本节主要介绍高精度加/减/乘/除以及位运算。

高精度

高精度加法

c++
#include <iostream>
#include <vector>
using namespace std;

//C = A + B
vector<int> add(vector<int>& A, vector<int>& B)//使用&，避免再次拷贝
{
	vector<int> C;

	int t = 0; //进位
	for (int i = 0; i < A.size() || i < B.size(); i++)
	{
		if (i < A.size()) t += A[i];
		if (i < B.size()) t += B[i];
		C.push_back(t % 10);
		t /= 10; //如果t>=10，进位
	}

	if (t) C.push_back(1); //进位
	return C;
}

int main()
{
	string a, b;
	vector<int> A, B;

	//表示数字时，倒着表示
	//'0'的ASCII码为48，如'9'-'0'=57-48=9
	cin >> a >> b; //a="123456"
	for (int i = a.size() - 1; i >= 0; i--) A.push_back(a[i] - '0'); // A = [6, 5, 4, 3, 2, 1]
	for (int i = b.size() - 1; i >= 0; i--) B.push_back(b[i] - '0'); // -'0'将字符转化成整数


	auto C = add(A, B);//auto == vector<int>

	for (int i = C.size() - 1; i >= 0; i--) printf("%d", C[i]);
	return 0;
}

高精度减法

c++
#include <iostream>
#include <vector>
using namespace std;

//判断是否有 A >= B
bool cmp(vector<int>& A, vector<int>& B)
{
	if (A.size() != B.size()) return A.size() > B.size();
	for (int i = A.size() - 1; i >= 0; i--)
		if (A[i] != B[i])
			return A[i] > B[i];
	return true;
}

//C = A - B
vector<int> sub(vector<int>& A, vector<int>& B)//使用&，避免再次拷贝
{
	vector<int> C;
	for (int i = 0,t=0; i < A.size(); i++)
	{
		t = A[i] - t;//减去借位
		if (i < B.size()) t -= B[i];//判断B[i]是否存在
		C.push_back((t + 10) % 10);//如果t>=0，则为t；如果t<0，则为t+10。用(t+10)%10考虑

		//判断是否需要借位
		if (t < 0) t = 1;
		else t = 0;
	}

	while (C.size() > 1 && C.back() == 0) C.pop_back(); //去掉前导0

	return C;
}

int main()
{
	string a, b;
	vector<int> A, B;

	//表示数字时，倒着表示
	cin >> a >> b; //a="123456"
	for (int i = a.size() - 1; i >= 0; i--) A.push_back(a[i] - '0'); // A = [6, 5, 4, 3, 2, 1]
	for (int i = b.size() - 1; i >= 0; i--) B.push_back(b[i] - '0');

	if (cmp(A, B))
	{
		auto C = sub(A, B);

		for (int i = C.size() - 1; i >= 0; i--) printf("%d", C[i]);
	}
	else
	{
		auto C = sub(B, A);

		printf("-");
		for (int i = C.size() - 1; i >= 0; i--) printf("%d", C[i]);
	}

	return 0;
}

高精度乘法

c++
#include <iostream>
#include <vector>
using namespace std;

// C = A * b 高精度*低精度
vector<int> mul(vector<int>& A, int b)
{
	vector<int> C;

	int t = 0; //进位
	for (int i = 0; i < A.size() || t; i++)
	{
		if (i < A.size()) t += A[i] * b;
		C.push_back(t % 10);
		t /= 10;
	}

	return C;
}

int main()
{
	string a;
	vector<int> A;
	int b;

	cin >> a >> b;
	for (int i = a.size() - 1; i >= 0; i--) A.push_back(a[i] - '0');

	auto C = mul(A, b);

	for (int i = C.size() - 1; i >= 0; i--) printf("%d", C[i]);

	return 0;
}

高精度除法

c++
#include <iostream>
#include <vector>
using namespace std;

// C = A / b 商是C，余数是r
vector<int> div(vector<int>& A, int b, int& r) //r是引用
{
	vector<int> C;
	r = 0;
	for (int i = A.size()-1; i >=0; i--)
	{
		r = r * 10 + A[i];
		C.push_back(r / b);
		r %= b; //余数是模除数
	}

	reverse(C.begin(), C.end());

	while (C.size() > 1 && C.back() == 0) C.pop_back(); //

	return C;
}

int main()
{
	string a;
	vector<int> A;
	int b;

	cin >> a >> b;
	for (int i = a.size() - 1; i >= 0; i--) A.push_back(a[i] - '0');

	int r;
	auto C = div(A, b, r);

	for (int i = C.size() - 1; i >= 0; i--) printf("%d", C[i]);
	cout << endl << r << endl;

	return 0;
}

位运算

c++
#include <iostream>
using namespace std;

int lowbit(int x)
{
    return x & -x;
}

int main()
{
    int n;
    cin >> n;
    int x, res;
    while (n--) {
        cin >> x;
        res = 0;
        while (x) x -= lowbit(x), res++;
        cout << res << ' ';
    }
    return 0;
}