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Introduction to Longest Common Subword Problem
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Today, we're going to delve into the longest common subword problem. Can anyone tell me what a 'subword' is?
Is it just any part of a word that can match another word?
Exactly! A subword is a segment of a word. For example, in 'secretary' and 'secret', 'secret' is a subword. Now, what do you think the challenge is in finding these subwords?
We need to compare the words to see which parts match?
Correct! And what's tricky about it?
There could be many positions to check, so it might take a long time!
Exactly! That's why we often need efficient algorithms like dynamic programming.
To remember the concept of subwords, think of the mnemonic 'SeC' for 'Segment comparison'.
SeC, I like that! It’s easy to remember.
Great! Now, let’s explore the brute-force approach to this problem.
Brute-force Algorithm
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To find the longest common subword using a brute-force method, we look at every position in both words. What do you think this would require?
Checking each character and comparing them?
Yes! If we denote the lengths of two words as m and n, the time complexity becomes O(m*n). What does that mean, in practice?
It means it could take a long time, especially with long words!
Exactly! Let's not forget that if we keep matching characters and encounter a mismatch, that's where our search terminates. Can someone summarize our discussion on brute-force?
We check all pairs of characters and keep track of matches until we find the longest one.
Well done! We can use the acronym MISMATCH to remember what happens when characters don’t match.
Dynamic Programming Approach
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Let's shift gears to a more efficient way using dynamic programming. Does anyone know what that means?
It involves breaking the problem down into smaller subproblems?
Exactly! The longest common subword can be determined through smaller segments. If we define LCW(i, j) as our subproblem, what happens when characters match?
We add 1 to the solution of the subproblem LCW(i+1, j+1).
Perfect! And what if they don’t match?
We just go to the next position, right?
Yes, we can't extend the match any further there. You can remember this logic using the mnemonic 'MATCH' for when characters match and 'NO MATCH' to signify the absence of commonality.
That’s a good way to remember it!
Constructing the Dynamic Programming Table
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Now, let's discuss how we can create a dynamic programming table. Starting points for our table would be?
We initialize the first row and column with zeros because there’s no common subword if either word is empty!
Correct! As we fill the table based on our LCW calculations, we’ll track the maximum value found. How does this help us?
The maximum value indicates the length of the longest common subword!
Great! You can remember this using the acronym MAX – it helps you recall to find the maximum length in the table.
I like that! It’s concise and easy to remember.
Excellent! Remember, filling in this table is our key step to efficiently finding the longest common subwords!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section elaborates on finding the longest common subword between two sequences, detailing the problem definition, brute-force approach, and how dynamic programming can optimize the solution. It emphasizes the recursive structure of subproblems in finding common subwords.
Detailed
Code Implementation of Dynamic Programming Algorithm
This section addresses the longest common subword problem, where we seek to find the length of the longest segment common to two sequences (or words). The initial focus is understanding what constitutes a 'subword', as well as the brute-force method that involves checking every possible position in both sequences to determine matches.
Key Concepts:
- Subwords are segments of a given word, exemplified by words such as "secret" and "secretary" or "bisect" and "trisect".
- The brute-force algorithm has a worst-case time complexity of O(m * n^2), where m and n are the lengths of two strings, making it inefficient for longer inputs.
- The section progresses toward a dynamic programming solution, highlighting an inductive structure that allows breaking down larger problems into manageable subproblems. The recursive nature is encapsulated in defining a subproblem as LCW(i, j), where this depends on whether characters in corresponding positions are equal or not. If equal, it reduces to the problem of finding the common length at positions (i+1, j+1).
- A dynamic programming table captures lengths, and the completion of this table allows for easy retrieval of the longest common subword when needed. An implementation is briefly outlined using an iterative approach rather than recursion, ensuring efficiency and clarity in execution.
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Audio Book
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Introduction to the Longest Common Subword Problem
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Let us suppose we are given two words for the moment, let us just assume there going in English or a language like that, we given two strings and we want to find the length of the longest common subword between them. In a subword, it is just a segment.
Detailed Explanation
This chunk introduces the concept of the longest common subword problem, which involves finding the length of the longest segment (subword) that appears in both of two given strings. A subword is simply defined as a continuous part of the string.
Examples & Analogies
Imagine you are trying to find common phrases in two different articles. If both articles contain the phrase 'deep learning', then that phrase is like the common subword that connects them.
Formulating the Problem
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So, the focus at the moment is to get the length of the subword... my main goal at the moment is to compute not the subword itself, but the length.
Detailed Explanation
This chunk highlights that the primary objective is not to actually identify the common subword, but rather to determine the length of that subword. Length gives us a quantifiable measure of similarity between two strings.
Examples & Analogies
Think of it as measuring the height of trees in two forests. You may not care about which trees are alike, but you want to know how similar the forests are based on tree height.
Brute Force Algorithm Explanation
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Here is a Brute force algorithm... So, therefore, this will be O of m times n into n or O m n square.
Detailed Explanation
This chunk discusses the brute force approach to find the longest common subword. It encompasses comparing all possible positions in both strings. The computational complexity of this method is analyzed, indicating it can become slow for longer strings since it operates in O(m * n²) time.
Examples & Analogies
Imagine trying to find matching items in two stores by checking each item one by one. If there are many items, this method can take an incredibly long time.
Inductive Approach Observation
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If I have a i equal to b j, I need to look at a i plus 1, b j plus 1... which gives us a handle on the inductive structure.
Detailed Explanation
The focus here is on leveraging an inductive approach. If characters from both strings match, you can build on that by checking the next characters. If they don't match, the search for a longer common subword must revert. This creates a recursive dependency in the calculation.
Examples & Analogies
Consider matching pairs of dominoes: when two dominos connect (match), you can keep extending the chain. If they don’t connect, you must start a new sequence.
Boundary Condition Handling
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So, we will allow a m plus one position and an n plus one position... then we can say that there cannot be a common subword, because there are no letters to use.
Detailed Explanation
This chunk discusses boundary conditions in the algorithm. When one of the words is exhausted (reaches the end), it is impossible to have a common subword, and thus the length of common subword is 0 under these conditions.
Examples & Analogies
Think of a race where one runner cannot continue past the finish line. Once a runner stops, they cannot match pace with the other: that race ends for them.
Dynamic Programming Solution Introduction
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So, we could write a memorized recursive function for that, but we will directly try to compute a dynamic programming solution for this problem.
Detailed Explanation
This section introduces the dynamic programming approach for solving the longest common subword problem. It emphasizes building solutions from previously computed values (subproblems), rather than recalculating existing solutions, thus saving time.
Examples & Analogies
Imagine solving a jigsaw puzzle: once you find that two pieces fit together, you remember that connection rather than forgetting it and searching for it again each time.
Implementation Details
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Here is some code for this particular algorithm... the maximum common subword of length 0.
Detailed Explanation
This chunk provides practical code examples of the dynamic programming algorithm discussed, explaining how the array is filled out and the logic behind it. It emphasizes column-wise processing of the problem.
Examples & Analogies
If you think of a painting by numbers, every area corresponds to a number that will dictate what color goes there. By filling one section at a time, you gradually see the complete image, just like filling out the matrix yields the answer.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Subwords are segments of a given word, exemplified by words such as "secret" and "secretary" or "bisect" and "trisect".
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The brute-force algorithm has a worst-case time complexity of O(m * n^2), where m and n are the lengths of two strings, making it inefficient for longer inputs.
-
The section progresses toward a dynamic programming solution, highlighting an inductive structure that allows breaking down larger problems into manageable subproblems. The recursive nature is encapsulated in defining a subproblem as LCW(i, j), where this depends on whether characters in corresponding positions are equal or not. If equal, it reduces to the problem of finding the common length at positions (i+1, j+1).
-
A dynamic programming table captures lengths, and the completion of this table allows for easy retrieval of the longest common subword when needed. An implementation is briefly outlined using an iterative approach rather than recursion, ensuring efficiency and clarity in execution.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example of longest common subword: Comparing 'secretary' and 'secret', we identify 'secret' as a subword of length 6.
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In comparing 'bisect' and 'trisect', the subword 'isect' is common, demonstrating how segments may overlap.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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When subwords match, don't be late, find the length and celebrate!
📖 Fascinating Stories
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Picture two friends, A and B, each carrying strings. They try to find shared parts, measuring lengths until victory springs!
🧠 Other Memory Gems
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MISMATCH – for when characters fail to line up and we move on!
🎯 Super Acronyms
MAX – find the maximum subword length in our dynamic programming table.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Dynamic Programming
Definition:
An optimization method used to solve problems by breaking them down into simpler subproblems, storing results of subproblems to avoid redundant calculations.
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Term: Subword
Definition:
A contiguous segment of a word that can match a segment in another word.
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Term: Longest Common Subword Problem
Definition:
The problem of finding the length of the longest contiguous segment that appears in both of two strings.
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Term: Brute Force Algorithm
Definition:
A straightforward approach to solving a problem by attempting every possible option.