Longest Palindrome After Substring Concatenation I
Time O(n * m) · Space O(n + m) · Official statement on LeetCode
Solutions
// Time: O(n * m)
// Space: O(n + m)
// manacher's algorithm, dp
class Solution {
public:
int longestPalindrome(string s, string t) {
const auto& manacher = [](const string& s) {
const auto& preProcess = [](const string& s) {
if (empty(s)) {
return string("^$");
}
string ret = "^";
for (int i = 0; i < size(s); ++i) {
ret += "#" + s.substr(i, 1);
}
ret += "#$";
return ret;
};
string T = preProcess(s);
const int n = size(T);
vector<int> P(n);
int C = 0, R = 0;
for (int i = 1; i < n - 1; ++i) {
int i_mirror = 2 * C - i;
P[i] = (R > i) ? min(R - i, P[i_mirror]) : 0;
while (T[i + 1 + P[i]] == T[i - 1 - P[i]]) {
++P[i];
}
if (i + P[i] > R) {
C = i;
R = i + P[i];
}
}
return P;
};
const auto& longest_palindrome = [&](const auto &s) {
vector<int> result(size(s) + 1);
const auto& P = manacher(s);
for (int i = 1; i + 1 < size(P); ++i) {
result[(i - P[i]) / 2] = P[i];
}
return result;
};
reverse(begin(t), end(t));
const auto& p1 = longest_palindrome(s);
const auto& p2 = longest_palindrome(t);
int result = 0;
vector<vector<int>> dp(size(s) + 1, vector<int>(size(t) + 1));
for (int i = 0; i < size(s); ++i) {
for (int j = 0; j < size(t); ++j) {
dp[i + 1][j + 1] = s[i] == t[j] ? dp[i][j] + 2 : 0;
result = max(result, dp[i + 1][j + 1] + max(p1[i + (s[i] == t[j] ? 1 : 0)] , p2[j + (s[i] == t[j] ? 1 : 0)]));
}
}
return result;
}
};
Beginner Explanation
What is Longest Palindrome After Substring Concatenation I?
Longest Palindrome After Substring Concatenation I (LeetCode #3503) is a Medium problem that primarily trains dynamic programming.
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 manachers algorithm and dynamic programming.
- Only then translate the idea into code.
Why this problem matters
It sits in the sweet spot of interview difficulty: multiple valid approaches, clear trade-offs. Official solution notes mention: Manacher's Algorithm, DP.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Longest Palindrome After Substring Concatenation I
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 manachers algorithm and dynamic programming.
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 * m)) and space (O(n + m)) 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 * m) time and O(n + m) space.
Pattern focus: manachers algorithm and dynamic programming
Use the pattern as a checklist:
- manachers algorithm — confirm the invariant holds after each step
- dynamic programming — confirm the invariant holds after each step
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 * m) |
| Space | O(n + m) |
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 Longest Palindrome After Substring Concatenation I
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for manachers algorithm and dynamic programming — 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 manachers algorithm and dynamic programming:
- 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: dynamic programming.
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 Longest Palindrome After Substring Concatenation I in a second language (cpp, python).
- Drill 3–5 more problems tagged dynamic programming.
- 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 manachers algorithm and dynamic programming 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
Longest Palindrome After Substring Concatenation I (#3503) — Medium. Pattern: manachers algorithm and dynamic programming. Complexity: O(n * m) time / O(n + m) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Longest Palindrome After Substring Concatenation I?+
The reference solutions aim for O(n * m) time and O(n + m) space. Always re-derive complexity from the code you write in the interview.
What pattern does Longest Palindrome After Substring Concatenation I use?+
It primarily maps to manachers algorithm and dynamic programming, within the broader topic of dynamic programming.
Is Longest Palindrome After Substring Concatenation I good for interviews?+
Yes — as a Medium 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/longest-palindrome-after-substring-concatenation-i/