[MUSIC] >> So in the previous video we talked about selection sort which is one algorithm for sorting data. By the end of this video, you'll have seen another algorithm for sorting data. And we're going to play a little game. Instead of talking with the algorithm at a high level, what we're going to do is present some code. That I promise you will sort the data, but what I'd like to do is work with you to figure out how to describe that code in high-level terms. And this are really useful skill that helps with reverse engineering code and also working in big teams, when other people are implementing different pieces of the code. And so then, once we'll have traced the code and figured out what it does. We'll be able to describe alternate algorithms for sorting and compare their relative merits. So let's start with a code, and let's see what we have here. So we're gonna call this method mysterySort until we solve the mystery together. And at this point there aren't a lot of comments to help guide us, we're just seeing some Java statements and so let's see what we've got here and we'll step through with a particular example. So we see that we've got a variable being declared and we've got a for loop that does something and has an inner while loop, okay. Doesn't help us so much so far, but let's work through an example. And again, it's really useful to work through some concrete examples when we're working through these questions to help guide our understanding. And then later, we wana generalize. We wanna think about the full generality. But first, let's make sure that we understand what's happening at a small scale. Okay, so we're working with the array of integers that we talked about last video as well. So we've got six elements and we're going to trace through and step through the execution of our mystery method to see what happens. Okay, so at the beginning in our for loop, we are initializing the value pos to one. And what's going to happen the first time we run through this for loop is that currInd. Which you might guess means current index, is going to be initialized to the same values position. And so what we see on the left of the code here, is that we've got a memory model. So think way back to the earlier module, one of the first modules that we did in this course where we talked about using memory models to help us reason through the change in values of variables as we run through code. And so we've got boxes for each variables and at the beginning each of those contains the values of the initial values of the variables namely, one. Okay, now what happens is, we look into the condition for the while loop and we see, do we have to execute the body of the while loop? While we're checking whether the current index is greater than zero, so yes, it's 1. And we wanna check whether the array element at position currIndex is less than the array element at position currIndex-1. So, currIndex is 1, currIndex minus 1 is 0. So we're comparing the numbers 7 and 16, and we're checking whether 16 is less than 7? It's not, and so we don't need to execute the body of the while loop. We go right back up to the top of the for loop, after incrementing the pos variable or position variable. Okay, so now position is two and the currIndex is initialized at the top of the body of the for loop to be two. And again, we have to check the condition for the while loop and we look, is the current index positive? Yes, okay and now we wanna compare the value of the array element at position 2 and compare that to the value of the array element at position 1. And basically look at what's happening here, we're asking whether they're in the correct relative position to one another. We're asking whether 66 is less than 16? Or 16 is less than 66? And in the array as it stands right now, 16 and 66 are the correct relative positions, so we don't need to do anything. All right, so far we haven't done much except for updating our helper variables, pos and currInd and we do that once more. Okay, we step up to the next execution of the body of the for loop and so now the pos variable has value of 3 and the current index has the same value of that at that the beginning. And now we check whether the array element at 43, i'm sorry at position 3 which is 43 is less than the array element at position 2, which is 66 and it is. So 43 is less than 66, which means that these two array elements are out of order. They need to be swapped in order to be in the correct location relative to one another. And so that's what's happening in the body of the while loop, we're swapping these two positions. And we also need to update the current index variable. Okay, so we've swapped the two positions, now we have 43 in position 2. We have 66 in position 3, and the current index is two. And we go back to the top of the while loop and we check whether 16 and 43 are in the correct relative positions? They are, so we don't need to do a thing in this execution of the body of the for loop. We increment, pos. We update currInd, and we go again and now we're checking whether 66 and 97 are in the correct relative position? They are, awesome, keep going, update pos, update currIndex. And we check whether 97 and 51, the elements at position 4 and 5 are in the correct relative position and they're not. 97 is greater than 51, and so what we need to do is we need to swap them. We update the current index, and now we check again. And what we're checking is whether 66 and 51 are in the correct relative positions? They're not, so we need to swap them and update current index. Okay current index is now three, 51 and 66 are in the correct relative positions. And now we check whether 43 and 51 are in the correct relative positions? And they are, so sweet, we're done. Okay, so what that means is, at each point of the execution of the for loop. At each time we've gone through the for loop, what we know is that we've guaranteed that an increasing part of the array is going to be in sorted order. So, if you think about what happens and what the array looks like at the end of the body of the for loop when pos equals one. What we have here is that 7 and 16 are in their correct final locations. Okay, at the end of position 2, we have 7, 16, and 66 at the head of our array. Now these are not going to be the correct final locations of each of these elements. At the end when we finish sorting the array, 66 is not gonna be in that third position. But, what we do know, is that the first three elements of the array at this point. Are in the correct relative order, with respect to one another. And so what we're doing is we're growing an array that is sorted, even though there might need to be some elements that get put in, into that array later on as we go through the rest of the algorithm. Okay, so at the end of position, at the end of the for loop when we've got position equals 3. What we have is, that we have a list of four elements of incorrect sorted order, then we have two elements that we haven't worried about yet. And they may need to go somewhere and nudge some of these sorted elements around, but these four elements will never switch position relative to one another. Okay. So, what we have here is. A beginning of a description in higher level terms of what this code is doing. And this transition between looking at code and being able to trace it's execution one step at a time, to being able to describe it in higher level terms is a really really hard cognitive leap but that is really important. And so it takes practice, keep at it. Work on it and you can see you'll get much better at it. And the one of the key strategies for doing it is tot take it little pieces at a time. Okay, and so what we're thinking about is we're thinking about this for loop. Where we're growing a piece of the array that is in relative sorted order. So it's sorted, and even though the elements in this part of the array may not be in the correct final location. Which is different from what we saw in the previous video by the way, they are still always going to be in the same relative order to one another. And then we're increasing the amount of the array that is in this relative sorted order each time. Okay, and so what's happening here is, in order to increase the piece of the array that's sorted each time within the body of the for loop. We're finding the correct location of the next element that we haven't thought about up until now relative to the first i-1 elements. And so then we want to nudge it into the sorted part of the array, nudge, nudge, nudge, nudge, nudge, until it gets into its correct relative location, leave it there. And then all of a sudden we have a nice, bigger part of the array that is sorted and we can go look at the next element that we haven't looked at yet. So this is actually a well known sorting algorithm. It's a mystery no longer. It's called Insertion Sort and that's because we keep inserting new elements into a previously sorted array, into the correct position relative to the other elements and then this sorted part grows and grows. All right, notice that this algorithm is quite different from the previous algorithm that we talked about, the selection sort algorithm that we talked about in the previous video. And that's because in the insertion sort algorithm, we have the elements that were taken care of, they get taken care of in a different way. We don't immediately put them where they're going to be. And they stay in that position forever after, and we know their location in the complete array from that point on. Rather, we're looking at relative locations and relative ordering and so we have two different algorithms. But they share some features and so with each algorithm, as we go through it. We ask similar questions about correctness, and performance, and similarly we would want to compare that selection sort algorithm to the insertion sort algorithms and other algorithms as well as they come. So, with insertion sort, the mystery sort that we were working through just now, how do we know that it's correct? Well, the argument that we were making before about the part of the array that is in relative sorted order, growing and growing and growing until we get to all of the elements in the array. Well, if we were able to make that precise, that would be a convincing argument that this algorithm is correct. Because no matter what we start with, at the end we get all of the elements in the array in sorted order and that's what we want. So then theres the question of can we do better? And when it comes to algorithms, we always want to do better but what is that even mean? And so the question of goodness of an algorithm has a few levels to it and has a lot of complexity to it. So the baseline, lowest bar of goodness that we want is that algorithm does what it's supposed to, that it's correct. Okay, both selection sort and insertion sort have met that bar. But then, we might want more. So for example, we want to be able to tackle huge reams of data and so we might want our algorithms to be really fast. And so we want to be able to process billions and trillions of pieces of information, very efficiently. And so we need to think about, how do we measure the speed of an algorithm? How do we talk about its performance in terms of time in general? When we don't a priori know what our data looks like. We wanted to be good for all sorts of pieces of data. And so, that will lead us to things like worst case analysis and asymptotic analysis that you'll see lots more about in the next course of this specialization. I want to also mention that there are other measures of goodness of an algorithm. For example, we might not want to waste a lot of memory while performing the execution of an algorithm. And so, we might want you to be very resource sensitive and not use up a lot of space. And so, there's a notion of space complexity that we would want to think about as well. And then we also might want to think about other properties that the algorithm could preserve or not preserve. So for example, if we want to sort a lot of data on dimensions. So you can think back about the airport data, we want to sort. First by name of country, and then by name of city. Then we would want our algorithm to behave in a way that's compatible with doing multiple task sorting. And so we might want to restrict which algorithms we consider, based on only those that have properties amenable to our use case. So goodness and can we do better? Are always relative questions, relative to how we plan to use the algorithms. Algorithms are actual living, breathing objects you can think about. And they don't exist in a vacuum, we want to use them and the measure of their goodness is only in terms of how we can use them and what they can tell us.