【codeforces 755D】PolandBall and Polygon

time limit per test4 seconds

memory limit per test256 megabytes

inputstandard input

outputstandard output

PolandBall has such a convex polygon with n veritces that no three of its diagonals intersect at the same point. PolandBall decided to improve it and draw some red segments.

He chose a number k such that gcd(n, k) = 1. Vertices of the polygon are numbered from 1 to n in a clockwise way. PolandBall repeats the following process n times, starting from the vertex 1:

Assume you’ve ended last operation in vertex x (consider x = 1 if it is the first operation). Draw a new segment from vertex x to k-th next vertex in clockwise direction. This is a vertex x + k or x + k - n depending on which of these is a valid index of polygon’s vertex.

Your task is to calculate number of polygon’s sections after each drawing. A section is a clear area inside the polygon bounded with drawn diagonals or the polygon’s sides.

Input

There are only two numbers in the input: n and k (5 ≤ n ≤ 106, 2 ≤ k ≤ n - 2, gcd(n, k) = 1).

Output

You should print n values separated by spaces. The i-th value should represent number of polygon’s sections after drawing first i lines.

Examples

input

5 2

output

2 3 5 8 11

input

10 3

output

2 3 4 6 9 12 16 21 26 31

Note

The greatest common divisor (gcd) of two integers a and b is the largest positive integer that divides both a and b without a remainder.

For the first sample testcase, you should output “2 3 5 8 11”. Pictures below correspond to situations after drawing lines.



【题目链接】:http://codeforces.com/contest/755/problem/D

【题解】



两个点之间如果连一条线;

则如果这条线没有穿过其他线;

则平面+1

否则平面+=1+穿过的线条数;

假设当前是第now个点;

则看一下now+1..now+k-1这个范围内的点连的线的条数;即为这条线穿过的其他线的次数;

但是如果k>n/2了,那么可能连线的时候,now+1..now+k-1这些点有出边,但是不会和这条线相交.

比如

5 3



可以看到这里的第二条线.

虽然5 1 中1号点有出边,但是不会和新连的第二条线有交点;

这种情况下可以让n=n-k;

这时k< n/2

且答案是不会变的;

如上图可以把图左右倒过来.



LL以及上面这个是HACK点

写个线段树维护;



【完整代码】

#include <bits/stdc++.h>
using namespace std;
#define lson l,m,rt<<1
#define rson m+1,r,rt<<1|1
#define LL long long
#define rep1(i,a,b) for (int i = a;i <= b;i++)
#define rep2(i,a,b) for (int i = a;i >= b;i--)
#define mp make_pair
#define pb push_back
#define fi first
#define se second
#define rei(x) scanf("%d",&x)
#define rel(x) scanf("%I64d",&x)

typedef pair<int,int> pii;
typedef pair<LL,LL> pll;

const int dx[9] = {0,1,-1,0,0,-1,-1,1,1};
const int dy[9] = {0,0,0,-1,1,-1,1,-1,1};
const double pi = acos(-1.0);
const int MAXN = 1e6+100;

int n,k;
LL sum[MAXN<<2];

LL query(int L,int R,int l,int r,int rt)
{
    if (L>R)
        return 0;
    if (L <= l && r<=R)
        return sum[rt];
    int m = (l+r)>>1;
    LL temp1=0,temp2=0;
    if (L<=m)
        temp1 = query(L,R,lson);
    if (m < R)
        temp2 = query(L,R,rson);
    return temp1+temp2;
}

void up_data(int pos,int l,int r,int rt)
{
    if (l==r)
    {
        sum[rt]++;
        return;
    }
    int m = (l+r)>>1;
    if (pos<=m)
        up_data(pos,lson);
    else
        up_data(pos,rson);
    sum[rt] = sum[rt<<1]+sum[rt<<1|1];
}

int main()
{
    //freopen("F:\\rush.txt","r",stdin);
    rei(n);rei(k);
    k = min(k,n-k);
    int now = 1;
    LL ans = 1;
    rep1(i,1,n)
    {
        int nex = now+k;
        LL temp;
        if (nex>n)
            temp = query(now+1,n,1,n,1)+query(1,nex-n-1,1,n,1);
        else
            temp = query(now+1,nex-1,1,n,1);
        ans += temp+1;
        cout << ans;
        if (i==n)
            puts("");
        else
            putchar(' ');
        up_data(now,1,n,1);
        if (nex>n)
            nex-=n;
        up_data(nex,1,n,1);
        now = nex;
    }
    return 0;
}