Codeforces

D. Om Nom and Necklace

time limit per test

1 second

memory limit per test

256 megabytes

input

standard input

output

standard output

One day Om Nom found a thread with n beads of different colors. He decided to cut the first several beads from this thread to make a bead necklace and present it to his girlfriend Om Nelly.

Om Nom knows that his girlfriend loves beautiful patterns. That's why he wants the beads on the necklace to form a regular pattern. A sequence of beads S is regular if it can be represented as S = A + B + A + B + A + ... + A + B + A, where A and B are some bead sequences, " + " is the concatenation of sequences, there are exactly 2k + 1 summands in this sum, among which there are k + 1 "A" summands and k "B" summands that follow in alternating order. Om Nelly knows that her friend is an eager mathematician, so she doesn't mind if A or B is an empty sequence.

Help Om Nom determine in which ways he can cut off the first several beads from the found thread (at least one; probably, all) so that they form a regular pattern. When Om Nom cuts off the beads, he doesn't change their order.

Input

The first line contains two integers n, k (1 ≤ n, k ≤ 1 000 000) — the number of beads on the thread that Om Nom found and number k from the definition of the regular sequence above.

The second line contains the sequence of n lowercase Latin letters that represent the colors of the beads. Each color corresponds to a single letter.

Output

Print a string consisting of n zeroes and ones. Position i (1 ≤ i ≤ n) must contain either number one if the first i beads on the thread form a regular sequence, or a zero otherwise.

Sample test(s)

Input

7 2
bcabcab

Output

0000011

Input

21 2
ababaababaababaababaa

Output

000110000111111000011

Note

In the first sample test a regular sequence is both a sequence of the first 6 beads (we can take A = "", B = "bca"), and a sequence of the first 7 beads (we can take A = "b", B = "ca").

In the second sample test, for example, a sequence of the first 13 beads is regular, if we take A = "aba", B = "ba".

---

思路：可以将AB看成一个串（设为C），然后问题就等价为：判断字符串是否由k个连续的C串以及C串的一个前缀组成（前缀可以为空）。

然后用Z算法求出z数组。

然后从1到n枚举C串的长度（设为len），判断z[0],z[len],z[len*2],....,z[len*(k-1)]的值是否都不小于len，如果都符合就标记最后一个匹配位置为true。

输出的时候维护一个last指针，表示C串的前缀最多能扩展到哪个位置。

复杂度o(n).

---

#include <iostream>
#include <stdio.h>
using namespace std;
#define MAXN 1000010

char s[MAXN];
bool ans[MAXN] = {0};
int z[MAXN] = {0};

int n, k;
bool check(int len)
{
    for(int i = 0, cnt = 0; i < n && cnt < k; i += len, cnt++)
    {
        if(z[i] < len)
            return false;
    }
    return true;
}

int main()
{
    freopen("in.txt", "r", stdin);
    cin >> n >> k;

    scanf("%s", s);

    // Z[i] is the length of the longest substring starting from S[i] which is also a prefix of S
    // s[0,n-1]
    int L = 0, R = 0;
    for(int i = 1; i < n; i++)
    {
        if(i > R)
        {
            L = R = i;
            while(R < n && s[R - L] == s[R]) R++;
            z[i] = R - L;
            R--;
        }
        else
        {
            int k = i - L;
            if(z[k] < R - i + 1) z[i] = z[k];
            else
            {
                L = i;
                while(R < n && s[R - L] == s[R]) R++;
                z[i] = R - L;
                R--;
            }
        }
    }
    z[0] = n;


    // 枚举AB串的长度
    for(int len = 1; len <= n; len++)
    {
        if(len * k > n)
            break;
        // 判断长度为len的AB串是否符合
        if(check(len))
        {
            int last = len * k;
            ans[last - 1] = true;
        }
    }

    int last = -1;
    for(int i = 0; i < n; i++)
    {
        if(ans[i] || i<=last)
            printf("1");
        else
            printf("0");
        if(ans[i] && i < n - 1)
        {
            int len = (i+1)/k;
            last = max(last, min(i+len, i+z[i+1]));
        }
    }

    return 0;
}