Longest Palindromic Substring

LC 5 is the cleanest illustration of why two algorithms with identical Big-O can ship to wildly different production fates. The 2D DP table and expand-around-centers are both O(n²); only one of them allocates a million booleans on the way. At LC 5’s stated n <= 1000 ceiling¹, the brute-force O(n³) approach hits ~10⁹ operations and times out; the two O(n²) approaches hit ~10⁶ and finish in well under a millisecond. The choice between the two O(n²)s is what the chapter is about.

The lesson is one sentence. Every palindrome has a center: either one character (odd length) or the gap between two adjacent characters (even length), so iterating over the 2n - 1 candidate centers and pushing outward at each is sufficient. No table, no allocation, no by-length fill order to remember.

The problem

Given a string s, return any longest contiguous substring of s that reads the same forward as backward. Constraints: 1 <= s.length <= 1000, alphabet limited to digits and English letters. If multiple maximum-length palindromes exist, the judge accepts any of them.¹

LC 5 wants the substring itself, not its length. It is a small detail that costs candidates an obvious extra five minutes when they only tracked max_len. Verbatim wording at LC 5 Longest Palindromic Substring.

Approach 1: brute force

Translate the statement directly. Enumerate every (i, j) with i <= j, check the substring for palindromicity in O(j - i), and track the longest hit.

# Brute force: O(n^3), what we're replacing
def longest_palindrome_brute(s: str) -> str:
    n = len(s)
    best_l, best_r = 0, 0

    def is_palin(lo: int, hi: int) -> bool:
        while lo < hi:
            if s[lo] != s[hi]:
                return False
            lo += 1
            hi -= 1
        return True

    for i in range(n):
        for j in range(i, n):
            if (j - i) > (best_r - best_l) and is_palin(i, j):
                best_l, best_r = i, j
    return s[best_l:best_r + 1]

The work that’s wasted is visible on the second-to-last line: is_palin(i, j) checks s[i] == s[j], then runs the entire palindromicity check on s[i+1..j-1] from scratch. That inner range is also a substring whose palindromicity the algorithm computes again at a different (i, j). Time O(n³), space O(1). At n = 1000 that’s roughly 5 × 10⁸ comparisons in the worst case, which TLEs on LeetCode’s ~1-second budget for Python and Java.²

The fix is to stop recomputing is_palin(i+1, j-1). Either materialize it in a table and read it once, or fold the palindromicity check into the iteration so that the work is reused implicitly. Both fixes get to O(n²); they differ on space and on what the iteration looks like.

Approach 2: the 2D DP table

Define is_palin[i][j] = True iff s[i..j] is a palindrome. The recurrence reads:

is_palin[i][j] = (s[i] == s[j]) AND (j - i < 2 OR is_palin[i+1][j-1])

Length-1 cells are trivially True; length-2 cells are True iff the two characters match. For length L >= 3, the cell depends on the strictly inner range of length L - 2, which is the row below and the column to the left.³

Critically, the fill order is by length, not by row. The cell is_palin[i][j] reads is_palin[i+1][j-1]. Filling top-to-bottom (for i in 0..n: for j in i..n) reads cells that have not been computed yet; the bug surfaces as a correct length-1/length-2 diagonal followed by every length-3+ palindrome reported as missing. The correct outer loop iterates over length L = 1, 2, 3, ..., n, then over the start index i, with j = i + L - 1. By the time you compute the length-L cell, every length-(L-2) cell already has its value.

Time O(n²), space O(n²). At n = 1000 that’s ~10⁶ operations and ~10⁶ booleans (~1 MB). It passes. The table version is also the natural primer for LC 647 (count palindromic substrings, same predicate) and LC 132 (min cuts, layers a 1D DP on top). The next approach has the same asymptotic time and zero allocation.

Approach 3: expand around centers (optimal)

Every palindrome has a center. For odd-length palindromes the center is a single character; for even-length palindromes the center is the gap between two adjacent characters. There are exactly 2n - 1 candidate centers (n character-centers plus n - 1 between-character centers). For each candidate, push two pointers outward as long as s[left] == s[right]. The longest expansion seen across all centers is the answer.⁴²

def longest_palindrome(s: str) -> str:
    if not s:
        return ""

    def expand(left: int, right: int) -> tuple[int, int]:
        while left >= 0 and right < len(s) and s[left] == s[right]:
            left -= 1
            right += 1
        return left + 1, right - 1

    best_l, best_r = 0, 0
    for i in range(len(s)):
        l1, r1 = expand(i, i)         # odd-length center at i
        l2, r2 = expand(i, i + 1)     # even-length center between i and i+1
        if r1 - l1 > best_r - best_l:
            best_l, best_r = l1, r1
        if r2 - l2 > best_r - best_l:
            best_l, best_r = l2, r2
    return s[best_l:best_r + 1]

# LC 5. Longest Palindromic Substring
from typing import Tuple


def longest_palindrome(s: str) -> str:
    """LC 5: longest contiguous palindromic substring of s.

    Expand around each of the 2n - 1 candidate centers (n character-centers
    plus n - 1 between-character centers). For each center, push outward
    while s[left] == s[right]; the loop terminates one step past the last
    valid match, so the maximal palindrome is s[left+1 : right] when the
    helper returns the corrected (left+1, right-1) pair.
    O(n^2) time, O(1) extra space.
    """
    if not s:
        return ""

    def expand(left: int, right: int) -> Tuple[int, int]:
        while left >= 0 and right < len(s) and s[left] == s[right]:
            left -= 1
            right += 1
        return left + 1, right - 1

    best_l, best_r = 0, 0
    for i in range(len(s)):
        l1, r1 = expand(i, i)            # odd-length center at i
        l2, r2 = expand(i, i + 1)        # even-length center between i and i+1
        if r1 - l1 > best_r - best_l:
            best_l, best_r = l1, r1
        if r2 - l2 > best_r - best_l:
            best_l, best_r = l2, r2
    return s[best_l:best_r + 1]

// LC 5. Longest Palindromic Substring
public final class Sol {

    /** LC 5. Return any longest palindromic substring of s. */
    public static String longestPalindrome(String s) {
        if (s.isEmpty()) return "";
        int bestL = 0, bestR = 0;
        for (int i = 0; i < s.length(); i++) {
            int[] odd = expand(s, i, i);
            int[] even = expand(s, i, i + 1);
            if (odd[1] - odd[0] > bestR - bestL) {
                bestL = odd[0];
                bestR = odd[1];
            }
            if (even[1] - even[0] > bestR - bestL) {
                bestL = even[0];
                bestR = even[1];
            }
        }
        return s.substring(bestL, bestR + 1);
    }

    private static int[] expand(String s, int left, int right) {
        while (left >= 0 && right < s.length() && s.charAt(left) == s.charAt(right)) {
            left--;
            right++;
        }
        return new int[] { left + 1, right - 1 };
    }

    private Sol() {}
}

// LC 5. Longest Palindromic Substring
#include <string>
#include <utility>

class Solution {
public:
    // Expand around centers: O(n^2) time, O(1) extra space.
    std::string longestPalindrome(const std::string& s) {
        if (s.empty()) return "";
        int bestL = 0, bestR = 0;
        const int n = static_cast<int>(s.length());
        for (int i = 0; i < n; ++i) {
            auto [l1, r1] = expand(s, i, i);
            auto [l2, r2] = expand(s, i, i + 1);
            if (r1 - l1 > bestR - bestL) { bestL = l1; bestR = r1; }
            if (r2 - l2 > bestR - bestL) { bestL = l2; bestR = r2; }
        }
        return s.substr(bestL, bestR - bestL + 1);
    }

private:
    std::pair<int, int> expand(const std::string& s, int left, int right) {
        while (left >= 0 && right < static_cast<int>(s.length()) && s[left] == s[right]) {
            --left;
            ++right;
        }
        return {left + 1, right - 1};
    }
};

// LC 5. Longest Palindromic Substring
package main

// longestPalindrome implements LC 5 by expanding around each of the 2n-1
// candidate centers; O(n^2) time, O(1) extra space.
func longestPalindrome(s string) string {
	if len(s) == 0 {
		return ""
	}
	bestL, bestR := 0, 0
	for i := 0; i < len(s); i++ {
		l1, r1 := expand(s, i, i)
		l2, r2 := expand(s, i, i+1)
		if r1-l1 > bestR-bestL {
			bestL, bestR = l1, r1
		}
		if r2-l2 > bestR-bestL {
			bestL, bestR = l2, r2
		}
	}
	return s[bestL : bestR+1]
}

func expand(s string, left, right int) (int, int) {
	for left >= 0 && right < len(s) && s[left] == s[right] {
		left--
		right++
	}
	return left + 1, right - 1
}

Interactive widget: Longest Palindromic Substring (LC 5): expand around centers, rendered on the website; the YAML spec describes the keyframes.

Static fallback: on s = "babad", the algorithm walks 9 centers (5 odd, 4 even). At i=1 the odd expansion finds "bab" and updates the running best to length 3. At i=2 the odd expansion finds "aba", also length 3; strict > rejects the tie, so the answer stays "bab". The remaining centers find nothing longer.

The expand helper is the load-bearing piece. Its loop terminates when the pointers either disagree or fall off the string, which means the last valid match was the iteration before. Returning (left + 1, right - 1) corrects for that one over-step. The dual call expand(i, i) and expand(i, i + 1) is what handles odd-length and even-length centers without the input-doubling trick that Manacher’s algorithm and the textbook “Slower algorithm” both use⁴.

Time O(n²): each of 2n - 1 centers expands at most n/2 steps, and the total work across all centers is bounded by n²/2 by an arithmetic-series argument.⁴ Space O(1) extra. The widget animates the "babad" trace because that input surfaces the strict-vs-non-strict update guard: the tie at i=2 disappears on shorter inputs.

Why this wins in practice over the table

Same asymptotic time. No allocation. No by-length fill order to memorize. Smaller constant: each iteration touches at most two characters and compares them; the table iteration touches three cells (is_palin[i+1][j-1], s[i], s[j]) and a write. The expand version stays in cache; the table version doesn’t.

Manacher’s algorithm gets to O(n) by exploiting palindrome mirroring: when a long palindrome centered at c already extends to the right boundary r, any position i < r inherits at least min(r - i, p[mirror_of_i]) of its radius for free, amortizing total work to linear via a “center never decreases, right boundary only increases” potential.⁵ At n = 1000 it is overkill; 10⁶ vs 10³ is sub-millisecond either way. The reason to know its name: at n >= 10⁵, expand-around-centers TLEs on the worst-case all-same input, and Manacher’s becomes load-bearing rather than optional. The full code is roughly four times longer than the centers method and is rarely typed in a 35-minute interview. Naming the algorithm and sketching the mirroring intuition is the expected response.

Edge cases

Empty string. LC’s constraint guarantees s.length >= 1, but production-safe code short-circuits at n == 0 to avoid an out-of-bounds read on s[0]. The Python reference returns "" immediately; the four sibling implementations do the same.

Even-length palindrome ("abba", "cbbd"). The classic catch for an odd-only implementation. expand(i, i) for any i finds at most a length-1 palindrome on "cbbd" (each character is a palindrome of itself); expand(1, 2) finds the length-2 "bb". Without the second call, the algorithm returns a length-1 substring on inputs that have an even-length answer.

All-same string ("aaaa"). Every range is a palindrome, so the answer is the full string. The odd center at the middle index walks all the way to both boundaries; the algorithm’s worst case for time is exactly this input, with each center expanding to half the string and total work at n²/2 matching the bound.⁴

Common bugs

Running only odd-length centers. Calling expand(s, i, i) for every i and skipping expand(s, i, i + 1) returns one of 'c', 'b', 'd' for "cbbd" and 'a' for "abba". The mental fix is the framing: a palindrome of even length anchors between two characters, not on one. Both anchorings are required.

Returning (left, right) instead of (left + 1, right - 1). The expand loop’s terminating condition is s[left] != s[right] or out-of-bounds. After the loop, left and right already point one step past the last valid match. Returning the raw (left, right) includes the mismatching characters; on "cbbd" the odd expansion at i = 1 would report length 4 (the whole string) instead of length 1. Compute the corrected window inside expand once; the rest of the algorithm consumes it cleanly.

Returning the length instead of the substring. Tracking only max_len while iterating answers a different question. LC 5 wants the substring; track (start, max_len) together (or (best_l, best_r) as the reference does) and slice once at the end. The textbook longest-palindromic-subsequence DP recurrence in CLRS Problem 14-2 stores the scalar length in the cell³, which trains the wrong reflex for LC 5; the witness has to be tracked alongside.

Computing length as r - l - 1 after expansion. When debugging an off-by-one, candidates often try to fix the expand return and then write length = r - l - 1 somewhere downstream to “subtract back the over-step”. Two corrections compound into a fresh bug. Pick one place to do the math (inside expand) and let the rest of the code work in true [bestL, bestR] coordinates.

Interview follow-ups

What if you need to count all palindromic substrings instead of returning the longest? That’s LC 647. Same expand-around-centers template. For each center, increment a counter on every successful match instead of tracking the longest. O(n²) time, O(1) space, two-line modification of the reference. Amazon asks this one a lot.⁴

What about the longest palindromic subsequence? Different problem despite the name collision: LC 516. The substring (LC 5) requires endpoints to match AND the inner range to be a palindrome: strict AND, propagating “no” along the chain. The subsequence relaxes to max(dp[i+1][j], dp[i][j-1]) when the endpoints disagree, because skipping one endpoint is allowed. On "bbbab", LC 5 returns "bbb" (length 3, contiguous); LC 516 returns 4 ("bbbb", skipping the 'a'). LC 516 is the canonical CLRS Problem 14-2 / 15-2 setup³ and does NOT yield to expand-around-centers. The disambiguation is the question’s whole point.

How would you partition the string into palindromes? That’s LC 131, backtracking on top of a palindrome check. Precompute the is_palin[i][j] table from Approach 2 in O(n²), then enumerate partitions by recursing from each position to every later position where the prefix is a palindrome. The expensive piece is the partition enumeration (exponentially many in the worst case), not the palindrome predicate; this is where the table earns its keep.

What if characters arrive one at a time? This is the original Manacher framing. When the algorithm has to maintain “longest palindrome so far” at every prefix, expand-around-centers does not extend cleanly: each new character potentially invalidates work done for centers near the right edge. Manacher’s 1975 paper title is “A new linear-time on-line algorithm for finding the smallest initial palindrome of a string”⁵; the on-line qualifier is the entire interview answer. The right-boundary r and center c advance with each character at amortized O(1) cost.

References