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We can represent each decimal number, potentially with repeating decimals, as a fraction. By converting the number represented by the string into a fraction (numerator and denominator) in a consistent way, we can directly compare the fractions. This involves algebraic manipulation to handle the repeating parts.
For example, 0.1(6) can be represented as 1/6. By converting the repeating part into a fraction, you maintain precision without dealing with floating point inaccuracies. Once both numbers are represented as fractions, compare the fractions as ratios of their numerators and denominators to determine if they represent the same number.
Time Complexity: O(n), where n is the length of the maximum string length, since we handle each character a constant number of times.
Space Complexity: O(1), as we're using a constant amount of additional space for our number components and calculations.
1using System;
2
3public class RationalNumberComparer {
4 private static long Gcd(long a, long b) {
5 return b == 0 ? a : Gcd(b, a % b);
6 }
7
8 private void SplitNumber(string s, out long intPart, out long nonRepeat, out long repeat, out int nonRepeatLen, out int repeatLen) {
9 intPart = 0;
10 nonRepeat = 0;
11 repeat = 0;
12 nonRepeatLen = 0;
13 repeatLen = 0;
14 bool intPartRead = true;
foreach (char c in s) {
if (char.IsDigit(c)) {
if (intPartRead) intPart = intPart * 10 + (c - '0');
else if (repeatLen == 0) nonRepeat = nonRepeat * 10 + (c - '0'); nonRepeatLen++;
else repeat = repeat * 10 + (c - '0'); repeatLen++;
} else if (c == '.') {
intPartRead = false;
} else if (c == '(') {
repeatLen = 1;
}
}
}
private void ToFraction(string s, out long numerator, out long denominator) {
long intPart, nonRepeat, repeat;
int nonRepeatLen, repeatLen;
SplitNumber(s, out intPart, out nonRepeat, out repeat, out nonRepeatLen, out repeatLen);
if (repeatLen == 0) {
numerator = intPart * (long)Math.Pow(10, nonRepeatLen) + nonRepeat;
denominator = (long)Math.Pow(10, nonRepeatLen);
} else {
long powNR10 = (long)Math.Pow(10, nonRepeatLen);
long powR10 = (long)Math.Pow(10, repeatLen);
numerator = intPart * powNR10 * (powR10 - 1) + nonRepeat * (powR10 - 1) + repeat;
denominator = powNR10 * (powR10 - 1);
}
long commonDiv = Gcd(numerator, denominator);
numerator /= commonDiv;
denominator /= commonDiv;
}
public bool IsEqualRational(string s, string t) {
ToFraction(s, out long numS, out long denS);
ToFraction(t, out long numT, out long denT);
return numS == numT && denS == denT;
}
public static void Main() {
RationalNumberComparer rnc = new RationalNumberComparer();
string s = "0.(52)", t = "0.5(25)";
Console.WriteLine(rnc.IsEqualRational(s, t) ? "true" : "false");
}
}
C# provides a similar syntax and approach to C++ and Java. The C# solution uses methods to split and process parts of the number. Here, each number is processed into parts in a similar manner, managing their fractional conversion and leveraging Math.Pow for powers of ten where necessary. The answer is decided based on the equality of the resulting fractions.
Instead of converting into a fraction, simulate the actual number by expanding it up to a certain number of places where we are confident they will differ if they are indeed different numbers. Focus on accurately reproducing the decimal expansion including the repeating part, and use this expanded version for comparison.
You could use string manipulation to handle the repeat section and use a loop to add the repeating digits until you achieve a sufficient significant length (e.g., 20-30 digits) for a robust comparison.
Time Complexity: O(p), where p is the precision we choose (20 in this case), because we repeatedly generate strings to be compared until this precision is achieved.
Space Complexity: O(p), primarily used to store sequences of the generated decimal until precision is satisfied.
1def expand_decimal(s, precision
This solution focuses on simulating the decimal expansion by repeating the portion inside parentheses enough times to ensure that, if the numbers are the same, they will coincide over a sufficient length to determine equality. The expand_decimal function repeats the periodic section until reaching a designated precision and then compares the results much like directly comparing two string values.