Problem:


Score : 1000 points

Problem Statement

There are N non-negative integers written on a blackboard. The i-th integer is A_i.

Takahashi can perform the following two kinds of operations any number of times in any order:

Select one integer written on the board (let this integer be X). Write 2X on the board, without erasing the selected integer.

Select two integers, possibly the same, written on the board (let these integers be X and Y). Write X XOR Y (XOR stands for bitwise xor) on the blackboard, without erasing the selected integers.

How many different integers not exceeding X can be written on the blackboard? We will also count the integers that are initially written on the board.
Since the answer can be extremely large, find the count modulo 998244353.

Constraints

1 \leq N \leq 6

1 \leq X < 2^{4000}

1 \leq A_i < 2^{4000}(1\leq i\leq N)

All input values are integers.

X and A_i(1\leq i\leq N) are given in binary notation, with the most significant digit in each of them being 1.

Input

Input is given from Standard Input in the following format:

N X
A_1
:
A_N

Output

Print the number of different integers not exceeding X that can be written on the blackboard.

Sample Input 1

3 111
1111
10111
10010

Sample Output 1

4

Initially, 15, 23 and 18 are written on the blackboard. Among the integers not exceeding 7, four integers, 0, 3, 5 and 6, can be written.
For example, 6 can be written as follows:

Double 15 to write 30.

Take XOR of 30 and 18 to write 12.

Double 12 to write 24.

Take XOR of 30 and 24 to write 6.

Sample Input 2

4 100100
1011
1110
110101
1010110

Sample Output 2

37

Sample Input 3

4 111001100101001
10111110
1001000110
100000101
11110000011

Sample Output 3

1843

Sample Input 4

1 111111111111111111111111111111111111111111111111111111111111111
1

Sample Output 4

466025955

Be sure to find the count modulo 998244353.



Sample input:
3 111
1111
10111
10010


Sample output:
4

Submission status: Wrong Answer

Solution code:
#!/usr/bin/env python3
def divmod(f, g):
    assert g
    h = 0
    for i in reversed(range(f.bit_length() - g.bit_length() + 1)):
        if f & (1 << (g.bit_length() + i - 1)):
            f ^= g << i
            h ^= 1 << i
    return h, f

def gcd(f, g):
    while g:
        q, r = divmod(f, g)
        f, g = g, r
    return f

import functools
def solve(n, x, a):
    # (g) = (a_1, ..., a_n) is a principal ideal since F_2[x] is a PID
    g = functools.reduce(gcd, a)

    # count h in F_2[x] s.t. h g <= x
    cnt = 0
    h = 0
    for k in reversed(range(x.bit_length() - g.bit_length() + 1)):
        bit = 1 << (g.bit_length() + k - 1)
        if (x & bit):
            cnt += 1 << k
        if (x & bit) != (h & bit):
            h ^= g << k
    cnt += 1  # zero
    return cnt % 998244353


def main():
    n, x = input().split()
    n = int(n)
    x = int(x, 2)
    a = [ int(input(), 2) for _ in range(n) ]
    print(solve(n, x, a))

if __name__ == '__main__':
    main()

Submission status: Accepted

Solution code:
# coding: utf-8
# Your code here!
import sys
read = sys.stdin.read
readline = sys.stdin.readline
sys.setrecursionlimit(10**5)

n,x = readline().split()
n = int(n)
x = int(x,2)
*a, = map(lambda x:int(x,2), read().split())

def gcd(a,b):
    c = 0
    while b:
        if a < b:
            a,b = b,a
            continue
        a ^= b<<(a.bit_length() - b.bit_length())
    return a

v = 0
for ai in a: v = gcd(ai,v)

MOD = 998244353
p2 = [1]
for _ in range(5000): p2.append(p2[-1]*2%MOD)

dv = v.bit_length()
dx = x.bit_length()

def dfs(x,y,d):
    if d < dv:
        return int(x >= y)
    else:
        bx = x>>(d-1)&1
        by = y>>(d-1)&1
        if bx and by:
            return (p2[d-dv] + dfs(x,y,d-1)) %MOD
        elif not bx and by:
            return dfs(x,y^(v<<(d-dv)),d-1)
        elif not by and bx:
            return (p2[d-dv] + dfs(x,y^(v<<(d-dv)),d-1)) %MOD
        else:
            return dfs(x,y,d-1)

print(dfs(x,0,dx))

Submission status: Accepted

Solution code:
import random

mod=998244353
N,X=input().split()
N=int(N)
A=[]
for i in range(N):
    A.append(int(input(),2))
A.sort()

a=A[-1]
M=max(len(X)-1,a.bit_length()-1)
data=[0]*(M+1)
n=a.bit_length()-1
for i in range(M-n,-1,-1):
    data[i+n]=a<<i

low=n
for i in range(0,N-1):
    a=A[i]
    flag=True
    while flag:
        n=a.bit_length()
        for j in range(n-1,low-1,-1):
            a=min(a,a^data[j])
        if a!=0:
            data[a.bit_length()-1]=a
            id=a.bit_length()-1
            low=id
            while data[id+1]==0:
                data[id+1]=min((data[id]<<1)^a,(data[id]<<1))
                id+=1
            else:
                a=data[id]<<1
        else:
            break

data2=[0]*(M+1)
for i in range(M+1):
    data2[i]=(data[i]!=0)

for i in range(1,M+1):
    data2[i]+=data2[i-1]

data2=[0]+data2

#print(data)
#print(data2)

x=0
ans=0
n=len(X)-1
for i in range(len(X)):
    if X[i]=="1":
        if x>>(n-i)&1==1:
            if data[n-i]:
                ans+=pow(2,data2[n-i],mod)
                ans%=mod
        else:
            ans+=pow(2,data2[n-i],mod)
            ans%=mod
            if data[n-i]:
                x=x^data[n-i]
            else:
                break
    else:
        if x>>(n-i)&1==1:
            if data[n-i]:
                x=x^data[n-i]
            else:
                break
        else:
            continue
else:
    ans+=1
    ans%=mod
print(ans)

Submission status: Accepted

Solution code:
import random

mod=998244353
N,X=input().split()
N=int(N)
A=[]
for i in range(N):
    A.append(int(input(),2))
A.sort()

a=A[-1]
M=max(len(X)-1,a.bit_length()-1)
data=[0]*(M+1)
n=a.bit_length()-1
for i in range(M-n,-1,-1):
    data[i+n]=a<<i

for i in range(0,N-1):
    a=A[i]
    flag=True
    while flag:
        n=a.bit_length()
        for j in range(n-1,-1,-1):
            a=min(a,a^data[j])
        if a!=0:
            data[a.bit_length()-1]=a
            id=a.bit_length()-1
            while data[id+1]==0:
                data[id+1]=min((data[id]<<1)^a,(data[id]<<1))
                if data[id+1]:
                    id+=1
                else:
                    flag=False
            else:
                a=data[id]<<1
        else:
            break

data2=[0]*(M+1)
for i in range(M+1):
    data2[i]=(data[i]!=0)

for i in range(1,M+1):
    data2[i]+=data2[i-1]

data2=[0]+data2

#print(data)
#print(data2)

x=0
ans=0
n=len(X)-1
for i in range(len(X)):
    if X[i]=="1":
        if x>>(n-i)&1==1:
            if data[n-i]:
                ans+=pow(2,data2[n-i],mod)
                ans%=mod
        else:
            ans+=pow(2,data2[n-i],mod)
            ans%=mod
            if data[n-i]:
                x=x^data[n-i]
            else:
                break
    else:
        if x>>(n-i)&1==1:
            if data[n-i]:
                x=x^data[n-i]
            else:
                break
        else:
            continue
else:
    ans+=1
    ans%=mod
print(ans)

Submission status: Accepted

Solution code:
import random

mod=998244353
N,X=input().split()
N=int(N)
A=[]
for i in range(N):
    A.append(int(input(),2))
A.sort()

a=A[-1]
M=max(len(X)-1,a.bit_length()-1)
base=[]
n=a.bit_length()-1
for i in range(M-n,-1,-1):
    base.append(a<<i)

for i in range(0,N-1):
    a=A[i]
    for j in range(M):
        for b in base:
            a=min(a,a^b)
        if a==0:
            break
        else:
            base.append(a)
            a=a<<1

data=[0]*(M+1)
data2=[0]*(M+1)
for b in base:
    data[b.bit_length()-1]=b
    data2[b.bit_length()-1]=1

for i in range(1,M+1):
    data2[i]+=data2[i-1]

data2=[0]+data2

#print(data)
#print(data2)

x=0
ans=0
n=len(X)-1
for i in range(len(X)):
    if X[i]=="1":
        if x>>(n-i)&1==1:
            if data[n-i]:
                ans+=pow(2,data2[n-i],mod)
                ans%=mod
        else:
            ans+=pow(2,data2[n-i],mod)
            ans%=mod
            if data[n-i]:
                x=x^data[n-i]
            else:
                break
    else:
        if x>>(n-i)&1==1:
            if data[n-i]:
                x=x^data[n-i]
            else:
                break
        else:
            continue
else:
    ans+=1
    ans%=mod
print(ans)

Submission status: Accepted

Solution code:
import sys
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines

# xor shift（F_2上の多項式）に関する gcd を計算する

from functools import reduce

X = int(readline().split()[1],2)
A = [int(x,2) for x in readlines()]

MOD = 998244353

def gcd(a,b):
    if a < b:
        a,b = b,a
    while b:
        LA = a.bit_length()
        LB = b.bit_length()
        a ^= b << (LA - LB)
        if a < b:
            a,b = b,a
    return a

g = reduce(gcd,A)

# g の "倍数" で、X以下のものを作る方法の数を数える
# 上位の桁を決めると、gの倍数は一意に決まる。
# よって、上位の桁がXに一致するときのみが問題

LX = X.bit_length()
Lg = g.bit_length()
answer = X >> (Lg - 1)

prod = 0
x = X; Lx = LX
while Lx >= Lg:
    prod ^= g << (Lx - Lg)
    x ^= g << (Lx - Lg)
    Lx = x.bit_length()

if prod <= X:
    answer += 1

answer %= MOD
print(answer)

Submission status: Accepted

Solution code:
import sys
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines

# xor shift（F_2上の多項式）に関する gcd を計算する

from functools import reduce

X = int(readline().split()[1],2)
A = [int(x,2) for x in readlines()]

MOD = 998244353

def gcd(a,b):
    if a < b:
        a,b = b,a
    while b:
        LA = a.bit_length()
        LB = b.bit_length()
        a ^= b << (LA - LB)
        if a < b:
            a,b = b,a
    return a

g = reduce(gcd,A)

# g の "倍数" で、X以下のものを作る方法の数を数える
# 上位の桁を決めると、gの倍数は一意に決まる。
# よって、上位の桁がXに一致するときのみが問題

LX = X.bit_length()
Lg = g.bit_length()
answer = X >> (Lg - 1)

prod = 0
x = X; Lx = LX
while Lx >= Lg:
    prod ^= g << (Lx - Lg)
    x ^= g << (Lx - Lg)
    Lx = x.bit_length()

if prod <= X:
    answer += 1

answer %= MOD
print(answer)

Submission status: Accepted

Solution code:
import sys

sys.setrecursionlimit(10 ** 6)
int1 = lambda x: int(x) - 1
p2D = lambda x: print(*x, sep="\n")
def MI(): return map(int, sys.stdin.readline().split())
def LI(): return list(map(int, sys.stdin.readline().split()))
def LLI(rows_number): return [LI() for _ in range(rows_number)]

def gcd(a,b):
    if b==0:return a
    if a<b:a,b=b,a
    k=a.bit_length()-b.bit_length()
    return gcd(b,a^(b<<k))

def main():
    md=998244353
    n,x=input().split()
    x=int(x,2)
    aa=[input() for _ in range(int(n))]
    aa=[int(a,2) for a in aa]
    #bitを多項式の係数と考えて多項式のGCDを求める
    g=aa[0]
    for a in aa[1:]:
        g=gcd(g,a)
    #print(bin(g))
    #GCDの桁数以上は自由に決められるのでその分を答えに
    ans=x>>(g.bit_length()-1)
    s=0
    # GCDの桁数以上をxと同じにした場合にx以下になるようであればその分も足す
    while 1:
        k=(x^s).bit_length()-g.bit_length()
        #print(k,bin(x^s),bin(x),bin(s))
        if k<0:break
        s^=g<<k
    if s<=x:ans+=1
    print(ans%md)

main()

Submission status: Accepted

Solution code:
N, X = input().split()
N = int(N); X = int(X, 2)
p = int(input(), 2)
for i in range(N-1):
    v = int(input(), 2)
    if p > v:
        p, v = v, p
    while 0 < p:
        q = (v ^ (p << (v.bit_length() - p.bit_length())))
        if p < q:
            p, v = p, q
        else:
            p, v = q, p
    p = v

lx = X.bit_length()
lp = p.bit_length()

MOD = 998244353
ans = (X >> (p.bit_length() - 1)) % MOD

q = p << (lx - lp)
b = 1 << (lx - 1)

x = 0; y = 0
for k in range(lx - lp, -1, -1):
    if b & (X ^ y):
        y ^= q
    b >>= 1; q >>= 1
if y <= X:
    ans = (ans + 1) % MOD
print(ans)

Submission status: Accepted

Solution code:
#!/usr/bin/env python3
def divmod(f, g):
    assert g
    h = 0
    for i in reversed(range(f.bit_length() - g.bit_length() + 1)):
        if f & (1 << (g.bit_length() + i - 1)):
            f ^= g << i
            h ^= 1 << i
    return h, f

def gcd(f, g):
    while g:
        q, r = divmod(f, g)
        f, g = g, r
    return f

import functools
def solve(n, x, a):
    # (g) = (a_1, ..., a_n) is a principal ideal since F_2[x] is a PID
    g = functools.reduce(gcd, a)

    # count h in F_2[x] s.t. h g <= x
    cnt = 0
    h = 0
    for k in reversed(range(x.bit_length() - g.bit_length() + 1)):
        bit = 1 << (g.bit_length() + k - 1)
        if (x & bit):
            cnt += 1 << k
        if (x & bit) != (h & bit):
            h ^= g << k
    cnt += (h <= x)
    return cnt % 998244353


def main():
    n, x = input().split()
    n = int(n)
    x = int(x, 2)
    a = [ int(input(), 2) for _ in range(n) ]
    print(solve(n, x, a))

if __name__ == '__main__':
    main()