Check if DFS Strings Are Palindromes

Check if DFS Strings Are Palindromes

What is this problem about?

The "Check if DFS Strings Are Palindromes interview question" is an advanced tree and string challenge. You are given a tree where each node has a character. For every node, you form a "DFS string" by concatenating the characters of its descendants in the order they are visited during a DFS. You need to determine, for every single node, if its corresponding DFS string is a palindrome.

Why is this asked in interviews?

Google asks the "Check if DFS Strings Are Palindromes coding problem" because it combines tree traversal with efficient string matching. It’s a "Hard" problem that requires Rolling Hashing or Manacher's Algorithm concepts to avoid $O(N^2)$ time. It evaluates a candidate's ability to handle hierarchical data and verify properties across a whole structure in linear or $O(N \log N)$ time.

Algorithmic pattern used

This problem follows the Tree DFS with Rolling Hash pattern.

1. Post-order Traversal: Use DFS to process each node. The DFS string of a node is the concatenation of DFS strings of its children plus its own character.
2. Hash Representation: For each node, compute a forward hash and a backward hash of its DFS string.
3. Hash Merging: A node's hash is the combined hash of its children's strings in order. You can use properties of rolling hashes to merge these in $O(1)$ or $O( ext{alphabet})$ time per node.
4. Validation: A string is a palindrome if its forward_hash == backward_hash.

Example explanation

Suppose node 1 has character 'a' and children 2 ('b') and 3 ('a').

• DFS visits 2, then 3, then 1 (post-order style usually).
• DFS string for 1: "ba" + "a" = "baa"? (Note: DFS order usually visits root then left then right, depends on the problem variant).
• If DFS string is "aba", the forward hash matches the backward hash.
• Palindrome check: aba == aba. Result: True.

Common mistakes candidates make

• String Concatenation: Actually building the strings for each node. This leads to $O(N^2)$ time and memory, which will fail for large trees.
• Collision Handling: Not using a large enough prime or a double-hash, leading to incorrect "True" results due to hash collisions.
• DFS Order: Misunderstanding the specific DFS visit order required by the problem (e.g., pre-order vs post-order).

Interview preparation tip

Master "Rolling Hashes" for trees. This "Hash Function interview pattern" is the only way to compare strings across thousands of nodes efficiently. Practice problems like "Longest Palindromic Substring" to understand the basics of fast palindrome checking.