[MUSIC] So now we're ready to analyze the performance of some of the sorting algorithms that we've already talked about. So by the end of this video, you'll be able to state and justify the asymptotic performance, so the big O classes for the running time, of both selection sort and insertion sort. And we'll be thinking about them, both in the best case and in the worst case. Now, for both of these sorting algorithms if you need a refresher just go back to the first course in the specialization. That's when we talked about them in quite a bit of detail. But, we'll be doing an overview now, as well. So, our goal is to fill in this table with big O classes that represent the best and worst case performance for these two sorting algorithms. So let's start with selection sort, and in selection sort we take a list or an array of data. And what we'd like to do is organize it so that it's presented in order from smallest to biggest based on some measure of smallest and biggest. And so our strategy is going to look at each position in the array in turn. And for each of those positions, find the right element to put in there. And so we look at all of the possible elements remaining in the array. And figure out which one of those is smallest and put it in this current position. Okay, so that's the algorithm. And so now, let's think ahead about what insertion sort algorithm is going to do. It's a little bit different. We're still going through all of the positions in the array in turn. But instead of looking for the correct elements to put in each time, what we're doing is looking at the current value that we're focused in and nudge it over to its correct position relative to the elements that are already listed. So we're still going step by step and increasing the part of the array that's sorted each time, but doing it in a slightly different strategy. And we'll see the impact of these different strategies on their performance of the two algorithms. So let's start with selection sort, and let's think about the performance of this sorting algorithm. So we actually used selection sort in a previous support video where we went through the code for selection sort and computed its performance. This was an example of nested for loops that were really interesting to analyze and we used some cool identities. But notice that when we did that analysis, we could say, this algorithm is big O of n squared, it's quadratic, and when we did that, it was before we said anything about best case, worst case, average case. We weren't even thinking about the input or how it was organized, because this sorting algorithm, if you think about it, doesn't really care what the input looks like when you first get it. It goes through and does the same sequence of steps no matter how its input is organized. So the number of steps, the performance of the sorting algorithm, only depends on the size of the array that it starts with. It doesn't depend on how the contents of the array are organized and that means that the best case and the worst case performance of this algorithm are exactly the same. It's the same as the average case. It's the same as any case, so for any array of size n, the best case and worst case performance of selection sort, is big O of n squared. Okay, so that's the one algorithm. Now let's think about the insertion sort algorithm, and try to fill in some of those questions marks, and we haven't analyzed it yet. So let's do it now, and what we're trying to do, as I mentioned earlier, is focus on one new element of the array at a time. And for that new element put it in its correct location relative to the earlier ones that have been sorted already. And so we assume that at each point we have that initial piece of the arrays already sorted. And so what we need to do is figure out where the next element goes and we do that by comparing that next element to, well, the highest element in the sorted part of the array and see, well should I swap their locations or not? And then maybe I'm gonna need to swap again, and maybe I need to swap again until the new element gets to its correct relative location based on the previously sorted pieces of the array. Okay, so what would the best case of this algorithm look like? And in order to do that, we need to focus in on that for loop. Where that for loop looks at each position in the array and says, I'm going to put the element in this position, in its correct location. And the piece that really depends on the values is the inner while loop. So, the inner while loop maybe gets executed a different number of times, based on the relative size of the consecutive pairs of elements in the array. So, let's think about the best case first. So, in the best case, what we would have, is that we don't need to make very many swaps at all. So we're looking at a new element in the array, and we'd like to find its correct relative position, relative to the earlier elements in the array. And if we're really lucky then it's already going to be in that position. Well what that would mean is that it's bigger than the biggest element in the already sorted part of the array. And so, what we need to check is that the current index has, the value of that current index is already bigger than the value at the last piece of the already sorted part of the array. So when does that happen? Well, it happens when the array that we start with is already sorted. Every time we look at a new element, it's already in its correct location relative to the earlier ones, so we don't need to do anything. The body of the while loop never executes. And so when we look at the performance in this case, that means that every time we try to start executing the while loop, all we need to do is do that just one test, and it's going to evaluate to false. And we go away from it, and we go to the next iteration of the for loop. And so that means that the body of the for loop in this case, when the array's already sorted, every time we go through the for loop, we just execute a constant number of instructions. And so all in all, we have n iterations of the for loop, each time doing just a constant amount of work, and so all in all we have just O(n) work. So when the array is already sorted, in this best-case lucky, lucky scenario, insertion sort just takes big O(n) time. Cool. That's much, much better than quadratic, which was selection sort. So that's interesting, okay. Is it gonna be good all the time? Is it a big win? Well, so let's think about the worst case. So what input might we get that would require us to run many, many steps? And remember the piece that we're focusing in on is that inner while loop. Where we're nudging the current value that we're looking at to its correct location within that first part of the array that's already sorted. So, if we think about what might make us really unlucky is if we have to do a lot of the nudging, a lot of the swaps over. And so, if we think of our arrays being really far from sorted, then we're gonna have to do a lot of work to keep moving these elements to their correct location, relative to the sorted part of the array. So, thinking about an array that's given to us in reverse sorted order, we're going to have to do a lot of work in this algorithm in order to get it to the correct form, namely going from one through five, rather than from five through one. So in this worst case analysis what we see is that when we are looking at, for example, at the second position, so position equals 1, then when we want to nudge 4 over to its correct location we have to do a swap. And then when we look at the next index at position 2, then in order to move 3 to its correct location we're going to have to swap it twice. And so every time we're going to have to swap all the way over to the head of the array and so all together what's going to happen is that on average we're gonna have about O(n) many swaps for each position. And that happens n many times as we go through all the positions. And so we get to O(n) squared amount of work. And so the analysis at the end was a little brief but it's really similar to what we do when we think about analyzing selection sort. So go back and revisit that support video if you'd like to see a little bit more detail. Now what we wanna keep in mind, though, is that the worse case for insertion sort happened when our input array was in reverse order. And so this is really illustrating how the structure of the input can determine the performance of our algorithm. And when we're thinking about worst-case analysis, we want to think about the kinds of inputs that will make us do the most amount of work. So here is our table so far for sorting algorithms. And in the next video we'll introduce a couple of new sorting algorithms just to expand our vocabulary, expand our toolbox of sorting algorithms and compare how they're run as well.