Equal Rational Numbers

To solve the 'Equal Rational Numbers' problem, convert the given strings into equivalent rational forms. Focus on understanding how repeating decimals behave. The solution involves comparing two rational numbers represented as strings with potential repeating parts.

Problem Statement

Given two strings, s and t, representing non-negative rational numbers, the task is to determine if they represent the same number. The strings may include parentheses to denote repeating decimal parts.

A rational number can be represented as a combination of an integer part, a non-repeating decimal part, and a repeating decimal part. The repeating portion is denoted within parentheses. The goal is to compare these representations for equality.

Examples

Example 1

Input: s = "0.(52)", t = "0.5(25)"

Output: true

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 details omitted.

Example 3

Input: s = "0.9(9)", t = "1."

Output: true

"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) = "".

Constraints

• Each part consists only of digits.
• The does not have leading zeros (except for the zero itself).
• 1 .length <= 4
• 0 .length <= 4
• 1 .length <= 4

Solution Approach

Handle Repeating Decimals

To compare the two numbers, first expand the repeating decimal into an infinite string by repeating the part inside the parentheses. This helps in dealing with the repeating section of the decimal and ensures accurate comparison.

Normalize the Numbers

Normalize the strings representing the numbers by converting them into a standard format. This may involve removing the parentheses and converting the number to a non-repeating decimal form. The goal is to make the two numbers comparable in a consistent way.

Compare the Numbers

After normalization, compare the two numbers. If they are identical, return true. Otherwise, return false. Pay close attention to the rounding issues that can arise when handling repeating decimals.

Complexity Analysis

Metric	Value
Time	O(1)
Space	O(1)

The time and space complexity of this solution is O(1) because the problem involves comparing two numbers with at most 4 characters each, ensuring constant time and space usage during computation.

What Interviewers Usually Probe

• The candidate correctly identifies how to handle repeating decimal parts.
• The candidate ensures that both numbers are normalized before comparison.
• The candidate efficiently handles edge cases like 0.9(9) vs. 1.0.

Common Pitfalls or Variants

Common pitfalls

• Forgetting to handle the infinite nature of repeating decimals, leading to inaccurate comparisons.
• Not normalizing the numbers correctly, which can result in a false conclusion of inequality.
• Overcomplicating the problem by focusing too much on string manipulations without recognizing the underlying mathematical nature.

Follow-up variants

• Handling more than two numbers and determining if all are equivalent.
• Modifying the problem to compare fractions with non-repeating decimals.
• Generalizing the solution to handle arbitrary precision numbers with repeating decimals.

FAQ

What is the primary challenge in the 'Equal Rational Numbers' problem?

The main challenge is properly handling repeating decimals and ensuring that the strings representing the numbers are normalized before comparison.

How can I compare two numbers with repeating decimals in LeetCode's 'Equal Rational Numbers'?

First, expand the repeating decimals, normalize the numbers, and then compare them for equality. This ensures that the representation of the repeating parts is accurately handled.

What is the time complexity of solving 'Equal Rational Numbers'?

The time complexity is O(1) because the inputs are constrained to small strings, ensuring constant time processing.

Can you compare numbers without expanding the repeating decimals in 'Equal Rational Numbers'?

No, you must account for the infinite nature of repeating decimals by expanding them to their full mathematical representation.

How does GhostInterview assist in solving this problem?

GhostInterview helps by providing a clear approach to handle repeating decimals and ensuring the numbers are normalized before comparison, reducing the chances of errors.