HDU 1394 Minimum Inversion Number (数据结构-线段树)
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Minimum Inversion Number
Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 9514 Accepted Submission(s): 5860
Problem Description
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.
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
101 3 6 9 0 8 5 7 4 2
Sample Output
16
Author
CHEN, Gaoli
Source
ZOJ Monthly, January 2003
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Ignatius.L
题目大意:
求逆序数。也就是给你一个序列,每次求逆序数,然再把第一个数放到这个序列的末尾,构成新的序列。问你这n个序列的最小的逆序数。
解题思路:
1、对于每个序列,其原来的逆序数记为 pre , 如果当前把该序列 第一个数 a[0] 移动到尾部,那么新序列的逆序数为 pre-a[i]+(n-a[i]-1)
因为序列中比a[i]大的数有 n-a[i]-1 个,比a[i]小的有 a[i]个。
因此只需求出第一个序列的逆序数,依次可以递推出这n个序列的逆序数,求出最小的即可
2、求第一个序列的逆序数的方法
(1)暴力算法,据说不会超时
(2)线段树,建 [0,n]这段树,对于数据 a[i] ,先查询 (a[i]+1,n) 这段的值也就是比a[i]大的数的个数也就是 逆序数,然后插入 (a[i],a[i]) 值为1
代码:
#include <iostream>#include <cmath>#include <vector>#include <cstdio>#include <algorithm>using namespace std;const int maxn=5100;struct tree{ int l,r,sum;}a[maxn*4];int data[maxn],n,m;void build(int l,int r,int k){ a[k].l=l; a[k].r=r; a[k].sum=0; if(l<r){ int mid=(l+r)/2; build(l,mid,2*k); build(mid+1,r,2*k+1); }}void insert(int l,int r,int k,int c){ if(l<=a[k].l && a[k].r<=r){ a[k].sum+=c; }else{ int mid=(a[k].l+a[k].r)/2; if(r<=mid) insert(l,r,2*k,c); else if(l>=mid+1) insert(l,r,2*k+1,c); else{ insert(l,mid,2*k,c); insert(mid+1,r,2*k+1,c); } a[k].sum=a[2*k].sum+a[2*k+1].sum; }}int query(int l,int r,int k){ if(l<=a[k].l && a[k].r<=r){ return a[k].sum; }else{ int mid=(a[k].l+a[k].r)/2; if(r<=mid) return query(l,r,2*k); else if(l>=mid+1) return query(l,r,2*k+1); else{ return query(l,mid,2*k) + query(mid+1,r,2*k+1) ; } }}void solve(){ int ans=0; build(1,n,1); for(int i=1;i<=n;i++){ ans+=query(data[i]+1,n,1); insert(data[i]+1,data[i]+1,1,1); //cout<<data[i]<<" "<<ans<<endl; } int tmp=ans; for(int i=1;i<=n;i++){ tmp-=data[i]; tmp+=n-data[i]-1; if(tmp<ans) ans=tmp; } cout<<ans<<endl;}int main(){ while(scanf("%d",&n)!=EOF){ for(int i=1;i<=n;i++) scanf("%d",&data[i]); solve(); } return 0;}
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