[MUSIC] So in the last video, we talked about how we're going to measure performance, based on counting operations. So we're going to count the number of operations that are performed every time we execute code. So, let's practice how to do that. By the end of this video, you'll be able to count the operations when you look at a particular code snippet. And think about what happens when we run it for a given input. All right, so the code that we're going to focus on is going to solve the answer, answer the question of looking for a particular letter in a word. And we'd like to think about how long that takes. So, for example, we're going to look at the word or two words, but it's a single string, San Diego, and we're going to search for the letter, lowercase a. Now in order to do that, Christine showed you a method in the previous lesson where we have the method hasLetter, which has input both the string, which is the word that we're looking in. And the letter that we're looking for, so that's a character. So we have those two inputs, and then what we're going to return is a Boolean which says true if we found the letter in the word and false if we didn't. Okay, so thinking about this algorithm, what we're really doing is a linear search. We're gonna go through the entire string, which remember, is an array of characters, and we're going to start at the beginning. And one by one, character by character, we're going to compare the current character to the letter that we're looking for. So we start at the beginning with our index being at 0 and then we keep checking whether the character at that index 0, is the same as the letter that we're looking for, which was input by the user. So we have our example word San Diego and we're looking for the letter a, remember. Okay, so we start by initializing our loop counter i, and that counts as a single operation. And now we check that our loop counter is less than the bound, and so we're comparing 0, the current value of i, to the word length, which is 9. And what we need to do now is say, okay, yes, that's correct, 0 is less than 9, and so we keep on going. But we have to increment the number of operations that we've done, because that test is an operation. All right, so now we've gone into the body of the for loop and inside the body of the for loop we have an if statement, a conditional branch. And what we're checking here is whether the letter at position 0, which was our current value of i, is the letter that we're looking for. But notice that even making that check, testing that condition, is an operation. And so we have to increment our operation count once more so, so far three operations. In this case, the current character is a capital S that's not the same as little a. And so we don't execute the body of the if branch at this point, we go up to the top of the for loop. We increment the counter i, oh, incrementing, that's another operation. And now we check that our current value of the counter, which is 1, is less than the word length. It is, and so we go back inside the for loop. But that means that we have to increment our operation count because we did another check. Okay, so we're now back inside the if and we're checking whether the current character, little a, matches the letter that we're looking for which is little a, all right? So this check evaluates to true, but before we go ahead and follow the logic, we remember that we have to increment our operation count. And now that we've decided that we have to go inside the if branch, we go in and we return true. Returning is an operation, and so our total number of operations is 7. Okay, so hopefully that trace through gives you an indication that we have to count each time that we execute a statement. Or that we evaluate a conditional branch, or a check, something that evaluates to a Boolean. Now, is this number of operations 7, going to be the same every time we run the method, hasLetter, on inputs of string and a character? And so what we have now, we have a couple of quizzes for you, to see what you think about what happens when we run this method in different contexts. So in those quizzes hopefully you got a sense of just how many variables there are for deciding how many operations get executed when we run a method on particular inputs. And so let's recap what you saw in those quizzes and let's think about what happens when we search for the letter x, still in that same word San Diego. So we're still looking at the hasLetter method and now we're thinking about just looking for a different character. So when we think about counting operations, I know that I think about groaning because I don't wanna do that same line by line counting each one again. That was kind of painful but I'd like to find a shortcut. So let's think instead of grouping a whole bunch of instructions, a whole bunch of operations into maybe something higher level, that then I can think about how that higher level construct repeats over and over. So let's think about what happens as we go through a single iteration of the for loop, what that body of the for loop is going to contribute to my operation count. And so, in a single iteration, what I need to do is, well I need to test whether my current loop counter is less than the bound. I need to, in that case, perform the contents of the body of the for loop. And then I'm going to need to increment my loop counter. Okay, so each of those contributes one instruction, in this particular case, to the operation count all in all. And this is, of course, assuming that in this current iteration of the for loop I haven't found the character that I'm looking for, because that would kick us out of the for loop. And so let's just think about what's happening in an iteration, somewhere in the middle of my algorithm, somewhere in the middle of the execution of the code. And so in that case, each iteration takes 3 operations. And so if we think back to looking for 'a', then we only really did 2 iterations of the for loop. We went through an iteration when our loop counter was at the 0, so we were comparing a to capital S, and then we had an iteration with our loop counter at position 1. And so then we were comparing a to a, and then we broke out of the for loop and were happy. Okay, so, when we had just the 2 iterations, we got 7 operations. And so now let's think about the how many iterations of the loop are there, when we're looking for the letter 'x'. Okay, so thinking back to the algorithm, we're going from the beginning to the very end of the word San Diego. And so we're going to be looking through every single position of the string, and that means that there's going to be an iteration of the for loop for each of the positions in the string. And there are 9 such positions because this is a length 9 word. Okay, now we can go ahead and do the calculations for the number of operations based on the 9 iterations. We see that there are going to be 29 operations. And now you might stop and say, whoa, hold on, I know my arithmetic, 9 times 3 is not 29, it's 27! Think about what happens at the very beginning, when we initialized the variable i, and at the very end, when we break out of the loop. And so, there's going to be some niggly little counts at the very beginning and the end that aren't exactly the same, it's just doing 3 times the number of iterations. You can do that calculation, I trust you, you can do that on your own. But big picture, we've got about 3 operations for each iteration. And then we've got 9 iterations of the for loop, as we're looking for that letter 'x' through the word San Diego. But in all of this, we're kinda sweeping under the rug, this question of what counts as an operation? What does it mean to have an operation? And so, when we first started out, I suggested that every time we execute a statement, that's an operation. And every time we evaluate a conditional statement and ask for the Boolean value of that conditional statement, then that's an operation. But you might ask, well how did you know? How did you know that when we returned to something that's an operation? What about other possibilities, or can I group operations? What if I have a single line that's both a variable declaration and an assignment, is that one operation or two? Who knows? Well so that's a really good question and it goes to the heart of what we wanna do when we're analyzing performance. What we'd like to do is not worry about this question, what we'd like to do is only worry about the pieces of the algorithm that we can control as algorithm designers. And not so much about the implementation of how a single line of Java code gets executed. And so a way to think about an operation, is a basic unit of execution that doesn't change as the input changes. So when we have a single line of code that is both a declaration and an assignment, then what we're doing in that single line of code doesn't depend on the input. How long it takes us to do that doesn't really depend on the input and so we can count that as a single operation. So a basic operation is something that doesn't depend on the size of the input, it doesn't depend on what happens with the input. And so what we'd like to do is not worry so much about whether we count such a declaration plus assignment as two instructions or one, and just think of it as a single operation. And the tools that we'll be developing in the next video will help us do even more of that kind of abstraction, where we're less worried about counting every single operation and every single line. And rather look at the whole scale performance of the algorithm, specifically as the input grows and the input size gets really, really big.