leetcode weekly 118 - Equal Rational Numbers

972. Equal Rational Numbers

Given two strings S and T, each of which represents a non-negative rational number, return True if and only if they represent the same number. The strings may use parentheses to denote the repeating part of the rational number.

In general a rational number can be represented using up to three parts: an integer part, a non-repeating part, and a repeating part. The number will be represented in one of the following three ways:

• <IntegerPart> (e.g. 0, 12, 123)
• <IntegerPart><.><NonRepeatingPart>  (e.g. 0.5, 1., 2.12, 2.0001)
• <IntegerPart><.><NonRepeatingPart><(><RepeatingPart><)> (e.g. 0.1(6), 0.9(9), 0.00(1212))

The repeating portion of a decimal expansion is conventionally denoted within a pair of round brackets.  For example:

1 / 6 = 0.16666666... = 0.1(6) = 0.1666(6) = 0.166(66)

Both 0.1(6) or 0.1666(6) or 0.166(66) are correct representations of 1 / 6.

Example 1:

Input: S = "0.(52)", T = "0.5(25)"
Output: true
Explanation:
Because "0.(52)" represents 0.52525252..., and "0.5(25)" represents 0.52525252525..... , the strings represent the same number.

Example 2:

Input: S = "0.1666(6)", T = "0.166(66)"
Output: true

Example 3:

Input: S = "0.9(9)", T = "1."
Output: true
Explanation: 
"0.9(9)" represents 0.999999999... repeated forever, which equals 1.  [See this link for an explanation.]
"1." represents the number 1, which is formed correctly: (IntegerPart) = "1" and (NonRepeatingPart) = "".

Note:

1. Each part consists only of digits.
2. The <IntegerPart> will not begin with 2 or more zeros.  (There is no other restriction on the digits of each part.)
3. 1 <= <IntegerPart>.length <= 4
4. 0 <= <NonRepeatingPart>.length <= 4
5. 1 <= <RepeatingPart>.length <= 4

---

Solution by neal_wu (Rank #1) in contest (Very elegant solution!)
Idea:

• Represent the number in "irreducible fraction" form  
• a and b can exceed 32-bit integer, so use 64-bit integer to store a and b
• Fraction can naturally represent repeat part

struct fraction {
    int64_t numer, denom;

    void reduce() {
        int64_t g = __gcd(numer, denom);
        numer /= g;
        denom /= g;
    }

    bool operator==(const fraction &other) const {
        return numer == other.numer && denom == other.denom;
    }
};

int64_t power10(int n) {
    return n == 0 ? 1 : 10 * power10(n - 1);
}

fraction to_fraction(string S) {
    size_t period = S.find('.');

    if (period == string::npos)
        return {stoll(S), 1};

    int64_t integer = stoll(S.substr(0, period));
    S = S.substr(period + 1);

    if (S.empty())
        return {integer, 1};

    size_t paren = S.find('(');

    if (paren == string::npos) {
        int n = S.size();
        int64_t p = power10(n);
        return {integer * p + stoll(S), p};
    }

    int64_t p = power10(paren);
    int64_t nonrepeating = paren == 0 ? 0 : stoll(S.substr(0, paren));

    string remaining = S.substr(paren + 1, S.size() - 1 - (paren + 1));
    int64_t rp = power10(remaining.size()) - 1;

Why rp = power10(remaining.size()) - 1? Because the repeating part is calculated by geometric series,
e.g. 0.(123)

    int64_t repeating = stoll(remaining);
    return {integer * p * rp + nonrepeating * rp + repeating, p * rp};
}

class Solution {
public:
    bool isRationalEqual(string S, string T) {
        fraction A = to_fraction(S);
        fraction B = to_fraction(T);
        A.reduce();
        B.reduce();
        return A == B;
    }
};