F - Fenwick Tree Gym - 100623F （树状数组规律）

F题 ：  http://codeforces.com/gym/100623/attachments

Problem F. Fenwick TreeInput file: fenwick.inOutput file: fenwick.outTime limit: 3 secondsMemory limit: 256 megabytesFenwick tree is a data structure effectively supporting prefix sum queries.For a number t denote as h(t) maximal k such that t is divisible by 2k. For example, h(24) = 3, h(5) = 0.Let l(t) = 2h(t), for example, l(24) = 8, l(5) = 1.Consider array a[1], a[2], . . . , a[n] of integer numbers. Fenwick tree for this array is the arrayb[1], b[2], . . . , b[n] such thatb[i] = Xij=i−l(i)+1a[j].Sob[1] = a[1],b[2] = a[1] + a[2],b[3] = a[3],b[4] = a[1] + a[2] + a[3] + a[4],b[5] = a[5],b[6] = a[5] + a[6],...For example, the Fenwick tree for the arraya = (3, −1, 4, 1, −5, 9)is the arrayb = (3, 2, 4, 7, −5, 4).Let us call an array self-fenwick if it coincides with its Fenwick tree. For example, the array above is notself-fenwick, but the array a = (0, −1, 1, 1, 0, 9) is self-fenwick.You are given an array a. You are allowed to change values of some elements without changing theirorder to get a new array a′ which must be self-fenwick. Find the way to do it by changing as few elementsas possible.InputThe first line of the input file contains n — the number of elements in the array (1 ≤ n ≤ 100 000). Thesecond line contains n integer numbers — the elements of the array. The elements do not exceed 109 bytheir absolute values.OutputOutput n numbers — the elements of the array a′. If there are several solutions, output any one.Examplefenwick.in fenwick.out63 -1 4 1 -5 90 -1 1 1 0 9Page 7 of 13

【题意】：给出一个数组a，让你任意修改其中的元素，使得修改后的数组a'满足  a[i] = b[i]（a b数组如题所述关系），要求修改的位置尽量少。输出修改后的数组。

【分析】

经队友分析，大胆的猜想：第偶数个数都不用改变，只修改第奇数位置。推导如下：

a[2] = b[2] = a[1]+a[2]，故必然 a[1]=0 ，a[2]暂可保持不变。

a[4] = b[4] = a[1]+a[2]+a[3]+a[4];     故a[1]+a[2]+a[3]=0，故a[3]=-a[2];

以此类推，

a[5]=0;

a[7]=-(a[6]+a[4])

.......

a[15]=-(a[14]+a[12]+a[8]);

如果会树状数组的话，这个规律很容易懂，看这个图

例如，x=11位置的数修改为即呢？  -a[10];

也就是x+1的lowbit之内的x的低位都要加起来，之和=0，然后可以解出a[x]

【代码】

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
const int N=1e6+5;

ll a[N],n;
int main()
{
    freopen("fenwick.in", "r", stdin);
  	freopen("fenwick.out", "w", stdout);

    cin>>n;
    for(int i=1;i<=n;i++)
    {
        scanf("%I64d",&a[i]);
    }
    for(int i=1;i<n;i++)if(i&1) //只改奇数
    {
        int x=i;
        a[i]=0;
        while(x)
        {
            x-=x&-x;
            if(x<=i+1-((i+1)&-(i+1)))break;
            a[i]-=a[x];
        }
    }
    for(int i=1;i<=n;i++)
        printf("%I64d%c",a[i],i<n?' ':'\n');

}