CF1692G 2^Sort

2^Sort

题面翻译

给你一个长度为 $n \ (\sum n < 2\cdot 10^5)$ 的数组 $a$，问你在这个数组中，有多少个长度为 $k + 1 \ (1\le k < n)$ 的区间，符合以下的条件：

$$2^0 \cdot a_i < 2^1 \cdot a_{i + 1} < 2^2 \cdot a_{i + 2} < \dotsi < 2^k \cdot a_{i + k}\\ \footnotesize{注：i 为这个区间开始的位置}$$

由tzyt翻译

题目描述

Given an array $a$ of length $n$ and an integer $k$ , find the number of indices $1 \leq i \leq n - k$ such that the subarray $[a_i, \dots, a_{i+k}]$ with length $k+1$ (not with length $k$ ) has the following property:

• If you multiply the first element by $2^0$ , the second element by $2^1$ , ..., and the ( $k+1$ )-st element by $2^k$ , then this subarray is sorted in strictly increasing order.

More formally, count the number of indices $1 \leq i \leq n - k$ such that $$2^0 \cdot a_i < 2^1 \cdot a_{i+1} < 2^2 \cdot a_{i+2} < \dots < 2^k \cdot a_{i+k}. $$$ $

输入格式

The first line contains an integer $t$ ( $1 \leq t \leq 1000$ ) — the number of test cases.

The first line of each test case contains two integers $n$ , $k$ ( $3 \leq n \leq 2 \cdot 10^5$ , $1 \leq k < n$ ) — the length of the array and the number of inequalities.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ( $1 \leq a_i \leq 10^9$ ) — the elements of the array.

The sum of $n$ across all test cases does not exceed $2 \cdot 10^5$ .

输出格式

For each test case, output a single integer — the number of indices satisfying the condition in the statement.

样例 #1

样例输入 #1

6
4 2
20 22 19 84
5 1
9 5 3 2 1
5 2
9 5 3 2 1
7 2
22 12 16 4 3 22 12
7 3
22 12 16 4 3 22 12
9 3
3 9 12 3 9 12 3 9 12

样例输出 #1

2
3
2
3
1
0

提示

In the first test case, both subarrays satisfy the condition:

• $i=1$ : the subarray $[a_1,a_2,a_3] = [20,22,19]$ , and $1 \cdot 20 < 2 \cdot 22 < 4 \cdot 19$ .
• $i=2$ : the subarray $[a_2,a_3,a_4] = [22,19,84]$ , and $1 \cdot 22 < 2 \cdot 19 < 4 \cdot 84$ .

In the second test case, three subarrays satisfy the condition: - $i=1$ : the subarray $[a_1,a_2] = [9,5]$ , and $1 \cdot 9 < 2 \cdot 5$ .

• $i=2$ : the subarray $[a_2,a_3] = [5,3]$ , and $1 \cdot 5 < 2 \cdot 3$ .
• $i=3$ : the subarray $[a_3,a_4] = [3,2]$ , and $1 \cdot 3 < 2 \cdot 2$ .
• $i=4$ : the subarray $[a_4,a_5] = [2,1]$ , but $1 \cdot 2 = 2 \cdot 1$ , so this subarray doesn't satisfy the condition.

思路：

主要是把公式好好思考一遍，从公式入手，分析前一项和当前项的关系，可以发现a[i]<2a[i+1]，所以这道题是求连续区间的满足这个条件且满足区间长度为k+1的这个问题。那么求一段区间连续的和用前缀和来写，我们只要把符合a[i]<2a[i+1]这个条件视为正确1，不满足为0，从k个长度的区间来看，一定有k-1个数对是满足这个需求的。所以这道题主要用前缀和就能搞定。

代码：

#include<iostream>
using namespace std;
int main(){
    int t;cin>>t;
    while(t--){
        int n,x;cin>>n>>x;
        int a[n+1],s[n+1];
        for(int i=1;i<=n;i++){
            cin>>a[i];//读入数据
        }
        //开始前缀和的预处理
        for(int i=1;i<=n;i++){
            bool flag=0;
            if(a[i]<2*a[i+1]){
                flag=1;//为合法数量
            }
            s[i]=s[i-1]+flag;
        }
        int ans=0;
        //然后统计满足区间的数量
        //长度为k+1的区间，实质上满足条件的数对只存在k+1-1个，也就是k个。
        for(int i=1;i<=n-x;i++)
            if(s[i+k-1]-s[i-1]==x)//i+k-1为右端点，i为左端点，共有i+k-1-i+1=k个数
                ans++;
        cout<<ans<<endl;
    }
}