Finding the Shortest Suffix
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Understanding Permutations
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Today, we'll dive into permutations. Can anyone tell me what a permutation is?
Isn’t it an arrangement of elements in a specific order?
Exactly! And we often deal with permutations when we talk about arranging numbers or letters. For instance, the arrangement of queens in the 8-Queens problem forms a permutation of column positions.
What about finding the next permutation? How is that done?
That's a great question! It involves identifying how to increment the current arrangement to get the next one in sequence.
Identifying the Shortest Suffix
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
To find the next permutation, we first need the shortest suffix that can be incremented. Can anyone guess why identifying the suffix is crucial?
Because if we find it correctly, we can rearrange it to form the next permutation?
Exactly! Specifically, we look for the longest part of the sequence that is already in descending order, as that can't be incremented.
So we need to find the last part where it stops increasing?
Precisely! Great observation!
Swapping Elements
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Once we have found this point, we then identify which element to swap. Can anyone share how we determine the swapping element?
Do we need to find the smallest larger element to the right of the point?
Exactly! We swap it with the element at our identified position, where we can increment.
And then we arrange the suffix afterwards in ascending order, right?
Correct! This is how we ensure we get the next permutation effectively.
Practical Application of Concepts
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Now, let’s apply what we've learned. Imagine our permutation is 'k o n j i m'. What do we do first?
First, we identify the longest suffix that is descending.
Yes, which is 'j i m'. Then we need to increment 'k'!
Good! And after swapping 'k' with the next larger element which is 'm', what’s next?
We rearrange the suffix in ascending order by reversing it.
Excellent! And what do we get as a result?
We get 'm i j k n o', which is the next permutation!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, we examine the process of generating the next permutation from a sequence of ordered elements, focusing on finding the shortest suffix that can be incremented. The method involves identifying a descending order suffix, swapping elements, and rearranging them to achieve the next lexicographical arrangement.
Detailed
Finding the Shortest Suffix
In this section, we discuss how to determine the next permutation of a sequence of elements by identifying the shortest suffix that can be incremented. The concept builds on the idea that a sequence's smallest permutation is its elements arranged in ascending order, while the largest permutation has them in descending order.
To find the next permutation, we must first identify the longest suffix that cannot be incremented, which corresponds to descending order. For example, if we have a sequence where part of it is already in descending order, this segment cannot be changed to a larger permutation. Therefore, we search for the first position (from the end) where an increment can be made, moving backwards until we find a point where the order stops increasing.
Once we have located this point, we swap it with the smallest larger element to its right. After making this swap, the subsequent suffix needs to be arranged in ascending order to ensure the next permutation is correctly defined. This systematic approach to generating permutations is key in solving problems such as the 8-Queens puzzle and other backtracking algorithms.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Understanding Suffixes
Chapter 1 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
If we want to find the next permutation we need to find as short suffix as possible that can be incremented. It is probably easiest to do it in terms of numbers but let us do it with letters.
Detailed Explanation
To find the next permutation, we initially look for the shortest suffix in a sequence that can be incremented. A suffix can be incremented if it is not in descending order. The goal is to identify this suffix to enable the next permutation generation. We can use either numbers or letters as our sequence for this process, but it is easier to visualize with letters.
Examples & Analogies
Think of a combination lock where the numbers can be incremented. If the lock shows '987', you know you cannot increase this number without starting over, much like how a descending order in suffixes cannot be incremented. In contrast, if the numbers are '789', you can easily adjust the last digit to generate '790' or similarly increment the sequence.
Identifying the Longest Suffix
Chapter 2 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
We want to find the longest suffix that cannot be incremented. So, a suffix that cannot be incremented is one which is as large as it could possibly be which means that it is already in descending order.
Detailed Explanation
A suffix that cannot be incremented is longer and in descending order, like 'onmji'. To identify this, you examine the string from the end backwards and see how long you can go while the characters are in descending order. Once you hit a character that forms an ascending order, that point marks where your suffix can change.
Examples & Analogies
Imagine a stack of books organized in descending order from largest to smallest. As long as the largest book is on top, you cannot add a new 'largest' without removing that one first. The stack ceases to function as intended if it’s in descending order, just like the suffix until you find that point of 'increasability'.
Finding the Increment Position
Chapter 3 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
If I want to change it and need to increment something and I mean to increment it, I cannot increment it within this red box so I must extend this to find the shortest suffix namely; suffix started with k where something can be incremented.
Detailed Explanation
Once we determine the longest suffix that cannot be incremented, we can identify the position just before this suffix. We will then find a suitable character to increment, which should be larger than the character at this position, ensuring we obtain the next valid permutation.
Examples & Analogies
Consider an elevator with floors numbered in order. If you want to go higher but find that you're currently at the 10th floor and the floors above are filled, you have to step back to the 9th floor to find another valid path upward. Choosing your position carefully allows you to make the next move effectively – similar to identifying the character you will increment from.
Rearranging for Next Permutation
Chapter 4 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Now among this suffix that begins with letter m I need the smallest one, that mean I rearrange the remaining letters k o n j i in ascending order to give me the smallest permutation to begin with m.
Detailed Explanation
After selecting the character to increment and swapping it with a larger character, the next step is to rearrange the tail end, or the suffix. This portion needs to be in ascending order to ensure the next permutation is indeed the smallest possible arrangement following the increment step.
Examples & Analogies
Think of a jigsaw puzzle. Once you've placed a piece (the incremented character), you need to rearrange the remaining pieces (the suffix) in the correct order (ascending) to complete the picture in a way that makes sense and uses as few moves as possible. Like solving this puzzle, each piece needs to fit snugly to ensure proper progression.
Algorithmic Steps for Finding Next Permutation
Chapter 5 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Algorithmically we can do it as follows, what we need to do is first identify the suffix that can be incremented. We begin by looking for a suffix that cannot be incremented namely we go backwards so long as it is in descending order.
Detailed Explanation
The algorithm first identifies the longest descending suffix by moving backward through the string. It finds the point where the sequence stops being in descending order, indicating the point where incrementing can occur. Once found, the next steps involve swapping characters and rearranging the suffix to derive the next permutation.
Examples & Analogies
This is akin to navigating a maze. You trace your steps backwards until you hit a dead end (descending suffix), then find an alternative route (incrementing a character) while knowing that each turn you make can lead you to a new but valid path through the maze (the next permutation).
Key Concepts
-
Permutation: An arrangement of elements in a specified order.
-
Suffix: A part of a sequence at the end that can be analyzed for its ordering.
-
Next Permutation: The algorithmic method to find the next arrangement in sequence.
-
Descending Order: The arrangement of elements from highest to lowest.
Examples & Applications
Given the permutation 'a b c', the next permutation is 'a c b'.
For '3 2 1', the next permutation is '1 2 3'.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
In a row and in a queue, permutations shine bright and new.
Stories
Imagine a librarian shuffling books on a shelf. She always wants the next title that follows alphabetically, ensuring no titles are overlooked.
Memory Tools
P.S. Next - Remember the 'Position Swap, Next Suffix' approach for permutations.
Acronyms
S.O.S. - Suffix Order Suffix for finding which part of the arrangement to change.
Flash Cards
Glossary
- Permutation
An arrangement of elements in a specific order.
- Suffix
A subsequence at the end of a sequence.
- Lexicographical Order
A generalization of the way lexicographers order words, often equated with dictionary order.
- Increment
To increase or change to a larger value or arrangement.
- Descending Order
An arrangement where the sequence decreases from left to right.
Reference links
Supplementary resources to enhance your learning experience.