AtCoder Beginner Contest 182D

D - Wandering

Time Limit: 2 sec / Memory Limit: 1024 MB

Score : $400$ points

Problem Statement

Given is a number sequence ${\mathrm{A1,A2,A3,\dots ,AN,}}^{}$ which may contain negative elements.
On a number line, there is a robot at coordinate $0$. It will do the following actions in order:

• Move ${\mathrm{A1}}^{}$ in the positive direction.

• Move ${\mathrm{A1}}^{}$ in the positive direction, and then move ${\mathrm{A2}}^{}$ in the positive direction.

• Move ${\mathrm{A1}}^{}$ in the positive direction, then move ${\mathrm{A2}}^{}$ in the positive direction, and then move ${\mathrm{A3}}^{}$ in the positive direction.

• Move ${\mathrm{A1}}^{}$ in the positive direction, then move ${\mathrm{A2}}^{}$ in the positive direction, then move ${\mathrm{A3}}^{}$ in the positive direction, $\dots \dots$, and then  ${\mathrm{AN}}^{}$ in the positive direction.

Find the greatest coordinate occupied by the robot from the beginning to the end of the process.

Constraints

• $\mathrm{1\le N\le 200000}$

• $-10^8\le Ai\le 10^8$

• All values in input are integers.

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Input

Input is given from Standard Input in the following format:

$N$ ${\mathrm{A1A2A3\dots AN}}^{}$

Output

Print the greatest coordinate occupied by the robot from the beginning to the end of the process.

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Sample Input 1 

3 2 -1 -2

Sample Output 1 

5

The robot moves as follows:

• Move $2$ in the positive direction, to coordinate $2$.

• Move $2$ in the positive direction, to coordinate $4$. Then move $-1$ in the positive direction, to coordinate $3$.

• Move $2$ in the positive direction, to coordinate $5$. Then move $-1$in the positive direction, to coordinate $4$. Then move $-2$ in the positive direction, to coordinate $2$.

The greatest coordinate occupied during the process is $5$, so we should print $5$.

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Sample Input 2

5

-2 1 3 -1 -1

Sample Output 2 

2

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Sample Input 3 

5

-1000 -1000 -1000 -1000 -1000

Sample Output 3 

0

In this case, the initial coordinate $00$ is the greatest coordinate occupied.

解题思路:题意很简单，先输入n表示执行n次操作，第二行输入每次操作的移动步数，机器人有两种走法,问你机器人在运行的过程中走到的最大的正方向的步数是多大

很明显是个模拟题机器人的每次移动都是从第1个指令运行到第i个指令，i从1到n，也就是总共会执行(n+1)*n/2次。看了下数据1e5直接T飞，这时候我们就要利用前缀和的关系

我们可以很容易的发现前缀和正好是机器人每次大循环的操作，这样我们就可以快乐的模拟了。

Code:

#include<cstdio>
#include<algorithm>
#include<cstring>
#include<iostream>
#define ll long long
using namespace std;

ll a[200005];

int main()
{
    int n;
    scanf("%d",&n);
    ll sum=0,cnt=0,ans=0,mx=0;
    for(int i = 0; i < n; ++i) {
        scanf("%lld",&a[i]);
    }
    for(int i = 0; i < n; ++i) {
        sum += a[i];//表示的是a[i]的前缀和 
        mx = max(mx,sum);//更新最大的前缀和 
        ans = max(ans,cnt + mx);//每次更新 
        cnt += sum;//表示的是前缀和的前缀和 
    }
    printf("%lld
",ans);
    return 0;
}