4.5 - Filling the LCS Table
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Introduction to LCS
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Today, we're going to explore the Longest Common Subsequence or LCS. The LCS is essential for comparing sequences, allowing us to determine how closely two sequences match by focusing on common letters.
Can you give an example of where we might use LCS?
Great question! One real-world application is in bioinformatics, where we compare DNA sequences. LCS helps us find the longest string shared between two DNA strands, even when some letters are omitted.
So, we want to find the longest sequence where letters match?
Exactly! We aim to find the longest sequence that appears in both strings while allowing some letters to be dropped.
How would we go about calculating that?
We will fill a table using dynamic programming, ensuring we check the relationships between letters in both sequences.
Can you explain dynamic programming again?
Sure! Dynamic programming breaks down a complex problem into simpler subproblems and solves each one only once, storing its solution, which we can refer to later.
Filling the LCS Table
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Let's talk about how we fill the LCS table. The table is m by n, with m being the length of one sequence and n being the length of the other. Each entry requires us to look at neighboring entries in the table.
What do you mean by neighbors?
Neighbors refer to the entries directly above, below, and to the left of the current cell we are filling. We use these values to determine the match length.
So every cell is computed quickly by just checking those three spots?
Exactly! This eliminates unnecessary computations and speeds up the process significantly.
And is the time complexity still O(m*n)?
Yes, that's correct! Although we are filling an m by n table, we're doing it efficiently thanks to dynamic programming.
What about the instances where letters don't match?
When letters do not match, we explore different subproblems for potential solutions and take the maximum of those subproblem results to ensure we have the best match.
Inductive Structure of LCS
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Now let's delve into the inductive structure of the LCS. If two letters match, we can include them in our sequence and look for the solution in the remaining parts.
But what if the letters don’t match?
If they do not match, we can't include both letters in our solution. Instead, we have to drop one and explore subproblems.
How do we know which one to drop?
That's where we create two separate subproblems, one dropping the first letter and one dropping the second, then we compare their results.
Can you summarize this idea?
Sure! If the letters match, include them in your count. If they don't, generate two possibilities to drop one letter each and determine which provides a larger subsequence.
Real-Life Applications and Code Implementation
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Finally, let's see how LCS is applied. In bioinformatics, researchers utilize LCS for comparative DNA analysis, which is crucial for understanding genetic relations.
And how about file comparisons?
Exactly! The DIFF command uses LCS to find differences between text files. It helps to determine where texts align and what changes need to be made.
Could you illustrate how the code might look?
Sure! The code initializes a matrix for the LCS table, filling it according to the rules we've discussed. It leverages dynamic programming for efficiency.
Is it difficult to implement in code?
Not at all! Once you understand the filling mechanism, the code follows logically. Always start with the base cases and fill by checking neighbors.
Introduction & Overview
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Quick Overview
Standard
In this section, we explore how to compute the Longest Common Subsequence (LCS) between two sequences using dynamic programming. The approach involves filling an m x n table where each entry is computed based on its neighboring entries while allowing for some letters to be dropped. We also discuss the significance of this method in fields such as bioinformatics and text comparison.
Detailed
In this section, we dive into the process of filling the Longest Common Subsequence (LCS) table using dynamic programming techniques. The LCS problem involves finding a subsequence of letters that appear in both sequences in the same order, while permitting the omission of letters. We illustrate how dynamic programming allows us to fill an m x n table efficiently, where each entry corresponds to the longest matching subsequence found up to that point. An optimal solution entails examining neighboring values in the table for each entry, resulting in a runtime complexity of O(m*n). The significance of understanding LCS is underscored through real-world applications, especially in bioinformatics for comparing DNA sequences and in computational tasks such as file comparison using commands like UNIX's DIFF. The inductive structure of LCS is also analyzed, revealing the recursive relationships that underlie the solution, demonstrating that if two letters match, they can be part of the solution. We also find that if they do not match, we can explore subproblems, emphasizing the need for a systematic approach to computing the solution.
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Understanding the Longest Common Subsequence
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Chapter Content
So, we have seen earlier that if we just look blindly at every position and try to scan the word starting that position, we get something which is an order m, n square. now, this solution when require as to fill in the table of size m time n, so obviously, every entries in the m times n table, we just have to look at a neighbors to fill it up. So, it is a constant time operation. So, m times n entries, we fill it m times n times.
Detailed Explanation
The Longest Common Subsequence (LCS) problem involves finding the longest sequence that can appear in both strings in the same order but not necessarily consecutively. To fill the LCS table, we consider all entries m by n, meaning we compare every character of one string with every character of another. The operation required to fill each entry is constant time since we only compare neighboring values, leading to a time complexity of O(m * n).
Examples & Analogies
Imagine you are trying to find all common words between two books in a library. Instead of reading both books line by line (which takes a long time), you can just scan for words that are in both and match them. This way, you are efficiently finding common phrases quickly, similar to how we fill out the LCS table.
Self-Drop in Subsequence Matching
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So, what if we do not look for an exact match, but we allow a self should drop some letters. So, we have a subsequence not a subword, it is allows us to drop some letters and now, if you want to know, after dropping some letters, what is the longest match we can find.
Detailed Explanation
In the LCS problem, we relax the condition of requiring exact matches by allowing for the deletion of characters (self-drop). This allows for a broader search for common sequences, meaning a longer subsequence can be derived from the original strings by dropping some letters. This introduces new matches that would not have been possible in a strict subword match, thus potentially increasing the length of the common subsequence.
Examples & Analogies
Think of it as playing a word game where you can form new words by leaving out certain letters. For example, if you drop 'o' from 'word', you can match it with 'ward'. This flexibility can lead to discovering longer common sequences that may not be immediately visible.
Applications of LCS Beyond Texts
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Before we proceed it useful to look at why the longest common subsequence problem is interesting. So, one of the area is an bio informatics. So, biologist are interested in identifying, how close two species are each other in the genetic sense.
Detailed Explanation
LCS has significant applications in various fields, particularly in bioinformatics, where researchers use it to analyze genetic sequences. By identifying common subsequences in the DNA of two organisms, scientists can infer how closely related they are genetically. This understanding can lead to discoveries about evolution, disease mechanisms, and potential treatments.
Examples & Analogies
Consider two family trees being traced. By looking for common ancestors (like looking for common letters in words), genealogists can determine how closely related two families are over generations, paralleling how LCS identifies relationships in genetic sequences.
Inductive Structure of LCS
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So, one might argue that, this is not the way, I want to go, supposing a 0, in fact, should be match to b 2, that would be the best solution. So, it is not good idea match a 0 to be b 0, but now notice that if a 0 is match to b 2, because it is the subsequence, then anything to the write, say a 1 is match to something it must be further to the right.
Detailed Explanation
The inductive approach to solving the LCS involves analyzing the substrings formed once a match is found or determining which letters to keep or drop. If the first character of both strings matches, it is safe to include it in the LCS and continue the search. However, if they do not match, a decision is made to either drop one character or include it in the search for matches to maximize the subsequence length, forming new subproblems.
Examples & Analogies
Consider assembling a puzzle where you've just found a piece that fits. If you put it in place, you can move ahead confidently, but if it seems like it doesn't fit, you might consider keeping it aside to see if you can find other pieces to match instead. This back-and-forth decision-making is akin to the process of dynamic programming in solving the LCS.
Filling the Dynamic Programming Table
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So, as usual let LCS i comma j, stand for the LCS of the problem starting at a i and b j. So, if a i equal b j as we say, LCS of i j is 1 plus LCS of i plus 1 j plus 1.
Detailed Explanation
The dynamic programming approach starts filling up entries in a grid. LCS[i][j] indicates the length of the longest common subsequence from strings a and b starting at positions i and j. If the characters are the same (a[i] == b[j]), it counts as 1 plus whatever LCS can be derived from the next characters. If not, we explore two possibilities: skipping one character from string a or one from string b to find the maximum LCS.
Examples & Analogies
Imagine building a staircase where each step represents a matching character. Each time you match a character, you step up, and if you encounter a mismatch, you have to decide whether to skip a step or choose another path to find the next step you can match, just as we build up the LCS using the grid.
Key Concepts
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Dynamic Programming: A method for solving complex problems by breaking them down into simpler subproblems.
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LCS Table: A matrix used to store lengths of Longest Common Subsequences.
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Subsequence vs. Substring: A subsequence allows for characters to be dropped while a substring does not.
Examples & Applications
For sequences 'ABCBDAB' and 'BDCAB', the LCS is 'BCAB' with a length of 4.
In the DNA sequences 'ATCGTA' and 'TCGATC', the LCS can highlight shared genes while allowing for omissions.
Memory Aids
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Rhymes
For matching letters, notice the dance, drop some too, give it a chance.
Stories
Imagine two friends, Alice and Bob, with different hobbies. They both enjoy painting and hiking, but Alice loves reading. Dropping the unread books, they discover their common interests, which reflect how LCS identifies commonalities by dropping some items from sequences.
Memory Tools
In LCS, think 'Match, Drop, Fill!' – prioritize matching, drop the non-matching letters, and fill the table!
Acronyms
Remember LCS as L=Longest, C=Common, S=Subsequence, a sequence that stays in line.
Flash Cards
Glossary
- Longest Common Subsequence (LCS)
A subsequence that appears in both sequences in the same order, allowing for some letters to be omitted.
- Dynamic Programming
An optimization method where complex problems are broken down into simpler subproblems, solved only once, and stored for future reference.
- Subproblem
A smaller instance of a larger problem that can be solved independently.
- Neighbors
Entries in a table that are directly adjacent to the current entry being analyzed.
- Time Complexity
An estimate of the running time of an algorithm based on the input size.
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