Given two binary strings, return their sum (also a binary string).

For example,
a = "11"
b = "1"
Return "100".

    public String addBinary(String a, String b) {
        if (a == null || b == null) {
            return null;
        }
        
        if (a.equals("")) {
            return b;
        }
        if (b.equals("")) {
            return a;
        }
        
        int lenA = a.length();
        int lenB = b.length();
        int i = lenA - 1;
        int j = lenB - 1;
        boolean flag = false;
        StringBuilder builder = new StringBuilder();
        while (i >= 0 && j >= 0) {
            char curA = a.charAt(i);
            char curB = b.charAt(j);
            if (flag) {
                if (curA == '0' && curB == '0') {
                    builder.append(1);
                    flag = false;
                }
                if (curA == '0' && curB == '1') {
                    builder.append(0);
                    flag = true;
                }
                if (curA == '1' && curB == '0') {
                    builder.append(0);
                    flag = true;
                }
                if (curA == '1' && curB == '1') {
                    builder.append(1);
                    flag = true;
                }
            } else {
                if (curA == '0' && curB == '0') {
                    builder.append(0);
                    flag = false;
                }
                if (curA == '0' && curB == '1') {
                    builder.append(1);
                    flag = false;
                }
                if (curA == '1' && curB == '0') {
                    builder.append(1);
                    flag = false;
                }
                if (curA == '1' && curB == '1') {
                    builder.append(0);
                    flag = true;
                }               
            }
            i--;
            j--;
        }
        while (i >= 0) {
            char curA  = a.charAt(i);
            if (flag) {
                if (curA == '0') {
                    builder.append(1);
                    flag = false;
                } else {
                    builder.append(0);
                    flag = true;
                }
            } else {
                builder.append(curA);
            }
            i--;
        }
        while (j >= 0) {
            char curB  = b.charAt(j);
            if (flag) {
                if (curB == '0') {
                    builder.append(1);
                    flag = false;
                } else {
                    builder.append(0);
                    flag = true;
                }
            } else {
                builder.append(curB);
            }
            j--;
        }
        if (flag) {
            builder.append(1);
        }
        return builder.reverse().toString();
    }