HDU 1394 Minimum Inversion Number(线段树:单点更新,区间求和或树状数组：求逆序对)

Posted 2023-02-07

 

 

The inversion number of a given number sequence a1, a2, ..., an is the number of pairs (ai, aj) that satisfy i < j and ai > aj. For a given sequence of numbers a1, a2, ..., an, if we move the first m >= 0 numbers to the end of the seqence, we will obtain another sequence. There are totally n such sequences as the following: a1, a2, ..., an-1, an (where m = 0 - the initial seqence) a2, a3, ..., an, a1 (where m = 1) a3, a4, ..., an, a1, a2 (where m = 2) ... an, a1, a2, ..., an-1 (where m = n-1) You are asked to write a program to find the minimum inversion number out of the above sequences.   Input The input consists of a number of test cases. Each case consists of two lines: the first line contains a positive integer n (n <= 5000); the next line contains a permutation of the n integers from 0 to n-1.   Output For each case, output the minimum inversion number on a single line.   Sample Input   10 1 3 6 9 0 8 5 7 4 2   Sample Output   16   Author CHEN, Gaoli   Source ​​ZOJ Monthly, January 2003 ​​   Recommend Ignatius.L   |   We have carefully selected several similar problems for you:  ​​1698​​ ​​1540​​ ​​1255​​ ​​2795​​ ​​1828​​

 

题意：给一个数字n，然后给出0~n-1的排列，求出逆序对数，然后，把a[1]放在a[n]后面，在求逆序对数，以此类推，求最小的逆序对数。

分析1（树状数组做法）：

首先树状数组求逆序对很好求，注意要加1，因为树状数组不能维护从0开始，关键是本题的一个规律。

1~n的排列）的排列第一个位置和最后一个位置是很特殊的：

    与第一个位置的匹配的逆序数就是比他的小的数的个数：a[1]-1

         与最后一个位置的匹配的逆序数就是比他的大的数的个数：n-a[1]

所以，如果将第一个移动到最后一个且排列前的逆序数为ans，则移动后的逆序数为ans-(a[1]-1)+n-a[1]

 

#include<stdio.h>
#include<string>
#include<string.h>
#include<cstdio>
#include<algorithm>
#include<iostream>
using namespace std;
typedef long long ll;
const int MAXN=200000+5;//最大元素个数
int n;//元素个数
ll c[MAXN],a[MAXN];//c[i]==A[i]+A[i-1]+...+A[i-lowbit(i)+1]
//返回i的二进制最右边1的值
int lowbit(int i)

    return i&(-i);

//返回A[1]+...A[i]的和
ll sum(int x)
    ll sum = 0;
    while(x)
        sum += c[x];
        x -= lowbit(x);
    
    return sum;

//令A[i] += val
void add(int x, ll val)
    
    while(x <= n)
        c[x] += val;
        x += lowbit(x);
    

int main()

    while(scanf("%d",&n)!=-1)
    
        
        memset(c,0,sizeof(c));
        for(int i=1;i<=n;i++)
        
            scanf("%lld",&a[i]);
            a[i]++;
           
        int ans=0;
        for(int i=n;i>0;i--)
        
            ans+=sum(a[i]-1);
            add(a[i],1);
        
        int minn=ans;
        for(int i=1;i<n;i++)
        
            
            ans+=n-a[i]-(a[i]-1);
            minn=min(minn,ans);
        
        cout<<minn<<endl;
    
    return 0;

分析2(线段树做法）：

线段树比区间更强大了，正序倒序均可

#include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <cstring>
#define N 200010
#define ll long long
using namespace std; 
ll a[N];
int n,m; 
struct node  //线段树

    int l,r;
    ll dat;
tree[N<<2]; 
void PushUp(int x)   //线段树维护结点信息

    tree[x].dat=tree[x<<1].dat+tree[x<<1|1].dat;

void build(int x,int l,int r)  //线段树建树

    tree[x].l=l;
    tree[x].r=r;
    if (l==r)   //叶子节点
    
        tree[x].dat=0;
        return;
    
    int mid=(l+r)>>1;
    build(x<<1,l,mid);
    build(x<<1|1,mid+1,r);
    PushUp(x);
  
ll query(int x,int l,int r)  //线段树区间查询
  
    if (l<=tree[x].l&&r>=tree[x].r) 
        return abs(tree[x].dat);  //找到
        
    int mid=(tree[x].l+tree[x].r)>>1;
    
    ll ans=0;
    if (l<=mid) ans+=query(x<<1,l,r);
    if (r>mid)  ans+=query(x<<1|1,l,r); 
    return ans;
 
void update(int x,int y,ll k)  //线段树单点修改

    if (tree[x].l==tree[x].r) 
    
        tree[x].dat += k;
        return ;
    
    int mid=(tree[x].l+tree[x].r)>>1;
    if (y<=mid)  update(x<<1,y,k);
    else         update(x<<1|1,y,k);
    PushUp(x);

 
int main()

   
    while(scanf("%d",&n)!=-1)
    
        memset(tree,0,sizeof(node));
        for(int i=1;i<=n;i++)
        
            scanf("%lld",&a[i]);
            a[i]++;
        
        build(1,1,n);   
        int ans=0;
        for(int i=n;i>0;i--)
        
            ans+=query(1,1,a[i]-1);
            update(1,a[i],1);
        
        int minn=ans;
        for(int i=1;i<n;i++)
        
            
            ans+=n-a[i]-(a[i]-1);
            minn=min(minn,ans);
        
        cout<<minn<<endl;
    
    return 0;