POJ1990--POJ 1990 MooFest（树状数组）

Posted 2020-10-01 Pic

Time Limit: 1000MS
Memory Limit: 30000K

Total Submissions: 8141
Accepted: 3674

Description

Every year, Farmer John‘s N (1 <= N <= 20,000) cows attend "MooFest",a social gathering of cows from around the world. MooFest involves a variety of events including haybale stacking, fence jumping, pin the tail on the farmer, and of course, mooing. When the cows all stand in line for a particular event, they moo so loudly that the roar is practically deafening. After participating in this event year after year, some of the cows have in fact lost a bit of their hearing.
Each cow i has an associated "hearing" threshold v(i) (in the range 1..20,000). If a cow moos to cow i, she must use a volume of at least v(i) times the distance between the two cows in order to be heard by cow i. If two cows i and j wish to converse, they must speak at a volume level equal to the distance between them times max(v(i),v(j)).
Suppose each of the N cows is standing in a straight line (each cow at some unique x coordinate in the range 1..20,000), and every pair of cows is carrying on a conversation using the smallest possible volume.
Compute the sum of all the volumes produced by all N(N-1)/2 pairs of mooing cows.

Input

* Line 1: A single integer, N
* Lines 2..N+1: Two integers: the volume threshold and x coordinate for a cow. Line 2 represents the first cow; line 3 represents the second cow; and so on. No two cows will stand at the same location.

Output

* Line 1: A single line with a single integer that is the sum of all the volumes of the conversing cows.

Sample Input

4
3 1
2 5
2 6
4 3

Sample Output

57

Source

USACO 2004 U S Open

题意：

给定一个序列，序列两点之间的花费为 两点之间的距离*（两点之间价值大者），求所有N(N-1)/2个点对的花费之和

思路：

对某个点来说，其等级可以作为乘数的情况是 它与它左右两边比它的等级小的组成点对

设某个点，坐标为 Xc，它的等级为val，其左边比它小的有n1个，右边比它小的有n2个，左边比比它小的坐标为Xi（1<=i<=n1），右边比它小的坐标为 Xj（1<=j<=n2）

则经过推导，我们可以得到其对答案的贡献为：

(n1-n2) * Xc * val + val ( segma(Xi) – segma(Xj) )

所以我们需要统计两个类型的值：左右两边比它等级小的点的个数，左右两边比它等级小的点的坐标和。

这两个类型的值需要分别求出来，但只要知道了其中一个就可得到另外一个（知道了左边的就可以推导出右边的）

我们利用树状数组来统计这些值，具体做法如下：

将所有点按照等级排序，则某个点现在的标号即为比它小的所有点（包括左和右）的总个数。

我们使用两个树状数组，从左到右一次插入每个节点，一个树状数组统计左边比它等级小的点的个数，另一个统计点的坐标和

第一个树状数组好理解，主要困难在于构造第二个树状数组

只需要每次 update（x[i], x[i]）即可

这样的话, x[i] 插入之前，getsum（x[i]）就是左边比i点等级小的点的坐标和

代码：

  1 /*
  2 * @FileName: D:\代码与算法\2017训练比赛\七月训练四\1013.cpp
  3 * @Author: Pic
  4 * @Date:   2017-08-04 21:31:01
  5 * @Last Modified time: 2017-08-05 20:21:10
  6 */
  7 
  8 #include <iostream>
  9 #include <algorithm>
 10 #include <cstdio>
 11 #include <string.h>
 12 #include <queue>
 13 using namespace std;
 14 typedef __int64 ll;
 15 const int MAXN=20000+30;
 16 //注意，这道题的vol有相同的情况出现。这样的话就不能先按x坐标排序，统计左右比这个点小的个数，然后按照vol排序统计左右比这个点小的点的vol和
 17 //而是应该同步地统计这两个量，防止相同的情况
 18 struct BIT{
 19 	void init()
 20 	{
 21 		memset(Tree_sum,0,sizeof(Tree_sum));
 22 	}
 23 	ll Tree_sum[MAXN];//存储树状数组的数组
 24 	//int maxn=20000+30;  //树状数组的下标最大值
 25 	ll lowbit(ll x)   //lowbit函数, 找到x与与 *最近的一个末位连续0比他多的数* 的距离
 26 	{
 27 	    return x&(-x);
 28 	}
 29 	ll getsum(ll x)   //获取0至x区间的和
 30 	{
 31 	    ll sum=0;
 32 	    for(;x>0;x-=lowbit(x)){
 33 	        sum+=Tree_sum[x];
 34 	    }
 35 	    return sum;
 36 	}
 37 	void update(ll x,ll v)    //更新某点的值
 38 	{
 39 	    for(;x<=MAXN;x+=lowbit(x)){
 40 	        Tree_sum[x]+=v;
 41 	    }
 42 	}
 43 };
 44 BIT tr,tr2;
 45 struct node
 46 {
 47 	ll vol,x,id;
 48 }a[MAXN];
 49 ll n1[MAXN];
 50 bool cmp(node a,node b)
 51 {
 52     return a.vol<b.vol;
 53 }
 54 bool cmp1(node a,node b)
 55 {
 56 	return a.x<b.x;
 57 }
 58 int main(){
 59     //freopen("data.in","r",stdin);
 60 	ll n;
 61 	while(~scanf("%I64d",&n)){
 62 		for(int i=0;i<n;i++){
 63 			//scanf("%d%d",&a[i].vol,&a[i].x);
 64 			//cin>>a[i].vol>>a[i].x;
 65 			scanf("%I64d%I64d",&a[i].vol,&a[i].x);
 66 			a[i].id=i;
 67 		}
 68 		tr.init();
 69 		tr2.init();
 70 		sort(a,a+n,cmp1);
 71         sort(a,a+n,cmp);
 72         ll res=0,sum=0,sumn=0;
 73         for(int i=0;i<n;i++){
 74         	sumn=tr.getsum(a[i].x);
 75             ll suma=tr2.getsum(a[i].x);
 76             res+=((sum-suma-suma)*a[i].vol);
 77             res+=((2*sumn-i)*a[i].vol*a[i].x);
 78             //cout<<res<<endl;
 79             tr2.update(a[i].x,a[i].x);
 80             tr.update(a[i].x,1);
 81             sum+=a[i].x;
 82         }
 83         //printf("%lld\n",res);
 84         //cout<<res<<endl;
 85         printf("%I64d\n",res);
 86 	}
 87     return 0;
 88 }

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