A subsequence is palindromic if it is the same whether read...

A subsequence is palindromic if it is the same whether read left to right or right to left. For instance, the sequence \[ A, C, G, T, G, T, C, A, A, A, A, T, C, G \] has many palindromic subsequences, including $A, C, G, C, A$ and $A, A, A, A$ (on the other hand, the subsequence $A, C, T$ is not palindromic). Devise an algorithm that takes a sequence $x[1 \ldots . n]$ and returns the (length of the) longest palindromic subsequence. Its running time should be \[ O\left(n^{2}\right) \]

A subsequence is palindromic if it is the same whether read left to right or right to left. For instance, the sequence
\[
A, C, G, T, G, T, C, A, A, A, A, T, C, G
\]
has many palindromic subsequences, including $A, C, G, C, A$ and $A, A, A, A$ (on the other hand, the subsequence $A, C, T$ is not palindromic). Devise an algorithm that takes a sequence $x[1 \ldots . n]$ and returns the (length of the) longest palindromic subsequence. Its running time should be
\[
O\left(n^{2}\right)
\]

Key Concepts

Palindrome

A palindrome is a sequence of characters that reads the same forward and backward. This concept is crucial in many string manipulation problems where symmetry properties are exploited to simplify the solution or to find specific subsequences.

Subsequence

A subsequence is a sequence derived from another sequence by deleting some or no elements without changing the order of the remaining elements. This concept allows solutions to consider non-contiguous character selections, which is often essential in pattern matching and optimization problems.

Longest Palindromic Subsequence Problem

This problem involves finding the longest subsequence within a given sequence such that the subsequence is a palindrome. It is a classic optimization problem that demonstrates how to handle constraints by choosing non-adjacent elements while maintaining the original order.

Dynamic Programming

Dynamic Programming is a method for solving problems by breaking them down into overlapping subproblems and storing the solutions of these subproblems to avoid redundant computations. For the longest palindromic subsequence problem, dynamic programming is used to build up solutions incrementally using a table that records the longest palindromic subsequence length for various subintervals of the sequence.

Algorithm Complexity Analysis

In the context of the longest palindromic subsequence problem, algorithm complexity analysis is important to ensure the solution meets the required running time of O(n^2). This involves reviewing the dynamic programming approach's nested loops and state transitions to confirm that the overall time or space requirements are within acceptable limits.

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