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Introduction to LCS Table Filling
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Today, we will learn about the process of filling the LCS table, which is crucial for solving the longest common subsequence problem. Can anyone tell me what LCS stands for?
It stands for Longest Common Subsequence.
Correct! Now, can anyone think of why we need to fill a table for this problem?
To systematically compute the lengths of common subsequences?
Exactly! We will use previous solutions to build up the answer for the entire sequences.
Understanding Dependencies
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Letβs talk about dependencies. Why is knowing which cells depend on which so important?
Because it helps us fill the table correctly!
Right! For each cell, we have at most three dependencies, which are crucial for determining its value. Can anyone name one?
The cell to the left?
Good! We also look at the cell below and the diagonal left-below. Remember, we draw from these cells based on whether characters match or not.
Initialization and Value Propagation
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How do we initialize the LCS table?
The first row and first column should be set to zero.
Exactly! This represents the case where one of the sequences is empty. From there, as we fill the table, what happens to the maximum value?
It propagates to the cell at the top left!
Correct! That top left corner gives us the length of the longest common subsequence.
Tracing the Resulting LCS
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Once we've filled the table, how can we find the actual longest common subsequence?
We can trace back through the table!
Exactly! Following the diagonal steps indicates where matches occurred. What happens if there's a tie in values?
We might have more than one longest common subsequence?
That's right! There can be multiple LCSs with the same length.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, we explore how to implement a dynamic programming technique to fill the LCS table, reviewing crucial dependency conditions and how to trace back the results for reconstructing the longest common subsequence from the filled table.
Detailed
Filling the LCS Table
The process of filling the LCS (Longest Common Subsequence) table involves systematically populating a matrix based on certain dependencies derived from two sequences, typically represented as arrays. This section starts with a basic overview of the problem, highlighting the fundamental recursive relation: if two characters match, the length of the longest common subsequence increases by one, plus the result of the subproblem. If they do not match, the larger value between two possible subproblems is chosen (dropping either character from one of the sequences).
Key aspects include:
- Dependencies: Each cell of the matrix has dependencies on three others (its left, below, and diagonal left-below neighbors), which need to be evaluated for filling out the current cell.
- Zero Initialization: The first row and first column are initialized to zero, especially indicating base cases where at least one sequence is empty.
- Value Propagation: As values are filled, the maximum value gets propagated to the top-left cell (0, 0), which ultimately holds the length of the longest common subsequence.
- Tracing Back: After filling the table, we can trace back the decisions made to find the actual subsequence by following paths through the table, indicating matches.
- Python Implementation: A brief discussion on the Python code illustrates the implementation of this logic through functions to populate the LCS table while maintaining efficiency.
Understanding these principles is essential for solving the LCS problem effectively using dynamic programming.
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Audio Book
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LCS Basic Principles
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This in general will take us deeper in the words. So, we said a 0 b 0 will require solved it for a 1 and b 0 or a b a 0 and b 1. So, in general we have a i and b j right.Again since we have aiand b j then you willuse the samelogicif a iis equal to b j thenit is oneplus the rest. So, this is the good case, if a i is not equal to b j then what we do is we look at the same thing, we drop b j and solve it and symmetrically we drop a i and solve it and take the betterof the two.Wetakemax of thesolution fromi andthe solution from j plus 1.
Detailed Explanation
In this chunk, we discuss the foundational logic behind calculating the Longest Common Subsequence (LCS). The basic idea is that when comparing elements from two sequences (letβs call them 'a' and 'b'), if the elements at position 'i' in 'a' and 'j' in 'b' (denoted as a[i] and b[j]) are equal, then it indicates that we can extend our common subsequence by one from previous values, hence we add 1 to the result derived from the previous indices (i-1, j-1). If the elements are not equal, we need to consider the maximum result by dropping one of the elements (either from sequence 'a' or 'b') and solving the subproblem. Therefore, the result is the maximum of the two potential subsequences derived from a[i-1 with b[j] and a[i with b[j-1] plus 1). This means we have a method to systematically build our solution row by row or column by column until we arrive at the length of the longest common subsequence.
Examples & Analogies
Think of it like two friends comparing their photo collections. If they find that a photo (say a beach photo) is the same in both collections, they can celebrate this as part of their shared experiences and count it as a match. If the next photos donβt match (like a mountain vs. a city photo), they have to decide whether to ignore the mountain photo or the city photo, and see if there are more matches further along in the collections. Each decision influences the count of their shared experiences, just as dropping elements influences the LCS count.
Understanding Dependencies
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So, here the dependency is slightly more complicated because depending on the case, I either have to look at i plus 1 j plus 1 or I have to look at i plus 1 j or i j plus 1. So, I had for this square, I had looked at its right neighbor, right diagonal neighbor and the bottom neighbor, but once again the ones which have no dependency appear. So, earlier we had for longest common subword we had only this dependency this mean that even a square likethis hadno dependencies because there is nothing to its bottom right.
Detailed Explanation
This chunk elaborates on how dependencies are formed based on the current state of the LCS table. When computing the values of a cell in the table, multiple previous cells need to be looked at, specifically the one directly to the left, directly above, and the one diagonally above and left (representing the matching case). This interplay can create a situation where certain cells can't be filled until their dependencies have been addressed. This shift adds complexity to when and how cells are populated, especially because it requires a proper observation of the context surrounding each cell.
Examples & Analogies
Imagine building a house. You canβt put a roof on a house before the walls are up. Similarly, in our LCS table, a cell canβt be filled until its 'walls' (dependencies from other cells) are correctly set. Just like youβd need to finish constructing the base of one wall before you can start working on the next wall, in LCS, you must resolve dependencies in the correct order to ensure accurate results.
Filling the LCS Table
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Now, how do we trace out the actual solution when the solution grows, whenever we increment the number? So, we can ask why is this 4? So, we say that this is 4 not because we did plus 3 because s is not equal to b, we did 4 because we got the max value from here, why is this 4 again i is not equal to s. So, we got the max value from here why is this 4, Oh s is equal to x. We must have got it by 3 plus 1.
Detailed Explanation
In this chunk, we focus on the mechanics of tracing back through the LCS table to reconstruct the actual longest common subsequence. As we compute values in the table, we also keep track of how those values were derivedβwhether through a match (adding 1) or by taking the maximum of surrounding values. This tracing enables us to understand why a particular value appears in the table and helps in reconstructing the sequence that leads to this value. Itβs all about keeping a path that shows the decisions made during the table filling process.
Examples & Analogies
Imagine you are solving a mystery. As you uncover clues (similar to filling in the LCS table), you might take notes (tracer values) about how you arrived at each conclusion. When you reach the final answer, you can look back at your notes to uncover the chain of logic that led you there. This helps not only in confirming your answer but also in understanding how all pieces fit together to tell the full story.
Using Python for LCS
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So here is the python code is not very different from the earlier one. We can just see we have just initialize the last row and the bottom row on the last column and then as before you walk up row by row, column by column and filling using the equation and in this case, we do not have to keep track of the maximum value and keep updating because the maximum value automatically propagates to the0 0 value.
Detailed Explanation
This chunk discusses the implementation aspect of the LCS algorithm using Python code. The structured approach allows programmers to follow a systematic fill of the table, taking care of initialization and filling it out based on previous calculations. Since the last column and bottom row are set to zero to signify base cases, it allows for clear propagation of the maximum values towards the cell at (0,0) which ends up giving the overall result for the LCS length.
Examples & Analogies
Consider a chef preparing a dish. They first set out all their ingredients (in this case, initializing the rows/columns) and then follow a recipe, adding ingredients in a specific order (like iterating through rows and columns). By the time theyβre ready to taste the dish, the accumulated flavors (the maximum values from the LCS table) come together to provide the final taste of the mealβthe LCS length.
Performance Considerations
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Just like the longest common subword, here once again we are filling in a table of size m times n. Each entry only requires you to look at most do three other entries. So, one to the right one to the bottom right in that the one below. So, it is a constant amount of work. So, mn entries constant amount of work per entry, this takes timem times n.
Detailed Explanation
In the final chunk, we examine the efficiency of the LCS algorithm. The time complexity is determined by the size of the table which is mn, where m and n are lengths of the two sequences compared. Each entry in the table requires a fixed amount of work, specifically looking at three other entries, leading to a predictable and manageable computation time of O(mn). Understanding this helps in assessing the algorithm's performance, particularly for larger inputs.
Examples & Analogies
Think of a team of workers, each responsible for assembling a part of a product at various stations. Each worker needs to check the status of a few neighboring stations to complete their task. Therefore, the workload is well-defined and can be planned efficiently; similarly, in the LCS calculation, the predictable dependency leads to efficient processing times.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Dynamic Programming: A method for solving problems by dividing them into smaller subproblems.
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LCS Table: A matrix filled with lengths of longest common subsequences derived from two sequences.
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Dependencies: Each cell in the LCS table requires values from up to three other cells to compute its value.
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Zero Initialization: The process of setting the first row and column of the table to zero, representing base cases.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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If we compare sequences 'ABC' and 'AC', the LCS is 'AC'. In the LCS table, the values would help us find that 'A' and 'C' match.
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For sequences 'AGGTAB' and 'GXTXAYB', the LCS is 'GTAB'. The LCS table shows how we derived this by tracing the maximum values.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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In sequences long, the common we seek, fill the table so the answer won't be weak.
π Fascinating Stories
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Imagine two friends tracking their favorite books. They list them, and through a magical table, they unearth the titles they both cherish, each step revealing another shared loveβa journey through their memories!
π§ Other Memory Gems
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Remember LCS as 'Longest, Common, Sequence' to quickly recall the steps for the longest common subsequence.
π― Super Acronyms
For dependencies, think 'ABDβAbove, Below, Diagonal'βthe key directions for all cells.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: LCS
Definition:
Longest Common Subsequence; a sequence that appears in the same relative order but not necessarily consecutively in both sequences.
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Term: Dynamic Programming
Definition:
An optimization technique used to solve complex problems by breaking them down into simpler subproblems.
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Term: Dependencies
Definition:
Relationships between different cells in the LCS table that determine how they are filled based on other cells' values.
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Term: Initialization
Definition:
Setting initial values in the LCS table, often to zero, to represent base cases.
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Term: Value Propagation
Definition:
The process of passing calculated values to other cells in the LCS table as it's being filled.