有趣的逆序数——HDU1394
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Minimum Inversion Number
Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)Total Submission(s): 11466 Accepted Submission(s): 7049
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
题意:就是给出n个数,如1,3,6,9,0,8,5,7,4,2这n=10个数,把第一个数放到最後面,其他不变,求该数列的逆序数的个数;之后又把第一个数放到最後面,。。。。
这样进行n-1次,求逆序数最小的序列的逆序数有多少个?
思路:只需要求出一个序列的逆序数,其他的可以根据sum=sum-a[i]+n-a[i]-1 得出;i为1~n-1,a[1]表示第二个数;求逆序数这里采用边输入边计算的方法,就比如这个例子:
输入1,就看2~9的范围已经输入了多少个数,这里是0个
输入3,就看4~9的范围已经输入了多少个数,这里是0个
.。。。
输入0,就看1~9的范围已经输入了多少个数,这里有1,3,6,9这4个
很明显,用线段数或树状数组能轻松解决;
还有就是归并排序能更快地解决!
数状数组:
#include<cstdio>#include<cstdlib>#include<cmath>#include<map>#include<queue>#include<stack>#include<vector>#include<algorithm>#include<cstring>#include<string>#include<iostream>const int MAXN=5000+10;using namespace std;int c[MAXN];int a[MAXN];int lowbit(int x){return x&(-x);}void updata(int x, int num){while(x<=MAXN){c[x]+=num;x+=lowbit(x);}}int Sum(int x){int res=0;while(x>0){res+=c[x];x-=lowbit(x);}return res;}int main(){ //freopen("in.txt","r",stdin); int n; while(scanf("%d", &n)==1) { memset(c,0,sizeof(c)); int i,j; int sum=0; for(i=1; i<=n; i++){ scanf("%d", &a[i]); j=a[i]+1;sum+=Sum(n+1)-Sum(j);0 updata(j,1); } int ans=sum; for(i=1; i<n; i++){ sum=sum-a[i]+n-a[i]-1; if(sum<ans) ans=sum; } printf("%d\n", ans); } return 0;}
线段数:
#include<cstdio>#include<cstdlib>#include<cmath>#include<map>#include<queue>#include<stack>#include<vector>#include<algorithm>#include<cstring>#include<string>#include<iostream>const int MAXN=5000+10;using namespace std;int tree[MAXN*4];int sum;int a[MAXN];void update(int left, int right, int rt, int val){ if(left==right){ tree[rt]=1; return; } int mid=(left+right)>>1; if(val<=mid) update(left, mid, rt<<1, val); else update(mid+1, right, rt<<1|1, val); tree[rt]=tree[rt<<1]+tree[rt<<1|1];}void query(int left, int right, int rt, int l, int r){ if(l<=left && right<=r){ sum+=tree[rt]; return; } int mid=(left+right)>>1; if(r<=mid) query(left, mid, rt<<1, l, r); else if(l>mid) query(mid+1, right, rt<<1|1, l, r); else{ query(left, mid, rt<<1, l, r); query(mid+1, right, rt<<1|1, l, r); }}int main(){ //freopen("in.txt","r",stdin); int n; while(scanf("%d", &n)==1) { memset(tree,0,sizeof(tree)); int i,j; sum=0; for(i=1; i<=n; i++){ scanf("%d", &a[i]); j=a[i]+1; query(1,n,1,j,n); update(1,n,1,j); } int ans=sum; for(i=1; i<n; i++){ sum=sum-a[i]+n-a[i]-1; if(sum<ans) ans=sum; } printf("%d\n", ans); } return 0;}
归并排序:
#include<cstdio>#include<cstdlib>#include<cmath>#include<map>#include<queue>#include<stack>#include<vector>#include<algorithm>#include<cstring>#include<string>#include<iostream>const int MAXN=5000+10;using namespace std;int a[MAXN];int A[MAXN];int t[MAXN];int sum;void merge_sort(int *A,int x,int y,int *t){ if(y-x>1){ int m=x+(y-x)/2; int p=x,q=m,i=x; merge_sort(A,x,m,t); merge_sort(A,m,y,t); while(p<m || q<y){ if(q>=y || (p<m && A[p]<=A[q])) t[i++]=A[p++]; else{ t[i++]=A[q++]; sum+=m-p; } } for(i=x;i<y;i++){ A[i]=t[i]; } }}int main(){ //freopen("in.txt","r",stdin); int n; while(scanf("%d", &n)==1) { int i,j; sum=0; for(i=1; i<=n; i++){ scanf("%d", &a[i]); } memcpy(A,a,sizeof(a)); merge_sort(a,1,n+1,t); int ans=sum; for(i=1; i<n; i++){ sum=sum-A[i]+n-A[i]-1; if(sum<ans) ans=sum; } printf("%d\n", ans); } return 0;}
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