The Question

Palindrome Number

Given an integer x, determine whether it is a palindrome. An integer is a palindrome when it reads the same backward as forward. For example, 121 is a palindrome while 123 is not. Implement the solution without converting the integer to a string. Constraints: - The input x is a signed 32-bit integer: -2^31 <= x <= 2^31 - 1. - Time complexity: O(log n). - Space complexity: O(1).

Python

Math

Questions & Insights

Clarifying Questions

How should negative numbers be handled? (e.g., is -121 a palindrome?)

Are there any constraints on the size of the integer? (e.g., standard 32-bit signed integer range?)

Is converting the integer to a string allowed? (Interviewers often forbid this to test mathematical logic.)

How should we treat numbers ending in zero? (e.g., 120 reversed is 021).

Assumptions:

Negative numbers are not palindromes because the negative sign doesn't appear at the end when reversed.

We should aim for an

O(1)

space solution, meaning we should avoid string conversion.

The input is within the standard 32-bit signed integer range.

Thinking Process

Initial Filter: Any negative number is immediately disqualified. Additionally, any non-zero number ending in zero cannot be a palindrome (as no integer starts with leading zeros).

Reverse Half Strategy: Instead of reversing the whole number (which might cause overflow in fixed-width integer languages), we reverse only the trailing half of the digits.

Iteration Logic: We extract the last digit of the original number using modulo (% 10) and append it to a new variable reversed_num. We then divide the original number by 10.

Termination Condition: We stop when the original number is less than or equal to the reversed_num, indicating we have reached or passed the middle digit.

Final Comparison: For even-length numbers, the two halves must be equal (x == reversed_num). For odd-length numbers, the middle digit is stored in the last position of reversed_num, so we compare x == reversed_num // 10.

Implementation Breakdown

Problem Set

Functional Requirements: Return True if x is a palindrome, False otherwise.

Constraints:

-2^{31} \le x \le 2^{31} - 1

Space complexity should be

O(1)

Time complexity should be

O(\log_{10} n)

Approach

Algorithm: Mathematical digit manipulation (Reversing half-integer).

Data Structure: None.

Complexity:

Time:

O(\log_{10} n)

because we divide the input by 10 in every iteration.

Space:

O(1)

as we only store a few integer variables.

Implementation

Wrap Up

Advanced Topics

Overflow Considerations: In Python, integers have arbitrary precision, so overflow isn't a runtime risk. However, in C++ or Java, reversing a full 32-bit integer could exceed the INT_MAX. Reversing only half the digits is a robust pattern that translates across all languages.

String Conversion Trade-offs: While str(x) == str(x)[::-1] is Pythonic and concise, it requires

O(n)

space for the string and slicing, which is less efficient for large-scale systems or embedded environments with memory constraints.

Bit Manipulation: For binary palindromes, bit manipulation (x & 1 and bit shifts) would be used, but for decimal palindromes, arithmetic operators (% and //) are the standard toolset.