Largest Palindrome Product
Time O(n * 10^N) · Space O(n) · Official statement on LeetCode
Solutions
// Time: O(n * 10^n)
// Space: O(n)
class Solution {
public:
int largestPalindrome(int n) {
if (n == 1) {
return 9;
}
// let x = 10^n-i, y = 10^n-j, s.t. palindrome = x*y
// => (10^n-i)*(10^n-j) = (10^n-i-j)*10^n + i*j
// assume i*j < 10^n (in fact, it only works for 2 <= n <= 8, not general),
// let left = (10^n-i-j), right = i*j, k = i+j
// => left = 10^n-k, right = i*(k-i)
// => i^2 - k*i + right = 0
// => i = (k+(k^2-right*4)^(0.5))/2 or (k+(k^2-right*4)^(0.5))/2 where i is a positive integer
const int upper = pow(10, n) - 1;
for (int k = 2; k <= upper; ++k) {
const int left = pow(10, n) - k;
auto s = to_string(left);
reverse(begin(s), end(s));
const int right = stoi(s);
const int d = k * k - right * 4;
if (d < 0) {
continue;
}
if (sqrt(d) == int(sqrt(d)) && k % 2 == int(sqrt(d)) % 2) {
return (static_cast<uint64_t>(left * pow(10, n)) + right) % 1337;
}
}
return -1;
}
private:
long long buildPalindrome(int n) {
string s = to_string(n);
reverse(s.begin(), s.end());
return stoll(to_string(n) + s);
}
};
// Time: O(10^(2n))
// Space: O(n)
class Solution2 {
public:
int largestPalindrome(int n) {
if (n == 1) {
return 9;
}
const int upper = pow(10, n) - 1;
const int lower = pow(10, n - 1);
for (int i = static_cast<uint64_t>(upper) * upper / pow(10, n); i >= lower; --i) {
auto candidate = buildPalindrome(i);
for (long long y = upper / 11 * 11; y >= lower; y -= 11) { // y must be divisible by 11 because even-number-length palindrome meets modulo 11 digit check
if (candidate / y > upper) {
break;
}
if (candidate % y == 0 && lower <= candidate / y) {
return candidate % 1337;
}
}
}
return -1;
}
private:
long long buildPalindrome(int n) {
string s = to_string(n);
reverse(s.begin(), s.end());
return stoll(to_string(n) + s);
}
};
Beginner Explanation
What is Largest Palindrome Product?
Largest Palindrome Product (LeetCode #479) is a Hard problem that primarily trains math.
How to think about it
- Restate the goal in your own words before coding.
- Work a tiny example by hand so the invariant becomes obvious.
- Identify the pattern — this problem aligns with general problem-solving.
- Only then translate the idea into code.
Why this problem matters
Hard problems force you to combine patterns and prove complexity carefully — interview gold. Official solution notes mention: Math.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Largest Palindrome Product
Opening (30–60 seconds)
- Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
- State a brute force so the interviewer knows you can solve it naively.
- Propose the optimal direction tied to general problem-solving.
Core solution narrative
- Define the state you track (pointers, DP cell, set membership, stack top, etc.).
- Explain the transition when you process the next element.
- Call out time (O(n * 10^N)) and space (O(n)) before coding.
- Code cleanly; narrate variable names.
What interviewers listen for
- Correctness on edge cases
- Complexity honesty
- Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)
Follow-up questions they may ask
- Can you solve it with less memory?
- What if the input stream is infinite / doesn't fit in RAM?
- How would tests look for adversarial inputs?
Optimized Approach
Optimized solution notes
The reference solutions on AlgoForge target O(n * 10^N) time and O(n) space.
Pattern focus: general problem-solving
Use the pattern as a checklist:
- Identify the dominant pattern and stick to one clear invariant
Multiple methods appear in the source solutions — compare them and explain when each is preferable.
Implementation tips
- Prefer readable names over micro-optimizations in interviews.
- Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
- After AC-level logic, re-scan for off-by-one and null checks.
Complexity Analysis
Complexity
| Measure | Bound |
|---|---|
| Time | O(n * 10^N) |
| Space | O(n) |
How to justify this in an interview
- Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
- Space: include hash maps, recursion stack, and output allocation when the problem asks for it.
If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.
Common Mistakes
Common mistakes on Largest Palindrome Product
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for general problem-solving — updating state too early or too late.
- Mutating input unexpectedly when the problem forbids it.
- Off-by-one in windows, ranges, or binary search bounds.
- Ignoring overflow / precision for integer arithmetic problems.
- Overengineering — jumping to an advanced structure when a simpler approach works.
Alternative Approaches
Alternatives
The source file includes more than one method. Compare:
- Primary optimized path — best complexity for typical interviews.
- Secondary approach — often brute force, sorting-based, or space-optimized variant.
Practice articulating when you would pick each (constraints, readability, follow-ups).
Edge Cases
Edge cases checklist
- Minimum input size
- Maximum input size / time limits
- Duplicates and already-sorted input
- Negative numbers / zeros (if applicable)
- Disconnected structures (graphs/trees)
- Single path vs branching recursion depth
Pattern Recognition
Spotting this pattern
Signal phrases that point to general problem-solving:
- Sorted input or ability to sort without changing the answer class
- Need for contiguous subarray / substring → consider sliding window
- Need for O(1) membership → hash set/map
- Optimal substructure + overlapping subproblems → DP
- Connectivity / components → graph DFS/BFS or Union-Find
Primary topics: math.
Follow-up Interview Questions
Follow-ups
- How does the solution change if the input is a stream?
- Can you solve it in-place?
- What if duplicates must be handled differently?
- How would you parallelize the approach?
- Design tests that would break a buggy implementation.
Practice Recommendations
What to practice next
- Re-solve Largest Palindrome Product in a second language (cpp, python).
- Drill 3–5 more problems tagged math.
- Teach the solution out loud in under 5 minutes.
- Add this problem to your revision calendar in 3 days and 14 days.
Visualization
Study checklist
- Read the official problem statement on LeetCode
- Solve on paper / whiteboard first
- Implement the general problem-solving approach
- Verify edge cases from the checklist
- State time and space complexity aloud
- Compare with the AlgoForge reference solution
- Schedule a revision session
Revision notes
Largest Palindrome Product (#479) — Hard. Pattern: general problem-solving. Complexity: O(n * 10^N) time / O(n) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Largest Palindrome Product?+
The reference solutions aim for O(n * 10^N) time and O(n) space. Always re-derive complexity from the code you write in the interview.
What pattern does Largest Palindrome Product use?+
It primarily maps to general problem-solving, within the broader topic of math.
Is Largest Palindrome Product good for interviews?+
Yes — as a Hard problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.
Where can I read the official statement?+
Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/largest-palindrome-product/