<p>Given two non-negative integers <code>num1</code> and <code>num2</code> represented as strings, return the product of <code>num1</code> and <code>num2</code>, also represented as a string.</p>

<p><strong>Note:</strong>&nbsp;You must not use any built-in BigInteger library or convert the inputs to integer directly.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre><strong>Input:</strong> num1 = "2", num2 = "3"
<strong>Output:</strong> "6"
</pre><p><strong class="example">Example 2:</strong></p>
<pre><strong>Input:</strong> num1 = "123", num2 = "456"
<strong>Output:</strong> "56088"
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= num1.length, num2.length &lt;= 200</code></li>
	<li><code>num1</code> and <code>num2</code> consist of digits only.</li>
	<li>Both <code>num1</code> and <code>num2</code>&nbsp;do not contain any leading zero, except the number <code>0</code> itself.</li>
</ul>

class Solution {
public:
    string multiply(string num1, string num2) {
        string ret;

        int row = num1.size();
        int column = num2.size();

        int array[row][column];

        for (int i = 0; i < row; i++) {
            
            int a = num1[i] - '0';
            for (int j = 0; j < column; j ++) {
                int b = num2[j] - '0';
                array[i][j] = a*b;
            }
        }

        int c = column -1;
        int r = row - 1;

        std::vector<int> vec;
        int append = 0;

        while (r >= 0 && c >= 0) {
            if (r == 0 && c ==0) {
                vec.insert(vec.begin(), array[r][c] + append);
                break;
            }

            int sum = 0;
            int tmpc = c;
            int tmpr = r;

            while(c < column && r >= 0) {
                sum += array[r][c];;
                c++;
                r--;
            }

            vec.insert(vec.begin(), (sum + append)%10);
            append = (sum + append) / 10;

            c = tmpc;
            r = tmpr;

            if (c > 0) {
                c--;
            }  
            else {
                r--;
            }
        }

        if (vec[0] == 0) {
            return "0";
        }

        stringstream ss;

        for (int i = 0; i < vec.size(); i++) {
            ss << vec[i];
        }

        ss >> ret;
        return ret;
    }
};
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