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SPEAKER 1: Good question.
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Could you make the whole list a big integer?
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You could, that and other tricks.
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But to do that, you're going to need to touch or somehow manipulate
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all of those values at least once to kind of do that, I would think.
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And so you've already spent n steps.
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AUDIENCE: [INAUDIBLE].
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SPEAKER 1: OK.
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So if you have a like guessing algorithm-- yes, this is sorted.
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That's constant time.
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You're just not always correct.
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So I should clarify, can you sort correctly n elements
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without necessarily looking at all n-- the short answer is no.
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There is like a fundamental limit of computation.
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And also, just human intuition here.
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You can't possibly claim to me and convince me, in a court of law even,
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that these n elements are sorted if you can't even
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testify to having looked at all n of those elements.
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It's just not possible given the computational model
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we've been discussing where a computer can only look at one element at a time.
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So sorting is out of the question.
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But an algorithm that takes omega of one step?
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For instance in the best case, when might an algorithm need only one step?
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How about something like linear search or even binary search?
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The running time of those algorithms was big O of n and big O of log n
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respectively.
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Linear search, binary search.
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What's the lower bound on those two algorithms' running times?
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AUDIENCE: [INAUDIBLE].
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SPEAKER 1: Yeah, it's omega of 1.
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But why?
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Derek wasn't so lucky.
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But AJ, last year, touched the number 50-- voila, 1 step.
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And that was actually, while sort of bad pedagogy for us,
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was a perfect manifestation of yes, in the best
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case the lower bound on your running time might indeed just be one step.
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Or two steps, but some constant number of steps.
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So we have a range of these options.
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Now, sometimes these ranges coincide.
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If your big O notation, so to speak, and your omega notation, so to speak,
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are one in the same values, you can claim that the algorithm
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has a running time in capital Theta.
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So theta just means when big O and omega are the same,
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you can instead use this formula and sort of convey two thoughts at once.
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But for now, the takeaway is that as best we've
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seen for something like sorting, the best we've done so far, at least
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in the worst case, is big O of n squared.
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And this doesn't feel very good, especially since sorting and searching
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is all around us.
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Anyone with an iPhone or an Android.
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I mean, there's so much data that we're carrying around in our pockets
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these days, let alone in the cloud and on our laptops and desktops,
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that it's just generally good practice to keep sorted.
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In your phone book, it would be pretty tedious
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if all of your friends and family were in random order in the contacts list.
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And you had to like scroll through the whole darn list, big O of n,
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to find someone.
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No, instead we have a search box, of course.
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And most of us probably wouldn't blindly scroll through the whole list.
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But even that search box, if you start typing in D-E-R to look up Derek,
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or S-M-I-T-H to find Smith.
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Well, how is iOS?
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How is Android finding Mike Smith or finding Derek?
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Those pieces of software themselves have algorithms that underneath the hood
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are hopefully leveraging smarter algorithms than we've even seen here.
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Because especially as you have more and more friends, more and more family
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members, even big O of n squared starts to feel kind of slow
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as we can see here.
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Let me go ahead and open up an example.
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So let's take a look at a demonstration.
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Here is a visualization that someone on the internet kindly put together.
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And it focuses on sorting numbers.
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But rather than just show the numbers textually as 1, 2, 3, 4, 5, 6, 7, 8,
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it instead shows them as bar.
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So the shorter the bar, the smaller the number.
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The taller the bar, the taller the number.
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And let me go ahead and increase the speed of this,
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just so it doesn't get too tedious.
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And let's take a quick look at bubble sort.
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Now, notice what's happening.
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The two red bars adjacently are the ones that are
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being compared and potentially swapped.
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And you'll notice that just like in our human demo, what is happening over
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there on the right-hand side?
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AUDIENCE: [INAUDIBLE].
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SPEAKER 1: Yeah.
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So the biggest elements seem to be bubbling up, if you will, to the top.
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Even though there's still a lot of work to be done to the left of those,
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we indeed have the ability here to see the list getting sorted progressively.
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But you know, I'm already getting bored.
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This algorithm doesn't feel very fast.
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It's very pretty.
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It's very correct, it would seem, but it's a little slow.
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Let's try another one to see if it's any better than that.
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That's the spoiler.
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That's what it should look like.
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Let's try our friend selection sort, which is already sorted.
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Let me stop this.
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Let me click on Randomize Array.
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And now click Selection Sort.
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