**Understanding Lexicographic Sorting**

Lexicographic sorting is a method of arranging data based on the alphabetical order of its elements. In lexicographic sorting, each element is compared character by character from left to right. This means that if two elements have the same first character, they are then compared based on their second characters, and so on. For example, in the words "Apple" and "Sir", both start with "A". According to lexicographic sorting rules, since 'C' comes before 'P', Sir comes before Apple. This concept is fundamental to understanding how data is sorted in various applications, including databases.

**Stability of Sorting Algorithms**

In addition to lexicographic sorting, there's a crucial concept in sorting algorithms known as stability. A stable sort algorithm preserves the relative order of equal elements. In other words, if two elements have the same value, their original order is maintained in the sorted output. This property is essential when sorting data based on multiple criteria, such as name and price.

When we start by sorting a dataset by name, this initial sorting operation sets the stage for subsequent sorting operations. As shown in the example with the products listed, if two or more products have the same price, they are sorted alphabetically by name. This ensures that the original order is preserved, making it easier to identify specific products or groups of products.

**Importance of Stability**

The importance of stability in sorting algorithms cannot be overstated, particularly when dealing with data that needs to be sorted based on multiple criteria. Imagine having a dataset where two or more products have the same price. If we were using an unstable sorting algorithm, these products could get jumbled up in the sorted output, making it difficult to identify them correctly.

In databases and other applications, stability is crucial when searching for specific data based on multiple conditions. For instance, if we're looking for all products that cost $31, knowing that they are sorted by price and then by name makes it easier to find exactly what we need. Moreover, when sorting data by multiple columns, having a stable sort algorithm ensures that the output remains ordered consistently.

**Real-World Applications**

Stable sorting algorithms have real-world implications in various applications, including databases. When designing database systems, developers must consider how data will be sorted and searched based on multiple criteria. If a sorting algorithm is not stable, it can lead to inconsistencies and errors, making it less reliable for production use.

In scenarios where data needs to be sorted by price, name, or other attributes, having a stable sort algorithm ensures that the output remains ordered consistently. This property allows developers to perform complex searches and queries on the data with confidence, knowing that the results will be accurate and reliable. Furthermore, when working with large datasets, stable sorting algorithms can significantly improve performance by reducing the number of comparisons needed to achieve sorted order.

**Conclusion**

In conclusion, lexicographic sorting and stability are essential concepts in understanding how data is sorted in various applications. Lexicographic sorting arranges data based on alphabetical order, while stability ensures that equal elements maintain their original order in the sorted output. The importance of stability lies in its ability to preserve the relative order of equal elements, making it a crucial property for reliable and efficient sorting algorithms.

In databases and other applications, stable sorting algorithms ensure that data is sorted consistently across multiple criteria, making it easier to search, query, and analyze large datasets. By understanding the concepts of lexicographic sorting and stability, developers can design more efficient and effective database systems that provide accurate results and improve overall performance.

"WEBVTTKind: captionsLanguage: eninteresting concept or I should say the property of certain algorithms is called stability right so stable sort is not the name of a sort right stability stability is a property stability is a property of sorting algorithms that we sometimes want right before we go and understand why we need it let us first understand what is stability in a sorting algorithm right so if you go to Wikipedia article for sorting algorithms like for sorting algorithms Here I am this is the page for sorting algorithms right and this is a subsection called stability there's a very nice diagram here which explains the concept right imagine I have cards like this right this is number seven with this black symbol this is 5 this is 2 and this is 5 now when I sort it right when I sort it based on these numbers based on these numbers in increasing order I can get two options right I can either see this where you have two five five seven or again I can get the same two five five seven but here if you notice the red five is is before the black five right notice that right in this in this see both these are sorted this is also sorted and this is also a sorted outcome both of them are sorted if you are sorting based on the number here both of them are sorted but there is one key distinction between this output and this output let's call this output one and this is output two in output two if you notice the what is the order see there are two fives here right which have the same value these are basically two two cards with exactly the same value now if the order in which they are initially present in for example initially in the input data this is the input data that I have right in the input data if you notice I have the red five before the black five right this is important I have the red five ahead of the black 5 right my red 5 is there before the black file in the output here in output 1 this is output 1 this is output 2 what are them are sorted in output 1 this order that the red 5 is before the black 5 is retained even in this output the red 5 is before the black 5 in this order right so whatever is there in the input this is same element whenever you have the same element this is only for same elements right because 5 and 5 the same element is repeating if the output the order of the output elements is same as the order of the input elements in other words this red is before the black in the input because the 5 is the same element here only in the case of same elements right even in the output the red 5 is before the black 5 this is said to be a stable sorting a state this this is what stability means right and if your algorithm if your algorithm if your algorithm ensures if you if your sorting algorithm whatever your sorting algorithm is since we are looking at insertion sort right if your sorting algorithm generates an output such that for the cases where you have the same element same value same element if the input order is same as the output order that algorithm is said to be stable now let's take another example take it take the example of what not stable means if your algorithm generate an output like this where n where you have the same element 5 you have the same element 5 here in the input right now here you have the red one before the black one but in the output in the output the black one is before the red one right so the order is messed up here so for four elements which have the same key value if the order is not preserved here the order is not preserved right such an algorithm is said to be not stable right again very look at the arrows this is their before this and even in the output this is that before this because these two values are the same in a step in a non stable sorting algorithm right this five the because it's the same five here and here the red one in the input is before the black one but an output the black one is before the red one which means the input order of a repeated element see this five is a repeated element right this five is a repeated element right so when you have a repeated element if the input order in which these two elements are there is preserved in the output it is said to be stable otherwise unstable or non stable right your insertion sort your insertion start is a stable algorithm your insertions start is a stable algorithm right you can simply verify this by going through the code right the insertion sort that we discussed right now is a stable algorithm very interesting property right so this diagram is a very very nice way to understand the concept of stability right so to summarize it let me write it down for you right so when you have repeated elements when you have repeated elements so when you have repeated elements in your input if the repeated elements appear in the same order in the same order as in the input as in the input right in the output so in the output so the key is this in the output repeated elements should appear in the same order as they are in the input but here the repeated element is a five right if it is so then the algorithm then the sorting algorithm is said to be stable so stability is a property not a type of algorithm just to be clear now you might wonder so we learnt what is stability right so we learned what is stability here right very important question but the next immediate question that comes to my mind is why is stability important where is it used how is it used so let me give you a real world example imagine a habit data which is stored in a table form like this right imagine I have some data like this right which happens a lot in databases suppose I have some data where I have the price of the data right a price of each item and the name of each item let's assume this is the data that I have right so let's assume I have $23 as the price for an apple probably a connect errors of some sort $36 for a lenovo all right I'm just taking an example here 31 dollars for an Acer connector of some sort again 31 dollars for an Asus connector $40 for a HP connector and 60 dollars for a Lenovo connector right so let's assume I first sought this so I sorted by name let's say suppose I sought by the name field just the way we have sorting we have we have learned about sorting by numbers you can also sort by sorry you can also sort by Alf by by characters right you can also sort alphabetically right we have seen sorting by numerical values similar thing can be done by sorting by alphabetically or sorting it's called it's also called as lexicographically right so sorting by name alphabetically alphabetically or lexicographically that's the that's the technical term now when I saw this what is that what is the so first basically what am i doing I'm sorting by name I'm not sorting by price so first step here is sorted by name when I sort by name what is the output that I would get let's write it down here right when I sorted by name I would get my price and name would be here right the first one which I would get is a sir with $31 right ASA $31 then Isis which is also $31 so then Apple sorry sorry sorry it is Apple also I'm sorry there is Apple here the next one is Apple an Apple device which is let's say 23 dollars because I am sorting by name I am not sorting by price if you see since both a both of these are s we compare the second character here C comes before P that's why a sir comes before Apple that's what lexicographic sorting means if you have the same first character compared the second character if you have the same second character to compare the third character so on so forth right so the third one would be as Seuss the fourth one will be HP v 1 will be Lenovo the sixth one also will be later all right so this is I'm sorting by name let's assume first I start by name in the second step let's say I sought by in the second step let's say I sought by price so I have already sorted by name remember this is already sorted by name this is already sorted by name now I am doing now I am doing a sort by price right when I do this this happens a lot in the real world right this happens a lot when you have data that you want to manipulate in your databases right so but so once I sort by name how do i how will my output look like let's do it sorry once I sort by price how the output look like so this is my price and this is my name right here here if you look at it when I sort by price right the lowest price is 23 which is Apple all right now I have two products which I have the same price 31 right I have the same price 31 both for Acer and Asus right now if I have a stable shot remember if my shot is stable if my shot is stable then the input order will be preserved in the output which means Acer will come Acer will come before Asus right because in this in the input in the input my thought because both both of them have the same value 31 31 right now if the input order in the input order 31 for Acer comes before 31 for Asus even here if you look at the input out order and output order match here right if it's a stable shot if it is not a stable shot these two could get jumbled up now let me tell you why this is important right so you have 36 dollars Lenovo then $40 HP and again $60 product which is then EV all right now now look at look at what is happening here this is lots of fun because you have sorted first by name this is your first sorting that you have done your second sorting is by price your final sorting that you get is such that it's obviously sorted by price but if something is like this where two items have the same price right which happens a lot when you when you have items which have the same price you're guaranteed that they're sorted by name because you have performed the sorting by name initially and this will this this property will only be useful will only will only be valid if it's a stable sort imagine if this sort by price was not stable then these two could have gotten jumbled up so in lots of databases when you store data you might want to say that first I want to sort it by price and if two items or more items have the same price I want them to be sorted based on the second property which is the name if they have the same price and to achieve that right this is this seems logical right because if you're searching for this information right if you're searching for all the products which are 31 dollars you can easily find it because they're sorted right that's one number two because this is they're also sorted by name so you have to sort things here one sorting the primary sorting is on the price but if the price is the same for two or more products they are now sorted alphabetically by name right that's because we first started by price was first sorted by name if this was not a stable sort and these two got jumbled up this whole sorting operation would have been useless because whatever because here whatever we are trying to do here which is sorting by alphabetic by by name alphabetically is no more preserved here right just because this sorting by price algorithm is a stable sort algorithm like insertion sort because it is so right this output is guaranteed to have sorted order when the price is the same this is the real-world implication on why we need stable sort algorithms or sorting algorithms which are stable right because when you want to sort by multiple columns of data right we might want to have a situation many times where if the products have the same price we want them to be sorted by the next column which is named right and so on so forth and this will be done for multiple columns not necessarily two columns you could have a table with tens of columns we could sort one after the other as long as all of your sorting algorithms are stable you're guaranteed to have this property preserved okay I hope you understood what is stability in a sorting algorithm and why stability is important right in the realinteresting concept or I should say the property of certain algorithms is called stability right so stable sort is not the name of a sort right stability stability is a property stability is a property of sorting algorithms that we sometimes want right before we go and understand why we need it let us first understand what is stability in a sorting algorithm right so if you go to Wikipedia article for sorting algorithms like for sorting algorithms Here I am this is the page for sorting algorithms right and this is a subsection called stability there's a very nice diagram here which explains the concept right imagine I have cards like this right this is number seven with this black symbol this is 5 this is 2 and this is 5 now when I sort it right when I sort it based on these numbers based on these numbers in increasing order I can get two options right I can either see this where you have two five five seven or again I can get the same two five five seven but here if you notice the red five is is before the black five right notice that right in this in this see both these are sorted this is also sorted and this is also a sorted outcome both of them are sorted if you are sorting based on the number here both of them are sorted but there is one key distinction between this output and this output let's call this output one and this is output two in output two if you notice the what is the order see there are two fives here right which have the same value these are basically two two cards with exactly the same value now if the order in which they are initially present in for example initially in the input data this is the input data that I have right in the input data if you notice I have the red five before the black five right this is important I have the red five ahead of the black 5 right my red 5 is there before the black file in the output here in output 1 this is output 1 this is output 2 what are them are sorted in output 1 this order that the red 5 is before the black 5 is retained even in this output the red 5 is before the black 5 in this order right so whatever is there in the input this is same element whenever you have the same element this is only for same elements right because 5 and 5 the same element is repeating if the output the order of the output elements is same as the order of the input elements in other words this red is before the black in the input because the 5 is the same element here only in the case of same elements right even in the output the red 5 is before the black 5 this is said to be a stable sorting a state this this is what stability means right and if your algorithm if your algorithm if your algorithm ensures if you if your sorting algorithm whatever your sorting algorithm is since we are looking at insertion sort right if your sorting algorithm generates an output such that for the cases where you have the same element same value same element if the input order is same as the output order that algorithm is said to be stable now let's take another example take it take the example of what not stable means if your algorithm generate an output like this where n where you have the same element 5 you have the same element 5 here in the input right now here you have the red one before the black one but in the output in the output the black one is before the red one right so the order is messed up here so for four elements which have the same key value if the order is not preserved here the order is not preserved right such an algorithm is said to be not stable right again very look at the arrows this is their before this and even in the output this is that before this because these two values are the same in a step in a non stable sorting algorithm right this five the because it's the same five here and here the red one in the input is before the black one but an output the black one is before the red one which means the input order of a repeated element see this five is a repeated element right this five is a repeated element right so when you have a repeated element if the input order in which these two elements are there is preserved in the output it is said to be stable otherwise unstable or non stable right your insertion sort your insertion start is a stable algorithm your insertions start is a stable algorithm right you can simply verify this by going through the code right the insertion sort that we discussed right now is a stable algorithm very interesting property right so this diagram is a very very nice way to understand the concept of stability right so to summarize it let me write it down for you right so when you have repeated elements when you have repeated elements so when you have repeated elements in your input if the repeated elements appear in the same order in the same order as in the input as in the input right in the output so in the output so the key is this in the output repeated elements should appear in the same order as they are in the input but here the repeated element is a five right if it is so then the algorithm then the sorting algorithm is said to be stable so stability is a property not a type of algorithm just to be clear now you might wonder so we learnt what is stability right so we learned what is stability here right very important question but the next immediate question that comes to my mind is why is stability important where is it used how is it used so let me give you a real world example imagine a habit data which is stored in a table form like this right imagine I have some data like this right which happens a lot in databases suppose I have some data where I have the price of the data right a price of each item and the name of each item let's assume this is the data that I have right so let's assume I have $23 as the price for an apple probably a connect errors of some sort $36 for a lenovo all right I'm just taking an example here 31 dollars for an Acer connector of some sort again 31 dollars for an Asus connector $40 for a HP connector and 60 dollars for a Lenovo connector right so let's assume I first sought this so I sorted by name let's say suppose I sought by the name field just the way we have sorting we have we have learned about sorting by numbers you can also sort by sorry you can also sort by Alf by by characters right you can also sort alphabetically right we have seen sorting by numerical values similar thing can be done by sorting by alphabetically or sorting it's called it's also called as lexicographically right so sorting by name alphabetically alphabetically or lexicographically that's the that's the technical term now when I saw this what is that what is the so first basically what am i doing I'm sorting by name I'm not sorting by price so first step here is sorted by name when I sort by name what is the output that I would get let's write it down here right when I sorted by name I would get my price and name would be here right the first one which I would get is a sir with $31 right ASA $31 then Isis which is also $31 so then Apple sorry sorry sorry it is Apple also I'm sorry there is Apple here the next one is Apple an Apple device which is let's say 23 dollars because I am sorting by name I am not sorting by price if you see since both a both of these are s we compare the second character here C comes before P that's why a sir comes before Apple that's what lexicographic sorting means if you have the same first character compared the second character if you have the same second character to compare the third character so on so forth right so the third one would be as Seuss the fourth one will be HP v 1 will be Lenovo the sixth one also will be later all right so this is I'm sorting by name let's assume first I start by name in the second step let's say I sought by in the second step let's say I sought by price so I have already sorted by name remember this is already sorted by name this is already sorted by name now I am doing now I am doing a sort by price right when I do this this happens a lot in the real world right this happens a lot when you have data that you want to manipulate in your databases right so but so once I sort by name how do i how will my output look like let's do it sorry once I sort by price how the output look like so this is my price and this is my name right here here if you look at it when I sort by price right the lowest price is 23 which is Apple all right now I have two products which I have the same price 31 right I have the same price 31 both for Acer and Asus right now if I have a stable shot remember if my shot is stable if my shot is stable then the input order will be preserved in the output which means Acer will come Acer will come before Asus right because in this in the input in the input my thought because both both of them have the same value 31 31 right now if the input order in the input order 31 for Acer comes before 31 for Asus even here if you look at the input out order and output order match here right if it's a stable shot if it is not a stable shot these two could get jumbled up now let me tell you why this is important right so you have 36 dollars Lenovo then $40 HP and again $60 product which is then EV all right now now look at look at what is happening here this is lots of fun because you have sorted first by name this is your first sorting that you have done your second sorting is by price your final sorting that you get is such that it's obviously sorted by price but if something is like this where two items have the same price right which happens a lot when you when you have items which have the same price you're guaranteed that they're sorted by name because you have performed the sorting by name initially and this will this this property will only be useful will only will only be valid if it's a stable sort imagine if this sort by price was not stable then these two could have gotten jumbled up so in lots of databases when you store data you might want to say that first I want to sort it by price and if two items or more items have the same price I want them to be sorted based on the second property which is the name if they have the same price and to achieve that right this is this seems logical right because if you're searching for this information right if you're searching for all the products which are 31 dollars you can easily find it because they're sorted right that's one number two because this is they're also sorted by name so you have to sort things here one sorting the primary sorting is on the price but if the price is the same for two or more products they are now sorted alphabetically by name right that's because we first started by price was first sorted by name if this was not a stable sort and these two got jumbled up this whole sorting operation would have been useless because whatever because here whatever we are trying to do here which is sorting by alphabetic by by name alphabetically is no more preserved here right just because this sorting by price algorithm is a stable sort algorithm like insertion sort because it is so right this output is guaranteed to have sorted order when the price is the same this is the real-world implication on why we need stable sort algorithms or sorting algorithms which are stable right because when you want to sort by multiple columns of data right we might want to have a situation many times where if the products have the same price we want them to be sorted by the next column which is named right and so on so forth and this will be done for multiple columns not necessarily two columns you could have a table with tens of columns we could sort one after the other as long as all of your sorting algorithms are stable you're guaranteed to have this property preserved okay I hope you understood what is stability in a sorting algorithm and why stability is important right in the real\n"