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Happy Birthday Tony

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.

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Happy Birthday Tony

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  1. Happy Birthday Tony Palindromes

  2. 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.

  3. 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

  4. 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.

  5. 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.

  6. Illustration

  7. 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?

  8. 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 points1<=N<=1000: 20 points

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