | 1; }
 
void push_up(int id)
{
    tree[id].sum = tree[ls(id)].sum + tree[rs(id)].sum;
}
 
void push_down(int id)
{
    if (tree[id].flip)
    {
        tree[ls(id)].flip ^= 1; tree[rs(id)].flip ^= 1;
        tree[ls(id)].sum = tree[ls(id)].r - tree[ls(id)].l + 1 - tree[ls(id)].sum;
        tree[rs(id)].sum = tree[rs(id)].r - tree[rs(id)].l + 1 - tree[rs(id)].sum;
        tree[id].flip = 0;
    }
}
 
void build(int id, int l, int r)
{
    tree[id].l = l; tree[id].r = r;
    tree[id].sum = tree[id].flip = 0;
    if (l == r) return;
    int mid = (l + r) >> 1;
    build(ls(id), l, mid);
    build(rs(id), mid + 1, r);
    push_up(id);
}
 
void update(int id, int l, int r)
{
    if (tree[id].l == l && tree[id].r == r)
    {
        tree[id].flip ^= 1;
        tree[id].sum = r - l + 1 - tree[id].sum;
        return;
    }
    push_down(id);
    int mid = (tree[id].l + tree[id].r) >> 1;
    if (r <= mid) update(ls(id), l, r);
    else if (l > mid) update(rs(id), l, r);
    else
    {
        update(ls(id), l, mid);
        update(rs(id), mid + 1, r);
    }
    push_up(id);
}
 
int query(int id, int l, int r)
{
    if (tree[id].l == l && tree[id].r == r) return tree[id].sum;
    push_down(id);
    int mid = (tree[id].l + tree[id].r) >> 1;
    if (r <= mid) return query(ls(id), l, r);
    if (l > mid) return query(rs(id), l, r);
    return query(ls(id), l, mid) + query(rs(id), mid + 1, r);
}
 
int main()
{
    t = read();
    for (int i = 1; i <= t; i++)
    {
        printf("Case #%d: ", i);
        n = read(); m = read();
        build(1, 1, n);
        while (m--)
        {
            x = read() + 1; y = read() + 1;
            update(1, x, y);
        }
        printf("%d\n", query(1, 1, n));
    }
    return 0;
}

②

對於每一對Li、Ri，我們將其定義爲：將Li之前的所有燈全都filp一次，再將Ri之前的所有燈全都filp一次，如圖，即爲將C段先翻轉一次，再將A段翻轉一次，這樣一來，C段的燈被翻轉了2次，即沒有變化，而B段則被翻轉了1次，達成了翻轉[Li, Ri]區間的效果。

故，我們將所有的 Li（1 ~ Li-1）和 Ri+1（要包括Ri，故爲Ri+1之前）保存在一個數組$v$中，再將其由小到大排序遍歷，維護在此座標之前亮着的燈的數量$ans$（翻轉一次即爲 $ans = v[i] - ans$）

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
 
const int maxn = 1e6 + 10;
int v[maxn];
 
inline const int read()
{
    int x = 0, f = 1; char ch = getchar();
    while (ch < '0' || ch > '9') { if (ch == '-') f = -1; ch = getchar(); }
    while (ch >= '0' && ch <= '9') { x = (x << 3) + (x << 1) + ch - '0'; ch = getchar(); }
    return x * f;
}
 
int main()
{
    int t = read();
    for (int i = 1; i <= t; i++)
    {
        int n = read(), m = read(), cnt = 0, ans = 0;
        while (m--)
        {
            v[cnt++] = read();
            v[cnt++] = read() + 1;
        }
        sort(v, v + cnt);
        for (int j = 0; j < cnt; j++) ans = v[j] - ans;
        printf("Case #%d: %d\n", i, ans);
    }
    return 0;
}