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Introduction to Longest Common Subsequence (LCS)
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Today, we are exploring the longest common subsequence, or LCS. Does anyone know what that means?
Is it the longest sequence that appears in both sequences?
Exactly! It's the longest sequence that can be derived from both original sequences without rearranging any of the characters. Letβs break down the strategy for solving it.
How do we actually find the LCS?
Matching Elements
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When we find a match between two elements, what should we do?
We should add one to the count and look at the remaining elements, right?
Correct! If `a_i` is equal to `b_j`, we take that match plus the LCS of the remaining elements, which can simplify our computation.
What if they do not match?
Handling Non-matching Elements
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Great question! If the elements do not match, we need to check two scenarios: drop the current element of the first sequence or the second sequence. Can anyone tell me what we do next?
We find the maximum of the two results from those scenarios?
Yes, and we add one to the maximum we get in a certain way. Thatβs how we can build our solution incrementally.
That makes sense! So, it's like choosing the best outcome from our options.
Use of Dynamic Programming
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By using dynamic programming, we can fill a table of size `m x n`. How many entries will that give us?
That's `m*n`, which sounds like a lot of work!
Indeed! However, each entry relies only on the three neighboring entries: to the left, below, and diagonally. So it's efficient in how we do it.
Tracing the Result
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Finally, how do we trace back to find our actual LCS once our table is filled?
We follow the diagonals where matches were made, right?
Yes, as the diagonal steps represent matches. We can sometimes have multiple valid subsequences as well!
Introduction & Overview
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Quick Overview
Standard
The material explores the methodology behind solving for the longest common subsequence (LCS) by examining dependencies between elements in sequences. It emphasizes the use of recursive relations to find common subsequences while considering both matches and mismatches between elements.
Detailed
Good Case and Solution Strategy
This section delves into the intricate strategy for determining the longest common subsequence (LCS) between two sequences. The core approach is built on recursive relations whereby for any two elements from the sequences, either the sequences contain matching elements or they do not. When they match, the problem simplifies to finding the LCS of the remaining elements plus one for the match. In cases where the elements do not match, the strategy involves recursively dropping one of the elements and finding the maximum of the two potential solutions.
The section discusses the grid-based approach commonly used in dynamic programming, where each entry depends on a small number of neighboring entries, simplifying the process of filling the solution matrix. It explains how to utilize a table of size m x n, where m and n are the lengths of the two sequences. As the algorithm progresses, it shows how one can trace back through the filled table to recover the LCS. Ultimately, the approach ensures efficiency while also being adaptable to non-unique solutions.
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Audio Book
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Understanding the Problem
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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.
Detailed Explanation
In dynamic programming, we often break down a complex problem into simpler sub-problems. Here, we reference sequences a and b represented as indices. The notation a[0] and b[0] means we begin comparing the first elements of both sequences. The strategy is to explore various conditions based on these elements, such as comparing a[1] with b[0] or relating a[0] with b[1]. This allows us to dynamically solve the problem based on their state.
Examples & Analogies
Imagine two students, Alice and Bob, each practicing a different subject. Alice starts with her first topic (a[0]) and Bob with his first topic (b[0]). If they both match on a topic, they move forward together; however, if not, they will evaluate different paths based on their studies to find the best possible solution, similarly to how we iterate through sequences here.
Matching Elements
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Again since we have ai and b j then you will use the same logic if ai is equal to bj then it is one plus the rest.
Detailed Explanation
If the elements at positions i of sequence a and j of sequence b are equal (a[i] == b[j]), we count that match as 1 plus whatever we can derive from the remaining parts of the sequences (i+1, j+1). This represents adding to our solution for every match found.
Examples & Analogies
Think of two puzzle enthusiasts who are matching pieces from their respective puzzles. If a piece from Aliceβs puzzle fits perfectly with Bobβs, it counts as a completed section, and they continue to match the next pieces as they build their puzzles together.
Handling Non-Matching Elements
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If a i is not equal to b j then what we do is we look at the same thing, we drop bj and solve it and symmetrically we drop ai and solve it and take the better of the two.
Detailed Explanation
When the elements a[i] and b[j] do not match (a[i] β b[j]), we explore different paths: first by excluding b[j] and then by excluding a[i]. We find optimal solutions by comparing the results of both exclusions and choosing the maximum. This step ensures we still account for all potentially matching subsequences.
Examples & Analogies
Imagine two chefs trying to create a dish from different ingredient lists. If one ingredient doesn't match, each chef tries one of their other ingredients to see which combination leads to a tastier dish, ensuring they maximize the flavor potential of their individual recipes.
Dynamic Programming Table 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.
Detailed Explanation
In crafting our solution, we need to consider dependencies from multiple directions. If we decide to increment both indices (i, j) because of a match, we also need to evaluate paths where only one index moves forward. This complexity arises as our algorithm needs to account for various permutations of matches and non-matches.
Examples & Analogies
Consider a team of detectives investigating multiple leads. Sometimes, they might find links between two suspects (i+1, j+1), but other times they might need to follow up with other leads (i+1, j) or (i, j+1). The detectives must strategically choose which path to pursue to uncover the truth.
Filling the Table
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Now, once we have this, we can fill up this part right and then again we can have two and three c entries we can fill up this.
Detailed Explanation
After processing the dependencies, we begin to fill our dynamic programming table cell by cell. Each cell reflects the solution to a sub-problem. By systematically filling up one cell after another and utilizing previous solutions in the table, we ensure that every piece of data contributes to solving the main problem effectively.
Examples & Analogies
Building a bridge with multiple arches. Each arch relies on the previous one being stable. By ensuring each arch is properly constructed before the next begins, the entire bridge becomes robust and well-supported, akin to the filled table contributing to the overall solution.
Tracing Back the Solution
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Now, how do we trace out the actual solution when the solution grows, whenever we increment the number?
Detailed Explanation
Post-computation, we need to backtrack to find the exact matches contributing to the longest common subsequence. This involves tracing back through our table to find where we increased values due to matches, leading us to reconstruct the actual sequence that makes up the solution.
Examples & Analogies
Imagine a treasure hunt where each clue leads to the next. After finding all the clues and reaching the treasure, teams can retrace their steps back to see how each clue helped them find their way to the prize, much like how we backtrack to derive our final sequence.
Dynamic Programming Time Complexity
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Just like the longest common subword, here once again we are filling in a table of size m times n.
Detailed Explanation
The process of solving the longest common subsequence requires a table with m rows and n columns (where m and n are the lengths of sequences a and b). Each cell in this grid can be filled based on existing computed values, leading to an overall time complexity of O(m*n), which informs us about the algorithm's efficiency.
Examples & Analogies
Think of a factory assembly line where each worker contributes to assembling a product step by step. If there are 10 sections (m) on the product and 5 workers (n), the assembly of products happens in a systematic manner involving all sections and workers, leading to a comprehensive total time for assembly similar to filling up our computational table.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Matching Elements: When elements in two sequences match, count that match and look at subsequent elements.
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Dynamic Programming: A technique that ensures each element's computation is efficient by leveraging previous calculations.
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Non-matching Elements: Recursively evaluate two scenarios - dropping one element from either sequence and comparing outcomes.
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Table Filling Dependency: The complexities involved in determining the longest subsequence through a systematic filling of a 2D table.
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Tracing Back the LCS: The method of following paths in the filled matrix to extract the original LCS.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example 1: Given sequences 'ABCBDAB' and 'BDCAB', the LCS is 'BCAB'.
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Example 2: For 'AGGTAB' and 'GXTXAYB', the LCS is 'GTAB'.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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If sequences match, add a plus one, / Then find the rest, your workβs almost done!
π Fascinating Stories
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Imagine two friends looking through old photos, finding common ones; each picture they agree on is a match in their LCS!
π§ Other Memory Gems
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Remember: 'M-A-C' for Match, Avoid Drop Draft, where M = Match, A = Add 1, C = Check max.
π― Super Acronyms
LCS
- Look for Common Subsequence!
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Longest Common Subsequence (LCS)
Definition:
The longest sequence that can appear in the same relative order but not necessarily consecutively in two sequences.
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Term: Dynamic Programming
Definition:
An optimization technique used in algorithms to solve problems by breaking them down into simpler subproblems.
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Term: Recursive Relation
Definition:
A way to define a sequence of numbers where each term is based on previous terms.
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Term: Table Filling
Definition:
The process of populating a matrix with values based on established relationships or dependencies.