Happy Birthday Tony Palindromes

Brute force Can search all possible pairs (i, j): 1<=i <= j <=n and check if the substring is a palindrome. Runs in O(n^3) time. Can use this to find all possible palindromes, number of palindromes, longest palindrome, etc.

Dynamic Programming Store bool isPal [1...n][1...n]; isPal[i][i] is always true. isPal[i][i+1] is true iff str[i] == str[i+1] In general, isPal[i][j] is true if isPal[i+1][j-1] is true and str[i]==str[j] O(n^2) time and O(n^2) memory Uses an unnecessary amount of memory Can find longest palindrome, count palindromes, etc., and can determine if a substring is a palindrome in O(1) time, but no real reason to use it

Searching From Center For each possible center (there are 2n-1 of them because the center doesn't have to be a letter), search until the two sides don't match. Runs in O(n^2) time: O(n) per center. Finds maximal palindrome from every center. Can this to find number of palindromes or longest palindrome, and can determine if a substring is a palindrome in O(1) time.

Manacher's algorithm An advanced linear time (O(n)) algorithm to find maximal palindrome at every center. Usually you insert a dummy character between each letter, so that each letter is now a valid center (remember there are 2n-1 centers). Takes advantage of symmetry. Usually you won't need it.

Illustration

Potw Tony is making a round birthday cake to celebrate his birthday. Tony has arranged N letters around the perimeter of his cake. He wishes to give a slice of cake (1 to N consecutive letters) to someone special, so he wants it to be as large as possible. Since Tony likes palindromes, he wants the letters on the slice to form a palindrome. What's the largest slice Tony can cut?

Potw Input format: Line 1: a string of N letters Output format: Line 1: L, the length of the longest palindrome Sample input: abaacaba Sample output: (abaaba is the palindrome) 6 Constraints: 1<=N<=100: 10 points 1<=N<=1000: 20 points