[SCOI2007]降雨量（離散化+二分查找+模擬算法+RMQ（ST表實現））

[SCOI2007]降雨量

2007年四川省選

題目描述

我們常常會說這樣的話：“X年是自Y年以來降雨量最多的”。它的含義是X年的降雨量不超過Y年，且對於任意Y < Z < X，Z年的降雨量嚴格小於X年。例如2002，2003，2004和2005年的降雨量分別爲4920，5901，2832和3890，則可以說“2005年是自2003年以來最多的”，但不能說“2005年是自2002年以來最多的”由於有些年份的降雨量未知，有的說法是可能正確也可以不正確的。

輸入輸出格式

輸入格式：
輸入僅一行包含一個正整數n，爲已知的數據。以下n行每行兩個整數yi和ri，爲年份和降雨量，按照年份從小到大排列，即yi < yi+1。下一行包含一個正整數m，爲詢問的次數。以下m行每行包含兩個數Y和X，即詢問“X年是自Y年以來降雨量最多的。”這句話是必真、必假還是“有可能”。

輸出格式：
對於每一個詢問，輸出true，false或者maybe。

輸入輸出樣例

輸入樣例#1：
6
2002 4920
2003 5901
2004 2832
2005 3890
2007 5609
2008 3024
5
2002 2005
2003 2005
2002 2007
2003 2007
2005 2008
輸出樣例#1：
false
true
false
maybe
false
說明

100%的數據滿足：1<=n<=50000, 1<=m<=10000, -10^9<=yi<=10^9, 1<=ri<=10^9

思路

本題做題歷時三個半小時，終於打完了大模擬。
做這個題你首先需要掌握的幾個算法：
1.RMQ的任意一種實現方法（ST表，線段樹均可），這就意味着倍增和二分的思想至少要熟練掌握一個。
2.二分查找（用的比較多），所以二分思想是逃不掉的了。
3.模擬算法，以及一顆勇於堅持的心（堅持看下去，最後有大數據奉上）。
4.離散化
然後就開始講講算法，這裏我的RMQ是用的ST表實現的。
ST表在這個題裏面是用於區間查詢最大值，對於詢問的X,Y，有幾點我們需要注意，首先對於開區間（X,Y）內的最大值MAXX一定要小於Y的降雨量，其次就是對於左端點值，一定要大於Y的降雨量（詳見題目第一行），所以就有MAXX < A[Y] < A[X]，只有此時這句話纔有可能成立，反之一定爲false。然而true的情況就是當[X,Y]區間每一年都有值（我們暫且稱其爲區間是連續的），而且MAXX < A[Y] < A[X]，那麼它一定是true，剩餘的情況就是maybe，可能說起來比較容易，但是還有一些需要特判，在此我特判了15個，比較噁心。
二分查找就是對離散化數組按數組值找數組下標，像這樣的整數二分我們可以拿一個ans記錄當前可行解，避免出錯或陷入死循環。

代碼

#include<iostream>
#include<cstring>
#include<cstdio>
#include<cstdlib>
#include<algorithm>
#include<cmath>
using namespace std;

int i,j,m,n,temp;
int s,t,a[50001],b[50001],map[50001][20],yea[50001];

int r()
{
    int ans=0,f=1;
    char ch;
    while(ch<'0'||ch>'9')
    {
        if(ch=='-')
        {
            f=-1;
        }
        ch=getchar();
    }
    while(ch>='0'&&ch<='9')
    {
        ans*=10;
        ans+=ch-'0';
        ch=getchar();
    }
    return ans*f;
}

void work()
{
    int i,j;
    for(j=1;1<<j<=n;j++)
    for(i=1;i+(1<<j)-1<=n;i++)
    map[i][j]=max(map[i][j-1],map[i+(1<<j-1)][j-1]);
}

int question(int z,int y)
{
    int x=(int)(log(y-z+1)/log(2));
    return max(map[z][x],map[y-(1<<x)+1][x]);
}

int find1(int x)
{
    int le=1,ri=n,ans=0,mid;
    while(le<=ri)
    {
        mid=le+ri;
        mid>>=1;
        if(yea[mid]<x)
        {
            le=mid+1;
            continue;
        }
        if(yea[mid]>x)
        {
            ri=mid-1;
            ans=mid;
            continue;
        }
        if(yea[mid]==x)
        {
            return mid;
        }
    }
    return ans;
}

int find2(int x)
{
    int le=1,ri=n,ans=0,mid;
    while(le<=ri)
    {
        mid=le+ri;
        mid>>=1;
        if(yea[mid]<x)
        {
            le=mid+1;
            ans=mid;
            continue;
        }
        if(yea[mid]>x)
        {
            ri=mid-1;
            continue;
        }
        if(yea[mid]==x)
        {
            return mid;
        }
    }
    return ans;
}

bool find3(int x)
{
    int le=1,ri=n,ans,mid;
    while(le<ri)
    {
        mid=le+ri;
        mid>>=1;
        if(yea[mid]<x)
        {
            le=mid+1;
            continue;
        }
        if(yea[mid]>x)
        {
            ri=mid-1;
            continue;
        }
        if(yea[mid]==x)
        {
            return 1;
        }
    }
    if((le==ri&&yea[le]==x))
    return 1;
    else
    return 0;
}

int main()
{
    freopen("weather1.in","r",stdin);
    n=r();

    int last=-2017,x,y;
    for(i=1;i<=n;i++)
    {
        x=r();
        yea[i]=x;
        if(x==last+1)
        {
            a[i]=r();
            b[i]=b[i-1];
        }
        else
        {
            a[i]=r();
            b[i]=b[i-1]+1;
        }
        last=x;
    }

    for(i=1;i<=n;i++)
    map[i][0]=a[i];
    work();
    int aa,bb,qq,ea,eb,ec,ebb;
    m=r();
//   -------------------------------------------- 
//    int duandian=49;
    for(i=1;i<=m;i++)
    {
        aa=r();bb=r();
        aa++;
        if(aa>bb)
        {
            cout<<"false"<<endl;
            continue;
        }
        ea=find1(aa);
        eb=find2(bb-1);
        ec=find3(aa-1);
        if(ea>eb)
        {
            if(a[find1(aa-1)]<a[find2(bb)]&&ec&&find3(bb))
            cout<<"false"<<endl;
            else if(a[find1(aa-1)]>a[find2(bb)]&&ec&&find3(bb)&&aa-bb==0)
            cout<<"true"<<endl;
            else
            cout<<"maybe"<<endl;
            continue;
        }
        else if(ea<=eb) 
        {qq=question(ea,eb);
        ebb=find2(bb);
        if(qq>=a[find1(bb)]&&find3(bb))
        {
            cout<<"false"<<endl;
        }
        else if(ec&&a[find1(aa-1)]<=qq)
        {
            cout<<"false"<<endl;
        }
        else if(!ec&&a[find1(aa-1)]<=qq)
        {
            cout<<"maybe"<<endl;
        }
        else if(!find3(bb))
        {
            cout<<"maybe"<<endl;
        }
        else if(bb==yea[ebb])
        {
            if(a[ebb]>qq&&b[ebb]-b[ea-1]>0&&!ec)
            {
                cout<<"maybe"<<endl;
            }
            else if(a[ebb]>qq&&b[ebb]-b[ea-1]>0&&ec)
            {
                if(a[find1(aa-1)]<a[find2(bb)])
                cout<<"false"<<endl;
                else
                cout<<"maybe"<<endl;
            }
            else if(a[ebb]>qq&&b[ebb]-b[ea-1]==0&&ec)
            {
                if(a[find1(aa-1)]<a[find2(bb)])
                cout<<"false"<<endl;
                else
                cout<<"true"<<endl;
            }
            else if(a[ebb]>qq&&b[ebb]-b[ea-1]==0&&!ec)
            {
                cout<<"maybe"<<endl;
            }
            else if(a[ebb]>qq&&!find3(aa))
            {
                cout<<"maybe"<<endl;
            }
        }
        else if(bb!=yea[ebb])
        {
            cout<<"maybe"<<endl;
        }

        else 
        cout<<"false"<<endl;}
    }
    return 0;
}
/*
2002 2003 2004 2005 2007 2008
4920 5901 2832 3890 5609 3024
6
2002 4920
2003 5901
2004 2832
2005 3890
2007 5609
2008 3024
1
2030 2032
*/

測試數據

INPUT

8 
-8 1 
-2 2 
0 6 
1 5 
2 7 
13 1 
18 10 
19 14 
10000 
-19 -2 
-2 -1 
-18 -2 
8 12 
1 2 
2 13 
13 18 
0 1 
0 1 
0 1 

-19 18 
-17 -10 
13 18 
-15 18 
-6 13 
-2 18 
4 13 
-6 0 
-2 19 
13 18 

2 19 
-13 13 
-19 14 
3 13 
0 1 
-9 18 
18 19 
6 7 
-11 2 
-2 0 

-16 1 

-16 15 
-8 16 
-2 18 
1 2 
0 1 
-18 1 
2 13 
-5 19 
-11 -2 
-8 -2 
0 1 
5 18 
-8 -2 
-10 18 
-2 0 
13 18 
-8 18 
-3 -2 
-13 -2 
1 11 
18 19 
-7 1 
0 1 
0 1 
1 13 
13 18 
-2 0 
0 18 
-19 18 
-8 -2 
-5 2 
10 19 
0 1 
-13 -9 
-15 -10 
-17 19 
0 17 
-2 4 
-13 -8 
-2 0 
-2 0 
-2 0 
-18 -2 
-2 16 
2 13 
-19 1 
-14 13 
-11 -8 
0 19 
2 13 
-8 -2 
0 18 
-8 2 
13 18 
0 18 
-5 -2 
-13 19 
-14 -8 
-13 15 
-11 1 
-2 0 
13 18 
13 18 
-5 1 
13 18 
-3 19 
0 4 
2 6 
2 13 
-7 6 
-8 18 
-11 2 
-2 0 
-9 2 
-18 18 
-12 -7 
1 2 
13 19 
-14 18 
-11 5 
-8 0 
0 1 
-1 19 
-10 -2 
-2 0 
-8 -2 
1 14 
-7 16 
0 14 
-7 17 
7 13 
2 13 
-2 13 
18 19 
-2 11 
-8 -2 
1 2 
-8 -2 
1 2 
2 13 
18 19 
2 13 
2 13 
1 2 
-2 0 
-2 0 
-5 18 
-8 -2 
0 1 
-2 0 
-8 -2 
-2 0 
2 6 
-8 -2 
-8 -2 
2 13 
1 11 
-8 -2 
-8 -2 
14 15 
13 18 
-6 2 
2 13 
0 13 
-2 0 
0 1 
0 1 
13 18 
-15 19 
2 18 
-8 0 
2 13 
16 18 
2 16 
-2 12 
13 18 
-12 12 
-8 2 
1 18 
-2 0 
-8 -2 
1 2 
-13 -2 
2 13 
-15 -8 
-17 1 
-5 1 
6 12 
0 1 
-2 0 
-8 -2 
-15 -2 
0 1 
-19 13 
18 19 
-16 18 
-7 1 
-8 -5 
1 2 
0 1 
-8 4 
-12 12 
-17 7 
2 13 
1 2 
1 13 
-2 13 
18 19 
2 13 
4 15 
2 13 
-8 10 
-10 19 
6 15 
13 18 
-8 11 
-8 3 
-8 -2 
-8 2 
-8 -2 
-8 -2 
-13 13 
1 14 
-14 16 
13 18 
-8 1 
0 1 
5 19 
-2 0 
0 1 
1 2 
-13 16 
-16 -8 
-12 17 
13 18 
-8 -2 
13 18 
-2 0 
-2 0 
1 2 
-14 19 
1 2 
2 13 
-8 -2 
-2 18 
2 13 
1 2 
-2 0 
-11 0 
-6 0 
2 9 
1 2 
0 1 
2 19 
-8 -2 
8 9 
18 19 
2 13 
0 1 
-8 -2 
7 19 
-8 -2 
-15 -12 
-13 -8 
1 2 
-2 0 
18 19 
-5 5 
18 19 
0 1 
-11 13 
13 18 
2 13 
-14 2 
0 18 
1 2 
0 1 
-2 18 
18 19 
18 19 
2 13 
-9 13 
2 13 
-8 2 
2 13 
0 1 
13 18 
-16 1 
-2 16 
-8 -2 
5 18 
-8 -2 
2 13 
-1 3 
7 13 
-2 0 
0 13 
18 19 
-2 0 
-2 0 
-17 9 
-8 -2 
-10 2 
13 18 
13 18 
1 2 
13 18 
-7 1 
0 12 
1 2 
-17 13 
2 13 
18 19 
2 13 
-8 -2 
2 13 
-8 1 
-2 0 
-8 15 
0 1 
-2 0 
13 18 
13 18 
-9 -1 
8 18 
2 7 
2 17 
-18 15 
-8 -2 
-2 11 
13 18 
-8 -2 
2 13 
-9 13 
2 13 
2 13 
18 19 
18 19 
-8 -2 
-14 2 
-8 -2 
1 9 
-8 -2 
0 1 
-9 2 
7 16 
-9 19 
2 18 
0 13 
-8 -2 
0 1 
1 2 
-5 1 
-8 9 
-2 0 
10 18 
-8 19 
13 18 
13 18 
-9 8 
-8 -2 
17 18 
-8 2 
-16 -2 
-2 0 
-5 0 
13 19 
0 1 
0 1 
18 19 
-2 0 
-13 -8 
-18 0 
-2 0 
1 2 
0 1 
-2 3 
0 1 
-10 1 
1 2 
-10 13 
-8 -2 
18 19 
1 13 
-2 0 
-16 13 
1 2 
18 19 
0 1 
18 19 
2 13 
-7 1 
1 2 
13 18 
2 13 
1 2 
-17 1 
18 19 
2 13 
-8 18 
2 13 
-2 1 
9 13 
5 18 
-12 13 
2 13 
1 8 
0 1 
13 18 
1 2 
13 18 
-12 -1 
-2 0 
2 13 
13 18 
11 19 
-5 2 
-10 18 
2 13 
0 1 
7 8 
-8 -2 
0 1 
13 18 
3 4 
0 16 
0 1 
-11 14 
-7 2 
-11 11 
1 2 
-2 5 
-19 -2 
0 1 
0 18 
-2 1 
18 19 
-18 -14 
13 18 
-13 0 
2 13 
0 1 
13 18 
-8 18 
13 18 
0 2 
13 18 
-5 -3 
-10 -1 
18 19 
18 19 
-2 0 
0 8 
0 1 
18 19 
-8 18 
-17 -2 
-2 0 
-2 1 
13 18 
18 19 
18 19 
8 13 
-12 2 
-8 1 
13 18 
8 9 
-8 12 
1 14 
-9 -2 
2 16 
-2 0 
2 7 
-18 8 
-8 -2 
2 13 
-8 -2 
18 19 
15 19 
-8 -2 
-2 0 
0 10 
2 13 
-16 6 
-2 0 
1 2 
13 18 
1 2 
-11 0 
2 13 
-11 2 
13 18 
1 2 
-4 2 
1 2 
1 2 
5 10 
-2 0 
2 13 
-11 -9 
-2 13 
15 18 
-12 18 
18 19 
-2 0 
2 13 
-16 1 
-17 -8 
0 16 
2 13 
-2 0 
-2 0 
-12 1 
-8 2 
-8 -2 
-2 0 
13 18 
0 1 
4 7 
12 15 
-16 16 
-2 19 
-12 10 
10 18 
-8 -2 
-2 0 
1 2 
2 13 
2 16 
-8 6 
13 18 
-9 19 
-7 13 
-2 0 
18 19 
-2 3 
-15 18 
-2 0 
-13 -5 
-2 0 
-5 1 
-8 0 
0 1 
-8 -2 
-2 0 
4 9 
-2 13 
2 13 
1 15 
-8 13 
-17 -5 
-2 7 
18 19 
-12 -2 
13 18 
-5 11 
-12 -8 
0 1 
-8 18 
1 2 
-4 2 
-8 19 
-8 -2 
2 13 
-2 18 
-8 -2 
1 2 
2 13 
18 19 
0 16 
-2 0 
-8 14 
2 13 
-8 -2 
-8 1 
-18 -13 
-2 1 
2 6 
-2 0 
-2 19 
2 13 
1 2 
0 2 
-8 -2 
1 2 
-15 18 
0 1 
-2 0 
13 18 
-19 -3 
18 19 
14 18 
0 1 
1 2 
-2 0 
1 2 
-10 2 
-18 15 
-9 -8 
10 19 
-9 5 
0 1 
-13 0 
0 1 
-16 19 
-2 0 
2 13 
2 13 
18 19 
1 2 
13 18 
-19 -18 
-2 0 
-4 -2 
11 13 
2 13 
-18 19 
18 19 
2 13 
0 6 
0 1 
0 1 
0 16 
-8 0 
-14 0 
0 1 
-2 19 
13 18 
18 19 
-16 -4 
-2 0 
2 13 
-13 17 
-11 0 
18 19 
-15 17 
-2 0 
2 13 
9 19 
-2 18 
1 7 
-12 1 
2 13 
-13 -12 
-2 0 
2 14 
-2 13 
-8 -2 
-8 2 
-16 -7 
13 18 
-2 0 
-2 13 
-8 -2 
13 18 
1 2 
0 1 
18 19 
-8 -2 
1 2 
-2 19 
-8 -2 
0 1 
-18 3 
0 1 
-6 19 
-18 2 
0 7 
2 13 
-9 13 
11 18 
0 1 
-17 -8 
1 2 
1 2 
16 18 
2 15 
-2 19 
1 2 
-9 7 
2 13 
-19 2 
13 18 
-8 -2 
15 19 
1 2 
0 1 
1 2 
18 19 
-9 13 
18 19 
-8 19 
1 2 
1 13 
13 19 
13 18 
0 1 
13 18 
18 19 
18 19 
2 13 
2 13 
-2 0 
13 18 
13 19 
-2 0 
-2 0 
9 13 
2 13 
-2 0 
2 13 
-11 18 
-2 0 
17 19 
0 13 
-14 10 
-2 0 
-8 -2 
-1 0 
1 2 
0 11 
-8 -2 
2 13 
0 18 
0 1 
-2 0 
1 2 
-1 0 
-7 11 
13 18 
2 13 
3 15 
-3 1 
6 18 
0 12 
2 19 
2 13 
-2 0 
-19 -16 
13 19 
18 19 
18 19 
18 19 
-8 -2 
-8 -2 
0 1 
-4 13 
4 6 
-8 18 
-2 0 
-8 1 
13 18 
-16 14 
-2 13 
-17 13 
-2 0 
-2 0 
2 13 
-8 -2 
-16 -2 
-8 -2 
-8 -2 
-13 -8 
13 18 
-8 -2 
2 19 
1 14 
-18 0 
-9 -1 
0 1 
18 19 
0 18 
0 3 
13 18 
-11 2 
-8 11 
0 5 
0 8 
18 19 
1 2 
0 1 
-13 -8 
1 2 
-8 -2 
18 19 
13 18 
-2 0 
13 18 
-19 1 
10 19 
-14 13 
-6 2 
1 19 
-8 -2 
-3 4 
-4 0 
18 19 
18 19 
1 2 
1 2 
2 13 
-16 1 
-11 19 
-18 1 
13 18 
2 13 
2 13 
13 18 
0 1 
4 17 
1 2 
-2 19 
13 18 
-2 -1 
0 1 
-3 0 
-12 2 
0 1 
1 2 
1 2 
-8 18 
18 19 
-11 -6 
-7 16 
12 19 
-8 -2 
-11 -2 
2 13 
1 7 
13 18 
-8 -2 
0 8 
-16 2 
13 18 
-8 -2 
2 13 
-16 -7 
-5 2 
13 18 
0 1 
18 19 
-2 0 
-9 8 
2 13 
1 2 
18 19 
1 2 
17 18 
-8 -2 
13 18 
8 13 
-2 0 
2 13 
-8 -2 
0 1 
-6 0 
0 1 
2 13 
-17 7 
-2 0 
2 13 
0 1 
13 18 
-18 8 
18 19 
0 1 
2 13 
-2 1 
-2 0 
0 15 
0 1 
17 19 
-8 -2 
2 13 
-8 10 
1 2 
-8 -2 
-15 16 
-2 0 
11 18 
-16 5 
0 1 
-8 19 
-8 -2 
1 2 
-8 -2 
2 13 
1 2 
-8 18 
-5 1 
-8 2 
-1 2 
-18 -2 
13 18 
1 2 
-8 -2 
1 2 
-18 -3 
4 5 
-18 0 
-8 5 
0 1 
-2 8 
-2 18 
1 2 
13 18 
13 18 
-4 1 
0 1 
0 1 
13 18 
-18 19 
-10 3 
2 19 
0 1 
-2 0 
-8 18 
-8 6 
-14 18 
-15 -9 
-8 -2 
0 1 
-8 -2 
1 2 
-15 16 
-5 5 
2 13 
-16 -2 
-15 2 
13 18 
12 19 
-8 -2 
2 13 
-8 13 
-19 1 
-8 -2 
18 19 
-2 2 
-18 -2 
-18 -2 
0 1 
2 18 
-2 0 
-8 14 
2 13 
-16 3 
13 18 
-11 18 
2 13 
18 19 
2 19 
2 19 
1 18 
2 13 
-2 0 
13 18 
-3 2 
2 13 
9 15 
11 13 
-8 -2 
-2 19 
-8 4 
2 13 
13 19 
7 9 
2 18 
18 19 
18 19 
-19 -2 
18 19 
-8 18 
1 2 
-2 13 
13 18 
-10 19 
-2 1 
0 1 
2 13 
10 18 
13 18 
13 18 
-18 -2 
-18 7 
-1 0 
13 18 
0 1 
-2 0 
-11 -2 
-11 -2 
0 18 
1 2 
13 18 
-11 1 
-12 -8 
0 19 
2 13 
2 13 
13 19 
2 13 
2 13 
13 19 
-2 0 
0 1 
0 18 
-2 4 
1 2 
18 19 
-2 0 
2 3 
2 16 
2 13 
2 18 
5 13 
13 18 
-2 0 
-8 1 
-2 19 
-8 -2 
0 13 
-19 -15 
13 18 
0 1 
2 13 
-12 1 
-2 14 
-5 18 
1 2 
-8 8 
-8 1 
0 19 
0 6 
-2 7 
13 18 
-8 -2 
-8 -2 
-17 2 
13 18 
-16 13 
-19 19 
-2 0 
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