String Transmission


Problem Statement :


Bob has received a binary string of length N transmitted by Alice. He knows that due to errors in transmission, up to K bits might have been corrupted (and hence flipped). However, he also knows that the string Alice had intended to transmit was not periodic. A string is not periodic if it cannot be represented as a smaller string concatenated some number of times. For example, "0001", "0110" are not periodic while "00000", "010101" are periodic strings.

Now he wonders how many possible strings could Alice have transmitted.

Input Format

The first line contains the number of test cases T. T test cases follow. Each case contains two integers N and K on the first line, and a binary string of length N on the next line.

Output Format

Output T lines, one for each test case. Since the answers can be really big, output the numbers modulo 1000000007.


Solution :



title-img


                            Solution in C :

In C :







#include <stdlib.h>
#include <stdio.h>
int N,K;
int F[1000][1000],S[1000];
char s[1001];
int f(int x, int i, int j) {
    if(j>K) return 0;
    if(i==x) return 1;
    if(F[i][j]==-1) F[i][j] = (f(x,i+1,j+S[i])+f(x,i+1,j+N/x-S[i]))%1000000007;;
    return F[i][j];
}
int g(int x, int *p) {
    if((*p)!=0) return (g(x,p+1)-g(x/(*p),p+1))%1000000007;
    int i,j;
    for(i=0; i<x; i++) for(j=0; j<=K; j++) F[i][j] = -1;
    for(i=0; i<x; i++) {
        S[i] = 0;
        for(j=i; j<N; j+=x) S[i]+= (s[j]=='1')?1:0;
    }
    return f(x,0,0);
}
int main() {
    int i,j,k,T;
    int ps[170],l[500],p[5];
    for(i=0;i<500;i++) l[i] = 1;
    ps[0] = 2;
    for(i=3,k=1;i<1000;i+=2) if(l[i/2]) {
        ps[k++] = i;
        for(j=i*i/2;j<500;j+=i) l[j] = 0;
    }
    scanf("%d",&T);
    for(;T>0;T--) {
        scanf("%d %d %[01]",&N,&K,s);
        for(i=0,j=0; i<k; i++) if(N%ps[i]==0) p[j++] = ps[i];
        p[j] = 0;
        printf("%d\n",(g(N,p)+1000000007)%1000000007);
    }
    exit(0);
}
                        

                        Solution in C++ :

In  C ++  :







#include <cstdio>
#include <cstring>

const int mod = 1000000007;

#define MAXN 1005

int T, N, K;
char b[ MAXN ];

int c[ MAXN ][ MAXN ];
int cnt[ MAXN ][ 2 ];
int dp[ MAXN ];
int p[ MAXN ];

int main( void )
{
  scanf( "%d", &T );

  for( int i = 0; i < MAXN; ++i ) {
    for( int j = 0; j < MAXN; ++j ) {
      if( j > i ) continue;
      if( i == 0 ) { c[i][j] = 1; continue; }
      c[i][j] = c[i-1][j] + c[i-1][j-1];
      if( c[i][j] > mod ) c[i][j] -= mod;
    }
  }

  while( T-- ) {
    scanf( "%d%d", &N, &K );
    scanf( "%s", b );

    for( int i = 1; i < N; ++i )
      p[i] = 0;

    int periodic = 0;

    for( int i = 1; i < N; ++i ) {
      if( N % i != 0 ) continue;

      for( int j = 0; j < i; ++j ) 
	cnt[j][0] = cnt[j][1] = 0;

      for( int j = 0; j < N; ++j )
	cnt[j%i][1-b[j]+'0']++;

      for( int j = 0; j <= K; ++j )
	dp[j] = 1;

      for( int j = 0; j < i; ++j ) {
	for( int k = K; k >= 0; --k ) {
	  dp[k] = ( !cnt[j][0] || !cnt[j][1] ) ? dp[k] : 0;
	  if( k >= cnt[j][0] && cnt[j][0] ) dp[k] += dp[k-cnt[j][0]];
	  if( k >= cnt[j][1] && cnt[j][1] ) dp[k] += dp[k-cnt[j][1]];
	  if( dp[k] > mod ) dp[k] -= mod;
	}
      }

      p[i] = dp[K];

      for( int j = 1; j < i; ++j ) {
	if( i % j == 0 ) p[i] = p[i] + mod - p[j];
	if( p[i] > mod ) p[i] -= mod;
      }

      periodic += p[i];
      if( periodic >= mod ) periodic -= mod;
    }

    int total = 0;

    for( int i = 0; i <= K; ++i ) {
      total += c[N][i];
      if( total >= mod ) total -= mod;
    }

    int Sol = total - periodic + mod;
    if( Sol >= mod ) Sol -= mod;

    printf( "%d\n", Sol );
  }

  return 0;
}
                    

                        Solution in Java :

In  Java :








import java.io.*;
import java.util.HashSet;
import java.util.Set;
import java.util.StringTokenizer;

public class Solution implements Runnable {

    // leave empty to read from stdin/stdout
    private static final String TASK_NAME_FOR_IO = "";

    // file names
    private static final String FILE_IN = TASK_NAME_FOR_IO + ".in";
    private static final String FILE_OUT = TASK_NAME_FOR_IO + ".out";

    BufferedReader in;
    PrintWriter out;
    StringTokenizer tokenizer = new StringTokenizer("");

    public static void main(String[] args) {
        new Solution().run();
    }

    final long MOD = 1000000007L;

    int N_MAX = 1005;
    long[][] cnk = new long[N_MAX][N_MAX];

    private void solve() throws IOException {
        for (int n = 0; n < N_MAX; n++) {
            cnk[n][0] = 1;
            cnk[n][n] = 1;
            for (int k = 1; k < n; k++) {
                cnk[n][k] = cnk[n - 1][k - 1] + cnk[n - 1][k];
                cnk[n][k] %= MOD;
            }
        }

        /*
        System.out.println("int[][] COEFF = new int[][] {");
        for (int n = 0; n <= 1000; n++) {
            fastPrecalc(n);
        }
        System.out.println("};");
        */

        /*
        Random r = new Random();
        for (int n = 1; n <= 50; n++) {
            System.out.println("N=" + n);
            for (int d = 0; d <= 4; d++)
                for (int attempts = 0; attempts < 20; attempts++) {
                    long mask = r.nextLong() & (1L << 3);

                    String maskC = "";

                    long num = mask;
                    for (int j = 0; j < n; j++) {
                        maskC += num & 1;
                        num >>= 1;
                    }

                    long a = naive(n, d, maskC.toCharArray());
                    long b = fast(n, d, maskC.toCharArray());

                    if (a != b) {
                        System.err.println(n + " - " + d + " - " + maskC + "(" + a + " vs. " + b + ", delta " + (a - b) + ")");
                    }
                }
        }

        for (int n = 1; n <= 1000; n++) {
            String maskC = "";
            for (int j = 0; j < n; j++) {
                maskC += "1";
            }

            long b = fast(n, 0, maskC.toCharArray());
            if (b > 0) {
                System.err.println(n + " - " + b);
            }
        }
        */

        int tc = nextInt();
        for (int tcIdx = 0; tcIdx < tc; tcIdx++) {
            int n = nextInt();
            int maxD = nextInt();
            char[] c = nextToken().toCharArray();

            out.println(fast(n, maxD, c));
        }
    }

    static int gcd(int a, int b) {
        if (b == 0) {
            return a;
        } else {
            return gcd(b, a % b);
        }
    }

    static int MAX_GCD_NUM = 1005;
    static int[][] GCD_STATIC = new int[MAX_GCD_NUM][MAX_GCD_NUM];

    static {
        for (int i = 0; i < MAX_GCD_NUM; i++)
            for (int j = 0; j <= i; j++) {
                int g = gcd(i, j);
                GCD_STATIC[i][j] = g;
                GCD_STATIC[j][i] = g;
            }
    }

    @SuppressWarnings({"UnusedDeclaration"})
    private long notFastEnough(int n, int maxD, char[] c) {

        int cnt = 0;
        for (int period = 1; period < n; period++) {
            if (n % period == 0) {
                cnt++;
            }
        }

        int[] periods = new int[cnt];
        {
            int idx = 0;
            for (int period = 1; period < n; period++)
                if (n % period == 0) {
                    periods[idx++] = period;
                }
        }

        long[] periodicAnswer = new long[n];
        for (int i = 0; i < cnt; i++) {
            periodicAnswer[periods[i]] = countPeriodic(n, maxD, periods[i], c);
        }

        // calculate total
        long total = 0;
        for (int i = 0; i <= maxD; i++) {
            total += cnk[n][i];
            total %= MOD;
        }

        // calculate partial gcd
        int lBits = cnt / 2;
        int lBits2 = 1 << lBits;
        int[] lGcd = new int[lBits2];

        {
            for (int mask = 0; mask < lBits2; mask++) {
                int common = -1;
                for (int i = 0; i < lBits; i++)
                    if ((mask & (1 << i)) != 0) {
                        common = (common == -1) ? periods[i] : GCD_STATIC[common][periods[i]];
                        if (common == 1) {
                            break;
                        }
                    }
                lGcd[mask] = common;
            }
        }

        int rBits = cnt - lBits;
        int rBits2 = 1 << rBits;

        int[] rGcd = new int[rBits2];
        {
            for (int mask = 0; mask < rBits2; mask++) {
                int common = -1;
                for (int i = 0; i < rBits; i++)
                    if ((mask & (1 << i)) != 0) {
                        common = (common == -1) ? periods[i + lBits] : GCD_STATIC[common][periods[i + lBits]];
                        if (common == 1) {
                            break;
                        }
                    }
                rGcd[mask] = common;
            }
        }

        long cnt2 = 1L << cnt;
        for (long mask = 1; mask < cnt2; mask++) {
            // determine sign by the number of bits in the mask
            int sign = (bitCount(mask) & 1) == 0 ? 1 : -1;

            // fast calculate gcd by analysing left and right part
            int gcdLeft = lGcd[(int) (mask & (lBits2 - 1))];
            int gcdRight = rGcd[(int) (mask >> lBits)];

            int common;
            if (gcdLeft == -1) {
                common = gcdRight;
            } else if (gcdRight == -1) {
                common = gcdLeft;
            } else {
                common = GCD_STATIC[gcdLeft][gcdRight];
            }

            /*
            int common = -1;
            for (int i = 0; i < cnt; i++)
                if ((mask & (1 << i)) != 0) {
                    common = (common == -1) ? periods[i] : GCD_STATIC[common][periods[i]];
                    if (common == 1) {
                        break;
                    }
                }
                */

            total += sign * periodicAnswer[common];
            total %= MOD;
        }

        total %= MOD;
        total += MOD;
        total %= MOD;

        return total;
    }

    private long fast(int n, int maxD, char[] c) {

        int cnt = 0;
        for (int period = 1; period < n; period++) {
            if (n % period == 0) {
                cnt++;
            }
        }

        int[] periods = new int[cnt];
        {
            int idx = 0;
            for (int period = 1; period < n; period++)
                if (n % period == 0) {
                    periods[idx++] = period;
                }
        }

        long[] periodicAnswer = new long[n];
        for (int i = 0; i < cnt; i++) {
            periodicAnswer[i] = countPeriodic(n, maxD, periods[i], c);
        }

        // calculate total
        long total = 0;
        for (int i = 0; i <= maxD; i++) {
            total += cnk[n][i];
            total %= MOD;
        }

        int[] coeff = COEFF[n];
        if (coeff.length != cnt) {
            throw new IllegalStateException("INVALID STATE");
        }

        for (int i = 0; i < cnt; i++) {
            total += coeff[i] * periodicAnswer[i];
            total %= MOD;
        }

        total %= MOD;
        total += MOD;
        total %= MOD;

        return total;
    }

    int[][] COEFF = new int[][] {
      {},
      {},
      {-1,},
      {-1,},
      {0,-1,},
      {-1,},
      {1,-1,-1,},
      {-1,},
      {0,0,-1,},
      {0,-1,},
      {1,-1,-1,},
      {-1,},
      {0,1,0,-1,-1,},
      {-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,-1,},
      {-1,},
      {0,0,1,-1,-1,},
      {-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {-1,},
      {0,0,0,1,0,-1,-1,},
      {0,-1,},
      {1,-1,-1,},
      {0,0,-1,},
      {0,1,-1,0,-1,},
      {-1,},
      {-1,1,1,1,-1,-1,-1,},
      {-1,},
      {0,0,0,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,1,0,-1,-1,},
      {-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,1,0,-1,0,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,1,-1,0,-1,},
      {0,1,0,-1,-1,},
      {1,-1,-1,},
      {-1,},
      {0,0,0,0,0,1,0,-1,-1,},
      {0,-1,},
      {0,0,1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {-1,},
      {0,0,0,0,1,-1,-1,},
      {1,-1,-1,},
      {0,0,1,0,-1,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {-1,},
      {0,-1,0,1,0,1,1,-1,0,-1,-1,},
      {-1,},
      {1,-1,-1,},
      {0,1,0,-1,-1,},
      {0,0,0,0,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {-1,},
      {0,0,0,0,0,0,0,1,0,-1,-1,},
      {-1,},
      {1,-1,-1,},
      {0,0,1,-1,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,0,0,0,1,0,-1,0,-1,},
      {0,0,0,-1,},
      {1,-1,-1,},
      {-1,},
      {0,-1,0,1,1,0,-1,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,1,-1,0,0,-1,},
      {-1,},
      {0,0,-1,0,1,1,0,1,-1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,0,1,0,-1,-1,},
      {-1,},
      {0,0,1,-1,-1,},
      {0,1,-1,0,-1,},
      {0,0,0,0,1,-1,0,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,0,1,-1,0,0,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {1,-1,-1,},
      {-1,},
      {0,0,0,0,0,0,0,1,0,-1,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,1,0,-1,0,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,-1,0,0,1,0,1,0,1,-1,0,-1,-1,},
      {0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {0,0,-1,},
      {0,0,-1,1,0,1,0,-1,1,-1,-1,},
      {-1,},
      {0,0,0,0,0,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,-1,0,1,1,0,-1,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,1,0,-1,-1,},
      {0,0,1,-1,0,0,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,-1,1,0,0,1,1,-1,-1,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,0,0,0,0,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,1,-1,-1,},
      {0,1,-1,0,-1,},
      {-1,},
      {0,0,0,-1,0,1,1,1,-1,-1,-1,},
      {-1,},
      {0,0,1,-1,0,0,-1,},
      {0,1,-1,0,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {1,-1,-1,},
      {0,-1,0,1,1,-1,0,1,0,-1,-1,},
      {-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,1,0,-1,0,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,1,-1,-1,},
      {-1,},
      {0,1,-1,0,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {1,-1,-1,},
      {-1,},
      {0,0,0,-1,0,0,1,1,0,0,-1,1,0,-1,-1,},
      {0,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {0,1,-1,0,-1,},
      {0,1,-1,0,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {0,1,0,-1,-1,},
      {0,0,0,1,0,-1,0,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {-1,},
      {0,0,0,0,0,-1,0,0,1,0,1,0,1,-1,0,-1,-1,},
      {-1,},
      {-1,1,1,1,-1,-1,-1,},
      {1,-1,-1,},
      {0,0,1,-1,0,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {0,0,0,1,0,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,0,0,0,0,0,0,0,0,1,0,-1,-1,},
      {-1,},
      {1,-1,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {0,0,0,0,1,-1,0,-1,},
      {-1,},
      {0,0,-1,1,1,0,-1,0,1,-1,-1,},
      {-1,},
      {0,0,0,0,0,0,1,0,-1,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,-1,0,1,1,-1,0,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {0,0,0,1,0,-1,0,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,-1,1,-1,1,1,1,1,-1,1,-1,-1,-1,},
      {-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,0,0,0,0,0,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,-1,1,0,1,0,-1,1,-1,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,0,0,0,0,0,1,0,-1,0,-1,},
      {0,0,0,0,1,0,-1,-1,},
      {1,-1,-1,},
      {-1,},
      {0,-1,0,1,1,-1,0,1,0,-1,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {0,0,1,-1,0,0,-1,},
      {-1,},
      {0,0,-1,1,1,0,-1,0,1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,0,0,0,0,0,-1,0,0,0,1,0,1,0,1,-1,0,-1,-1,},
      {-1,},
      {0,0,1,-1,-1,},
      {0,0,0,0,-1,},
      {0,1,-1,0,-1,},
      {0,0,1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,0,1,-1,0,0,-1,},
      {1,-1,-1,},
      {0,0,0,0,1,-1,-1,},
      {-1,},
      {0,0,0,0,-1,0,0,1,0,1,0,0,-1,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {0,0,0,0,0,0,0,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,-1,1,0,1,0,-1,1,-1,0,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {-1,},
      {0,0,0,-1,0,1,0,1,0,-1,0,1,0,-1,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {-1,},
      {0,0,0,0,0,-1,0,0,1,1,0,1,-1,-1,-1,},
      {-1,},
      {0,0,0,1,-1,0,0,0,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {1,-1,-1,},
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      {1,-1,-1,},
      {0,0,0,-1,0,1,1,1,-1,-1,-1,},
      {0,0,0,0,1,0,-1,0,0,0,-1,},
      {1,-1,-1,},
      {0,0,-1,1,1,-1,0,0,1,-1,-1,},
      {-1,},
      {0,-1,1,0,1,-1,0,1,-1,0,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,0,0,-1,0,1,1,-1,0,0,0,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {0,-1,1,0,0,1,1,-1,-1,0,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,0,-1,0,1,1,1,-1,-1,-1,},
      {-1,},
      {0,0,0,1,-1,0,0,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,1,0,0,-1,1,0,-1,-1,},
      {-1,},
      {1,-1,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {0,0,-1,0,1,0,0,1,0,-1,1,0,-1,0,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {0,-1,0,1,1,0,-1,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,-1,},
      {-1,},
      {1,-1,-1,-1,1,-1,1,1,1,1,-1,1,-1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {-1,},
      {0,0,-1,1,1,-1,0,0,1,-1,-1,},
      {0,1,-1,0,-1,},
      {0,0,1,-1,0,0,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,1,0,-1,0,-1,-1,1,0,0,1,-1,1,0,1,-1,0,1,1,-1,0,-1,-1,},
      {1,-1,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {0,0,1,-1,0,0,-1,},
      {0,0,0,0,0,0,0,0,0,1,0,-1,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,0,0,-1,0,0,1,0,1,0,0,-1,0,0,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {0,1,-1,0,-1,},
      {-1,},
      {1,-1,-1,1,-1,1,-1,1,1,-1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,-1,0,1,1,-1,0,1,0,-1,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,0,1,-1,0,0,-1,},
      {-1,},
      {0,0,0,0,0,0,0,0,0,-1,0,0,1,1,0,1,-1,-1,-1,},
      {-1,},
      {0,-1,1,0,1,-1,0,1,-1,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,-1,0,1,0,1,0,-1,0,0,0,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,-1,0,1,0,1,1,-1,0,-1,-1,},
      {0,-1,1,0,1,-1,0,1,-1,0,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,0,1,-1,0,0,-1,},
      {0,0,-1,0,1,1,0,1,-1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,0,0,0,-1,0,1,1,0,-1,0,0,0,1,0,-1,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,0,1,0,-1,0,0,-1,},
      {0,1,0,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,-1,1,0,0,1,1,-1,-1,0,-1,},
      {0,0,1,-1,0,0,-1,},
      {1,-1,-1,},
      {-1,},
      {0,0,0,1,0,0,0,-1,0,-1,0,0,-1,0,1,-1,0,0,1,0,1,1,0,1,0,-1,1,-1,0,-1,-1,},
      {0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {0,0,1,-1,-1,},
      {0,0,-1,1,1,-1,0,0,1,-1,-1,},
      {0,0,1,-1,-1,},
      {0,0,0,1,-1,0,0,0,-1,},
      {1,-1,-1,},
      {0,0,-1,1,0,1,0,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,-1,0,1,1,-1,0,1,0,-1,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {0,-1,0,1,1,0,-1,1,0,-1,-1,},
      {0,0,1,-1,0,0,-1,},
      {-1,},
      {1,-1,-1,1,-1,-1,1,1,1,1,-1,-1,1,-1,-1,},
      {-1,},
      {0,-1,1,0,1,-1,0,1,-1,0,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {-1,},
      {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,1,-1,-1,},
      {0,-1,1,0,1,-1,0,1,-1,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,-1,1,1,1,-1,-1,1,1,1,-1,-1,-1,},
      {1,-1,-1,},
      {0,0,1,-1,0,0,-1,},
      {0,1,-1,0,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {0,0,0,1,0,-1,-1,},
      {0,-1,0,1,1,-1,0,1,0,-1,-1,},
      {-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,-1,0,0,1,0,0,1,0,0,-1,1,0,-1,0,-1,},
      {-1,},
      {0,0,0,0,0,0,0,0,-1,1,0,1,0,-1,1,-1,-1,},
      {-1,},
      {0,-1,1,0,0,1,1,-1,-1,0,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {-1,},
      {0,0,0,-1,0,1,1,-1,0,0,0,1,0,-1,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {0,0,0,0,1,0,-1,0,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,0,0,0,0,1,0,-1,0,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,1,0,1,-1,0,-1,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {0,0,1,-1,0,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,1,-1,0,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,-1,1,-1,1,1,1,1,-1,1,-1,-1,-1,},
      {-1,},
      {0,0,0,0,0,-1,0,1,0,1,0,-1,0,0,0,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {0,0,0,0,-1,0,1,1,0,0,-1,0,1,-1,-1,},
      {-1,},
      {0,0,-1,0,1,0,1,0,-1,0,1,0,-1,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,1,0,-1,-1,0,0,1,-1,0,-1,1,0,1,1,1,0,-1,-1,1,0,-1,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {0,0,0,0,1,0,-1,0,0,0,-1,},
      {-1,},
      {1,-1,-1,-1,1,1,1,-1,-1,1,1,1,-1,-1,-1,},
      {0,1,0,-1,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {0,0,0,0,0,0,0,-1,0,0,1,0,1,0,0,-1,0,0,0,1,0,-1,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,-1,1,0,1,-1,0,1,-1,0,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,1,-1,0,0,0,-1,},
      {0,0,0,0,-1,0,0,1,0,1,1,0,-1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,-1,0,1,1,-1,0,1,0,-1,-1,},
      {1,-1,-1,},
      {0,0,-1,1,0,1,0,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,0,-1,0,1,0,0,1,0,-1,1,0,-1,0,-1,},
      {-1,},
      {0,0,-1,1,1,-1,0,0,1,-1,-1,},
      {1,-1,-1,},
      {0,1,-1,0,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,1,0,1,-1,0,-1,-1,},
      {0,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {0,1,-1,0,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {1,-1,-1,1,-1,1,1,-1,-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,0,0,0,0,0,1,-1,0,0,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {-1,},
      {0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,-1,},
      {1,-1,-1,},
      {1,-1,-1,},
      {0,0,-1,0,1,1,0,1,-1,-1,-1,},
      {0,0,0,1,-1,0,0,0,-1,},
      {-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,0,0,0,0,0,-1,0,1,0,0,1,1,-1,-1,0,-1,},
      {0,1,-1,0,-1,},
      {1,-1,-1,},
      {-1,},
      {0,0,0,-1,0,1,1,-1,0,0,0,1,0,-1,-1,},
      {1,-1,-1,},
      {-1,1,1,1,-1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {0,-1,1,0,0,1,1,-1,-1,0,-1,},
      {1,-1,-1,},
      {0,0,1,0,-1,-1,0,0,-1,1,0,1,-1,1,0,1,-1,1,0,1,-1,-1,-1,},
      {-1,},
      {0,0,0,0,1,0,-1,0,0,0,-1,},
      {1,-1,-1,},
      {-1,1,1,-1,1,-1,-1,},
      {1,-1,-1,},
      {0,-1,0,1,1,-1,0,1,0,-1,-1,},
      {-1,},
      {1,-1,-1,},
      {0,0,1,-1,0,0,-1,},
      {0,0,0,0,0,0,0,0,0,0,1,0,-1,0,-1,},
    };

    private void fastPrecalc(int n) {

        int cnt = 0;
        for (int period = 1; period < n; period++) {
            if (n % period == 0) {
                cnt++;
            }
        }

        int[] periods = new int[cnt];
        {
            int idx = 0;
            for (int period = 1; period < n; period++)
                if (n % period == 0) {
                    periods[idx++] = period;
                }
        }

        if (cnt > 30) {
            System.err.println("N = " + n + ", CNT = " + cnt);
        }

        long cnt2 = 1L << cnt;
        long[] coeff = new long[cnt];
        for (long mask = 1; mask < cnt2; mask++) {
            // determine sign by the number of bits in the mask
            int sign = (bitCount(mask) & 1) == 0 ? 1 : -1;

            int common = -1;
            for (int i = 0; i < cnt; i++)
                if ((mask & (1L << i)) != 0) {
                    common = (common == -1) ? periods[i] : gcd(common, periods[i]);
                    if (common == 1) {
                        break;
                    }
                }

            int j = -1;
            for (int i = 0; i < cnt; i++)
                if (periods[i] == common) {
                    j = i;
                    break;
                }

            coeff[j] += sign;
        }

        System.out.print("  {");
        for (int i = 0; i < cnt; i++) {
            System.out.print(coeff[i] + ",");
        }
        System.out.println("},");
    }

    private long countPeriodic(int n, int maxD, int period, char[] c) {
        int[] zeroes = new int[period];
        int[] ones = new int[period];
        for (int i = 0; i < period; i++) {
            int j = i;
            while (j < n) {
                if (c[j] != '0') {
                    ones[i]++;
                } else {
                    zeroes[i]++;
                }
                j += period;
            }
        }

        long[][] d = new long[period + 1][maxD + 1];
        d[0][maxD] = 1;
        for (int i = 0; i < period; i++)
            for (int remD = 0; remD <= maxD; remD++)
                if (d[i][remD] > 0) {
                    // replace zeroes with ones
                    if (remD - zeroes[i] >= 0) {
                        d[i + 1][remD - zeroes[i]] += d[i][remD];
                        d[i + 1][remD - zeroes[i]] %= MOD;
                    }

                    // replace ones with zeroes
                    if (remD - ones[i] >= 0) {
                        d[i + 1][remD - ones[i]] += d[i][remD];
                        d[i + 1][remD - ones[i]] %= MOD;
                    }
                }

        long result = 0;
        for (int i = 0; i <= maxD; i++) {
            result += d[period][i];
            result %= MOD;
        }
        return result;
    }

    @SuppressWarnings({"UnusedDeclaration"})
    private long naive(int n, int maxD, char[] c) {
        return naive3(n, maxD, toMask(c));
    }

    private boolean isPeriodic(int n, long mask) {
        for (int period = 1; period < n; period++)
            if (n % period == 0 && isPeriodic(n, mask, period)) {
                return true;
            }
        return false;
    }

    private boolean isPeriodic(int n, long mask, int period) {
        for (int i = 0; i < period; i++) {
            boolean bitHere = (mask & (1L << i)) != 0;

            int j = i;
            while (j < n) {
                boolean bitThere = (mask & (1L << j)) != 0;
                if (bitHere != bitThere) {
                    return false;
                }
                j += period;
            }
        }

        return true;
    }

    @SuppressWarnings({"UnusedDeclaration"})
    private long naive1(int n, int maxD, long c) {
        long result = 0;
        long n2 = 1 << n;
        for (long mask = 0; mask < n2; mask++) {
            if (distinctBits(n, mask, c) <= maxD && !isPeriodic(n, mask)) {
                result++;
            }
        }
        return result;
    }

    @SuppressWarnings({"UnusedDeclaration"})
    private long naive2(int n, int maxD, long c) {
        Set<Long> periodic = new HashSet<Long>();

        // count only periodic
        for (int periodLength = 1; periodLength < n; periodLength++)
            if (n % periodLength == 0) {
                int periodTimes = n / periodLength;

                // brute force period mask
                long p2 = 1L << periodLength;
                for (long pMask = 0; pMask < p2; pMask++) {
                    // calculate overall mask
                    long totalMask = 0;
                    for (int i = 0; i < periodTimes; i++) {
                        totalMask <<= periodLength;
                        totalMask |= pMask;
                    }

                    // see if it is applicable
                    if (distinctBits(n, totalMask, c) <= maxD) {
                        periodic.add(totalMask);
                    }
                }
            }

        int total = 0;
        for (int k = 0; k <= maxD; k++) {
            total += cnk[n][k];
        }

        return total - periodic.size();
    }

    private long naive3(int n, int maxD, long c) {
        long total = 0;
        for (int k = 0; k <= maxD; k++) {
            total += cnk[n][k];
        }

        return total - periodicGen(0, n, maxD, c);
    }

    private long periodicGen(int pos, int n, int maxD, long c) {
        if (pos >= n) {
            return isPeriodic(n, c) ? 1 : 0;
        }
        long total = periodicGen(pos + 1, n, maxD, c);
        if (maxD > 0) {
            total += periodicGen(pos + 1, n, maxD - 1, c ^ (1L << pos));
        }
        return total;
    }

    static int BIT_COUNT_LIMIT_POW = 18;
    static int BIT_COUNT_LIMIT_MASK = (1 << BIT_COUNT_LIMIT_POW) - 1;
    static int[] BIT_COUNT = new int[1 << BIT_COUNT_LIMIT_POW];

    static {
        for (int i = 1; i < BIT_COUNT.length; i++)
            BIT_COUNT[i] = BIT_COUNT[i & (i - 1)] + 1;
    }

    @SuppressWarnings({"UnusedParameters"})
    private int distinctBits(int n, long m1, long m2) {
        return bitCount(m1 ^ m2);
    }

    private int bitCount(long mask) {
        return BIT_COUNT[(int) (mask & BIT_COUNT_LIMIT_MASK)] + BIT_COUNT[(int) ((mask >> BIT_COUNT_LIMIT_POW) & BIT_COUNT_LIMIT_MASK)];
    }

    private long toMask(char[] c) {
        long result = 0;
        for (char v : c) {
            result <<= 1;
            result += v - '0';
        }
        return result;
    }

    public void run() {
        long timeStart = System.currentTimeMillis();

        boolean fileIO = TASK_NAME_FOR_IO.length() > 0;
        try {

            if (fileIO) {
                in = new BufferedReader(new FileReader(FILE_IN));
                out = new PrintWriter(new FileWriter(FILE_OUT));
            } else {
                in = new BufferedReader(new InputStreamReader(System.in));
                out = new PrintWriter(new OutputStreamWriter(System.out));
            }

            solve();

            in.close();
            out.close();
        } catch (IOException e) {
            throw new IllegalStateException(e);
        }
        long timeEnd = System.currentTimeMillis();

        if (fileIO) {
            System.out.println("Time spent: " + (timeEnd - timeStart) + " ms");
        }
    }

    private String nextToken() throws IOException {
        while (!tokenizer.hasMoreTokens()) {
            String line = in.readLine();
            if (line == null) {
                return null;
            }
            tokenizer = new StringTokenizer(line);
        }
        return tokenizer.nextToken();
    }

    private int nextInt() throws IOException {
        return Integer.parseInt(nextToken());
    }

}
                    

                        Solution in Python : 
                            
In  Python3 :







T = int(input())
M = 1000000007

from math import factorial, sqrt


def nck(n, k):
    res = 0
    
    for i in range(k+1):
        res += factorial(n)//(factorial(i)*factorial(n-i))
    return res


def divisors(n):
    d1 = [1]
    d2 = []
    for i in range(2, int(sqrt(n)) + 1):
        if n % i == 0:
            d1.append(i)
            if i*i != n:
                d2.append(n//i)    
    d1.extend(d2[::-1])
    return d1
        

for _ in range(T):
   
    N, K = [int(x) for x in input().split()]
    S = input()
    if N == 1:
        print(N+K)
        continue
    
    total = nck(N, K)
    div = divisors(N)
    dp = [[0]*(N+K+1) for i in range(len(div))]
    is_periodic = False
    
    for i, d in enumerate(div):
        dp[i][0] = 1
        for offset in range(d):
            zeros = 0
            
            for j in range(offset, N, d):
                if S[j] == "0":
                    zeros += 1
            ones = N//d - zeros  
            
            prev = list(dp[i])           
            dp[i][:] = [0]*(N+K+1)
            
            for k in range(K+1):
                if prev[k]:
                    dp[i][k + zeros] += prev[k]
                    dp[i][k + ones] += prev[k]
        
        if dp[i][0]:
            is_periodic = True
        
        for i2 in range(i):                
            d2 = div[i2]            
            if d % d2 == 0:
                for k in range(K+1):
                    dp[i][k] -= dp[i2][k]
                        
        for k in range(1, K+1):
            total -= dp[i][k]
    
    print((total-is_periodic) % M)
                    

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Swap Nodes [Algo]

A binary tree is a tree which is characterized by one of the following properties: It can be empty (null). It contains a root node only. It contains a root node with a left subtree, a right subtree, or both. These subtrees are also binary trees. In-order traversal is performed as Traverse the left subtree. Visit root. Traverse the right subtree. For this in-order traversal, start from

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Kitty's Calculations on a Tree

Kitty has a tree, T , consisting of n nodes where each node is uniquely labeled from 1 to n . Her friend Alex gave her q sets, where each set contains k distinct nodes. Kitty needs to calculate the following expression on each set: where: { u ,v } denotes an unordered pair of nodes belonging to the set. dist(u , v) denotes the number of edges on the unique (shortest) path between nodes a

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Is This a Binary Search Tree?

For the purposes of this challenge, we define a binary tree to be a binary search tree with the following ordering requirements: The data value of every node in a node's left subtree is less than the data value of that node. The data value of every node in a node's right subtree is greater than the data value of that node. Given the root node of a binary tree, can you determine if it's also a

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Square-Ten Tree

The square-ten tree decomposition of an array is defined as follows: The lowest () level of the square-ten tree consists of single array elements in their natural order. The level (starting from ) of the square-ten tree consists of subsequent array subsegments of length in their natural order. Thus, the level contains subsegments of length , the level contains subsegments of length , the

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Balanced Forest

Greg has a tree of nodes containing integer data. He wants to insert a node with some non-zero integer value somewhere into the tree. His goal is to be able to cut two edges and have the values of each of the three new trees sum to the same amount. This is called a balanced forest. Being frugal, the data value he inserts should be minimal. Determine the minimal amount that a new node can have to a

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Jenny's Subtrees

Jenny loves experimenting with trees. Her favorite tree has n nodes connected by n - 1 edges, and each edge is ` unit in length. She wants to cut a subtree (i.e., a connected part of the original tree) of radius r from this tree by performing the following two steps: 1. Choose a node, x , from the tree. 2. Cut a subtree consisting of all nodes which are not further than r units from node x .

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