Palindromes Hot Sheet

Use this page when one static string is the whole world and the task is really about palindrome centers or longest palindromic substring.

Do Not Use When

• the string changes online
• the task is many arbitrary substring palindrome queries, not one full static scan
• exact one-pattern matching or suffix order is the real bottleneck

Choose By Signal

• one static string, longest palindromic substring, or all center radii -> manacher.cpp
• many substring palindrome checks or updates -> compare String Hashing hot sheet
• prefix/border reasoning is the real structure -> compare Z-Function
• the modeling still feels fuzzy -> reopen Palindromes / Manacher

Core Invariants

• d1[i] is the odd radius count at center i, so odd length is 2 * d1[i] - 1
• d2[i] is the even radius count at split i, so even length is 2 * d2[i]
• each pass keeps one current rightmost palindrome interval and reuses mirrored information inside it
• new expansion work only happens when the right boundary moves farther right

Main Traps

• confusing center index with split index for even palindromes
• treating d1[i] as a half-length without the -1 correction
• extracting substring bounds with the wrong left endpoint
• reaching for hashing when one exact static scan already solves the task

Exact Starters In This Repo

• exact static palindrome-radii starter -> manacher.cpp
• flagship repo note -> Longest Palindrome
• compare query-style route -> String Hashing hot sheet

Reopen Paths

• odd/even radii, mirror reuse, and bound formulas -> Palindromes / Manacher
• neighboring string-family chooser -> String cheatsheet
• template map -> Template Library