Unraveling the Power of Longest Common Subsequences
What is a Longest Common Subsequence?
Imagine having two sequences of elements, and you want to find the longest sequence that appears in both, in the same order. This is precisely what a longest common subsequence (LCS) is. But there’s a catch – the elements don’t have to be consecutive in the original sequences. To qualify as an LCS, the elements must be in ascending order in both sequences.
Understanding LCS with an Example
Let’s take two sequences: S1 and S2. A subsequence is a sequence derived by deleting some or no elements from the original sequence without changing the order of the remaining elements. A common subsequence means a subsequence that appears in both sequences in the same relative order. For instance, {B, C}, {C, D, A, C}, {D, A, C}, {A, A, C}, {A, C}, {C, D} are all common subsequences of S1 and S2. Among these, {C, D, A, C} is the longest common subsequence.
Finding the LCS using Dynamic Programming
Dynamic programming is a powerful technique to find the LCS efficiently. Before we dive in, make sure you’re familiar with dynamic programming. If not, take a moment to brush up on the concept.
The Dynamic Programming Approach
To find the LCS, follow these steps:
- Create a table: Build a table of dimension n+1*m+1, where n and m are the lengths of X and Y respectively. Initialize the first row and column with zeros.
- Fill the table: Fill each cell using the following logic:
- If the characters corresponding to the current row and column match, add one to the diagonal element and point an arrow to the diagonal cell.
- Otherwise, take the maximum value from the previous column and previous row element, and point an arrow to the cell with the maximum value.
- Repeat step 2: Continue filling the table until it’s complete.
- Find the LCS: The value in the last row and last column is the length of the longest common subsequence. Start from the last element and follow the direction of the arrow to find the LCS.
Why Dynamic Programming is More Efficient
Dynamic programming reduces the number of function calls by storing the result of each function call. This approach eliminates the need for redundant calls, making it more efficient than recursive algorithms. In the case of LCS, the dynamic algorithm has a time complexity of O(mn), whereas the recursive algorithm has a complexity of 2max(m, n).
Real-World Applications of LCS
The longest common subsequence has numerous applications in various fields, including:
- Compressing genome resequencing data
- Authenticating users through in-air signatures on mobile phones
Get Started with LCS Algorithms
Explore Python, Java, and C/C++ examples to implement the longest common subsequence algorithm and uncover its power in solving real-world problems.