题目：

参考解法：

法一：

This is the standard manual multiplication algorithm. We use two nested for loops, working backward from the end of each input number. We pre-allocate our result and accumulate our partial result in there. One special case to note is when our carry requires us to write to our sum string outside of our for loop.

At the end, we trim any leading zeros, or return 0 if we computed nothing but zeros.

string multiply(string num1, string num2) {string sum(num1.size() + num2.size(), '0');for (int i = num1.size() - 1; 0 <= i; --i) {int carry = 0;for (int j = num2.size() - 1; 0 <= j; --j) {int tmp = (sum[i + j + 1] - '0') + (num1[i] - '0') * (num2[j] - '0') + carry;sum[i + j + 1] = tmp % 10 + '0';carry = tmp / 10;}sum[i] += carry;}size_t startpos = sum.find_first_not_of("0");if (string::npos != startpos) {return sum.substr(startpos);}return "0";
}

法二：The key part is to use a vector to store all digits REVERSELY. after the calculation, find the rightmost NON-Zero digits and convert it to a string.

class Solution {
public:string multiply(string num1, string num2) {unsigned int l1=num1.size(),l2=num2.size();if (l1==0||l2==0) return "0";vector<int> v(l1+l2,0);for (unsigned int i=0;i<l1;i++){int carry=0;int n1=(int)(num1[l1-i-1]-'0');//Calculate from rightmost to leftfor (unsigned int j=0;j<l2;j++){int n2=(num2[l2-j-1]-'0');//Calculate from rightmost to leftint sum=n1*n2+v[i+j]+carry;carry=sum/10;v[i+j]=sum%10;}if (carry>0)v[i+l2]+=carry;}int start=l1+l2-1;while(v[start]==0) start--;if (start==-1) return "0";string s="";for (int i=start;i>=0;i--)s+=(char)(v[i]+'0');return s;}
};

分析：

两个数相乘，最后的结果的长度，最大为两个乘数的长度之和。