Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to Longest Common Subsequence
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Today, we are going to explore the Longest Common Subsequence, or LCS. Can anyone tell me what we might want to use LCS for?
Is it used for comparing DNA sequences?
Exactly! In bioinformatics, LCS can help identify similarities between DNA sequences by allowing for some characters to be dropped. This helps in aligning genetic information efficiently.
How is it different from looking for common words?
Great question! Unlike common subwords, LCS allows dropping letters, making it easier to find matches in long or disparate sequences. Remember: Just think of LCS as being less strict! LCS = Less Common Strictness! (using 'LCS' as a mnemonic).
Dynamic Programming Approach
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Next, we will discuss how to compute LCS using dynamic programming. Can anyone tell me what dynamic programming involves?
It’s about breaking down problems into smaller subproblems, right?
Exactly! In our case, we look at pairs of sequences and build solutions based on previously calculated values. We store the results in a table covering all letters of both sequences.
How do we fill that table?
We fill the table based on conditions—if the letters match, we add one to the diagonal value. If not, we take the maximum from the left or below. This results in an O(mn) time complexity, meaning it's efficient!
Practical Applications of LCS
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Let's talk about applications of LCS in real life. Why do you think it’s particularly useful in bioinformatics?
It helps find genetic similarities between species!
Correct! LCS can help identify which genes are similar, despite variations in actual sequences. Can anyone think of another use?
How about comparing text files?
Exactly! The UNIX 'diff' command uses LCS to find differences between files, helping developers see changes easily.
Inductive Structure of LCS
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Now, let’s understand the inductive structure of how we derive LCS. Can anyone tell me what happens if we find a matching letter?
We can combine them in our subsequence!
That’s right. If letters match, we include those in our solution and move to the next letters in both sequences. What if they don't match?
Then we need to find the maximum from the subproblems, right?
Exactly! We create two subproblems, dropping each letter one at a time, and compute the LCS for both scenarios to find the optimal length.
Tracing Back the Matches
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Finally, let’s learn how we can trace back to find the actual longest common subsequence. How do we do that?
Do we look at where the values in the table came from?
Yes! By tracing the path through our filled table, we can identify whether it was a match or a result from the maximum, thus reconstructing the sequence.
So, we use the diagonal steps to find matched letters?
Exactly! Remember, our goal is to pull together the steps that led us to our final answer. LCS and tracing—like mapping out our journey!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section discusses the method of identifying the Longest Common Subsequence (LCS) using dynamic programming or memoization techniques. It highlights the differences between subsequences and subwords, explains the inductive structure leading to LCS formulation, and mentions the practical applications of LCS, particularly in bioinformatics and text comparison.
Detailed
Detailed Summary of Tracing Back the LCS
In this section, we delve into the Longest Common Subsequence (LCS), a computational approach that generalizes the search for matches within sequences by allowing for the omission of characters. Unlike finding a longest common subword, which requires an exact correspondence of positions, LCS permits dropping letters to maximize the length of matching sequences. This flexibility enables the identification of broader correlations between sequences, exemplified through comparative applications such as genetic analysis in bioinformatics and text file comparison using the Unix DIFF command.
The recursive nature of LCS is examined, presenting an algorithm that builds upon an inductive structure, relying on two main cases: identifying matches or deciding to skip elements. The LCS algorithm is explained through a dynamic programming scheme that fills a table based on previous computations, leading to the conclusion with an overall complexity of O(mn). This section underscores LCS's pivotal role in many practical fields by demonstrating its implementation and tracing back how to retrieve the specific matches within the sequences.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Understanding Longest Common Subsequences
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, we can now look at a slightly more general problem than longest common subword in one which is more interesting computationally. 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
This chunk introduces the concept of the Longest Common Subsequence (LCS), which is a problem that extends the idea of finding the longest common subword. Unlike subwords, which require exact matches, subsequences allow for the dropping of letters. This means we can find matches even when some characters are not present. Essentially, the LCS problem asks: Given two strings, what is the longest sequence that appears in both, allowing for some characters to be omitted.
Examples & Analogies
Think of it like trying to find common themes in two different books. You don't need the exact sentences to match, but if certain phrases or ideas are similar, you can note them as part of a 'common storyline.' This flexibility reflects how subsequences work - you can 'drop' some words but still recognize a shared theme.
Applications of LCS
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, 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
This chunk highlights the importance of the LCS problem in various fields, particularly in bioinformatics. In biology, the strings being compared are typically sequences of DNA, which consist of a long string over an alphabet of size four (A, T, G, C). By finding the LCS between two DNA sequences, researchers can infer how closely related two species are based on their genetic similarities, even when certain genes are missing or are located in different positions.
Examples & Analogies
Imagine comparing two different species' DNA as reading two recipes for a dish. Both recipes may use similar ingredients (representing the common sequence), but some ingredients might be missing or reorganized. By understanding which ingredients are common, we can determine how closely related the dishes (or species) are, similar to finding the LCS in DNA.
Inductive Structure of LCS
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, the first thing to note is that if I am looking for this longest common subword between these two, supposing I find that a 0, in fact equal to b 0. Now, I claim that, I should combine them in the solution, and then look for a solution in the rest.
Detailed Explanation
In this portion, the discussion revolves around the inductive structure of finding the LCS. It posits that if the first characters of the two sequences match (denoted as a[0] and b[0]), then those characters should be included in the LCS and the search for other matches should continue from the next characters. This recursive approach reinforces the idea that building upon the known matches helps in effectively solving larger subproblems.
Examples & Analogies
Consider a jigsaw puzzle where you manage to fit the first piece perfectly (a[0] and b[0] matching). Once these pieces are combined, you can focus on fitting the next pieces (the remaining characters). This focused approach makes solving the rest of the puzzle simpler, just like checking the rest of the characters in LCS.
Handling Non-Matching Characters
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Now, if it is not equal, then one is not sure what you do, it is not s sound idea a to drop both. So, for instant supposing I have something like straw and astray, then just because the S does not match the a, does not mean that I should in both of them, I should keep that S alive to match with next S.
Detailed Explanation
The text explains the case when the characters being compared do not match. If a[0] does not equal b[0], you cannot simply ignore both characters. Instead, the algorithm needs to explore the possibility of matching one character while retaining the other for potential matches later in the sequences. This creates multiple subproblems to evaluate which character to keep, ensuring the longest valid match is found.
Examples & Analogies
Imagine you're assembling a team and have two candidates. Just like two letters in our sequences, not every feature of each candidate may match completely. You might want to keep one candidate for their unique skills while checking if the other can fill a role later on, emphasizing the importance of maintaining options while searching for the best combination.
Computing LCS Values
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, as usual let LCS i comma j, stand for the LCS of the problem starting at a i and b j.
Detailed Explanation
This chunk introduces the notation used for the LCS function. It defines LCS(i, j) as the longest common subsequence considering subsequences starting at indices i and j in sequences a and b respectively. The rules for computing the values are established, showing a recursive formula to derive lengths based on character comparisons and previous LCS results.
Examples & Analogies
Think of it as tracking delivery of packages. Each index represents a point in the delivery chain; LCS(i, j) helps determine how many packages can be successfully delivered together when starting from points i and j. If they match, you add to the tally; if not, you check both delivery routes to maximize efficiency.
Filling the LCS Table
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, this is similar to LCW, we basically fill out in m, n size table each entry in the table is easy to compute takes only constant among look at the three neighbors.
Detailed Explanation
This section describes how the Longest Common Subsequence (LCS) table is systematically filled out, similar to the process used for the Longest Common Word (LCW). Each entry in the m by n table can be computed by examining a constant number of neighboring entries, using the previously established rules for matching and dropping characters.
Examples & Analogies
Picture laying out a grid for a game like 'Connect Four.' Each spot on the grid can hold a piece based on your move rules. Similarly, each entry in the LCS table derives its value based on previous choices, allowing you to build steadily towards a winning combination (the longest common subsequence).
Traceback for LCS
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
And now as before you can trace back the path, why was each value filled, was it filled, because it to 1 plus i plus 1, j plus 1 or it goes to fill, because max with other two networks.
Detailed Explanation
Finally, this block discusses the traceback process used after computing the LCS values to determine the actual subsequence itself. By following the filled values in the table, one can retrace the steps taken to form the longest subsequence, identifying which characters were included during the process.
Examples & Analogies
Imagine you've just completed a riddle or puzzle. Retracing your thought process allows you to see how you arrived at the final answer. Likewise, when we trace back from the completed LCS table, we unveil the steps taken to derive the longest common subsequence from our original sequences.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Dynamic Programming: A technique that utilizes previously computed results to enhance efficiency in solving problems.
-
Subsequences vs. Subwords: LCS allows for the dropping of characters, making it broader in application than subwords.
-
Inductive Structure: LCS is derived through a combination of matching and skipping letters, structured recursively.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
A practical example of LCS can be found in DNA sequencing, where researchers seek to find similarities in gene patterns.
-
Another application is in software development, where tools compare file versions to identify changes or similarities.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
To find LCS in a race, drop some letters to find your place!
📖 Fascinating Stories
-
Imagine two friends, Alex and Jamie, who have been writing letters but sometimes skip some words. To see how much they share, they create a game where they drop letters but still make sense. That's their LCS journey!
🧠 Other Memory Gems
-
Think of LCS as 'Longest Chosen Sequence' to remember its function of selecting letters while allowing omissions.
🎯 Super Acronyms
For LCS think
- Look Closely at Sequences.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Longest Common Subsequence (LCS)
Definition:
A subsequence that appears in the same relative order in both sequences but not necessarily consecutively.
-
Term: Dynamic Programming
Definition:
A method for solving complex problems by breaking them down into simpler subproblems and storing results to avoid redundant calculations.
-
Term: Memoization
Definition:
An optimization technique where previously calculated results are stored for reuse in future computations.
-
Term: Subsequence
Definition:
A sequence derived from another sequence by deleting some elements without changing the order of the remaining elements.
-
Term: Computational Complexity
Definition:
A field of study that focuses on classifying computational problems according to their resource usage, typically in terms of time and space.