c++ - 位数组除法的余数C++

Question

我有两个长度分别为 200 和 10 的位数组。我想将第一个除以第二个并得到余数。我如何在 C++ 中使用位运算而不是将它们转换为十进制并使用模运算来做到这一点？

Answer 1

这是 std::bitset 的实现。

求余数涉及将除数向左移动，直到它大于或等于被除数，然后开始向右移动它，直到它回到原来的位置。对于移位除数的每个新值，如果它大于运行余数(从被除数开始)，减去它以获得新的余数。如果除数等于余数，则返回 0。一旦除数到达其原始的、未移动的位置，在从余数中减去之后，如果需要，则余数完成。

BsMod 函数接受被除数和除数参数，被除数参数被原地替换为余数，因此请确保该参数是左值，以便您可以获得结果。

默认测试(在 main 中)随机创建二进制字符串并打印出二进制结果。这有点难以验证，所以我做了另一个随机测试 (Test())，它使用整数值自动验证结果。

#include <iostream>
#include <bitset>
#include <string>
#include <ctime>
#include <cstdlib>

// subtracts b from a, replacing a with the result
template <typename A, typename B>
void Subtract(A &a, const B &b) {
    static const std::size_t minc = a.size() < b.size() ? a.size() : b.size();
    bool borrow = false;
    for(std::size_t i = 0; i<minc; ++i) {
        const bool dif = a[i] ^ b[i] ^ borrow;
        borrow = (a[i] && b[i] && borrow) || (!a[i] && (b[i] || borrow));
        a[i] = dif;
    }
    for(std::size_t i=minc; borrow && i<a.size(); ++i) {
        a[i] = borrow = !a[i];
    }
}

// Returns the index of the highest set bit in b
// Returns unsigned -1 if all bits are 0
template <typename B>
std::size_t HiBit(const B& b) {
    for(std::size_t i = b.size()-1; i+1; --i) {
        if(b[i]) return i;
    }
    // b is zero
    return ~std::size_t(0);
}

// Compare returns 1 if a>b, 0 if a==b or -1 if a<b
template <typename B>
int Compare(const B &a, const B &b) {
    const std::size_t high = a.size()-1;
    for(std::size_t i=high; i+1; --i) {
        if(a[i] != b[i]) {
            return int(a[i]) - int(b[i]);
        }
    }
    return 0;
}

// nr is changed from the dividend to the remainder
template <typename B>
void BsMod(B &nr, B d) {
    const std::size_t hi_n = HiBit(nr);
    const std::size_t hi_d = HiBit(d);
    if(hi_d > hi_n) return;                            // nr < d, keep n as r
    if(hi_d == hi_n && Compare(nr, d) == -1) return;   // nr < d, keep n as r

    const std::size_t dshift = hi_n - hi_d;
    d <<= dshift;

    for(std::size_t i=0; i<=dshift; ++i) {
        const int cmp = Compare(nr, d);
        if(cmp == 0) { nr = B(); return; } // d evenly divides nr, so r is 0
        if(cmp > 0) {  // nr > shifted d
            // the quotient would accumulate a 1 bit here, at the d shift position
            Subtract(nr, d);
        }
        d >>= 1;   // divide d by 2, shift back toward original position
    }
}

template <typename B>
unsigned long long bs_to_ull(const B& b) {
    unsigned long long result = 0;
    for(std::size_t i=0; i<sizeof(unsigned long long)*8; ++i) {
        result |= static_cast<unsigned long long>(b[i]) << i;
    }
    return result;
}

template <typename B>
void ull_to_bs(B& b, unsigned long long n) {
    b.reset();
    for(std::size_t i=0; i<sizeof(unsigned long long)*8; ++i) {
        if(n & ((unsigned long long)1 << i)) b.set(i, true);
    }
}

unsigned long long rand_ull() {
    unsigned long long r = 0;
    unsigned long long b = 0;
    for(int i=0; i<sizeof(unsigned long long); ++i) {
        r = r * 33 + rand();
        b ^= rand();
        b <<= 8;
    }
    return ((r << sizeof(unsigned long long)*4) | (r >> sizeof(unsigned long long )*4)) ^ b;
}

void Test(unsigned long long max=0, int max_iters=0) {
    typedef unsigned long long uit;
    typedef std::bitset<sizeof(unsigned long long)*8+8> bs;
    typedef unsigned long long uit;

    int iter_count = 0;
    for(;;) {
        uit a = rand_ull();
        uit b = rand_ull();
        if(max) {
            a %= max;
            b %= max;
        }
        if(rand() & 255) {
            while(b > a) b >>= rand() & 3;
        }
        if(!b) continue;

        bs bsa;
        ull_to_bs(bsa, a);
        bs bsb;
        ull_to_bs(bsb, b);

        BsMod(bsa, bsb);

        uit ibsm = bs_to_ull(bsa);
        uit m = a % b;

        std::cout << a << " % " << b << " = " << m << "    :    " << ibsm << '\n';

        if(ibsm != m) {
            std::cout << "Error\n";
            return;
        }

        ++iter_count;
        if(max_iters && iter_count > max_iters) break;
    }
}

std::string RandomBinaryString(unsigned bit_count) {
    std::string binstr;
    for(unsigned i=0; i<bit_count; ++i) {
        binstr +=  ((rand() >> (i%5)) ^ i) & 1 ?  '1' : '0';
    }
    return binstr;
}

void TrimLeadingZeros(std::string& s) {
    if(s.length() < 2 || s[0] != '0') return;
    for(std::string::size_type i=1; i<s.length()-1; ++i) {
        if(s[i] != '0') {
            s = s.substr(i);
            return;
        }
    }
    s = s.substr(s.length()-1);
}

int main() {
    srand((unsigned int)time(0));
    //Test(0, 10);  // test with integer values (which are easy to auto-validate)
    //return 0;

    std::string a = RandomBinaryString(200);
    std::string b = RandomBinaryString(10);

    static const int max_bitount = 220;
    typedef std::bitset<max_bitount> bs;

    bs bsa(a);
    bs bsb(b);

    // both arguments must have the same type (number of bits)
    // bsa gets replaced with bsa modulo bsb
    BsMod(bsa, bsb); 


    std::string c = bsa.to_string();
    TrimLeadingZeros(c);

    std::cout << a << "\n   mod\n" << b << "\n   ==\n" << c << '\n';
}