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24.9
You only need n steps for linear search.
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But what if you want to instead find an algorithm that's logarithmic?
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What algorithm that we used might be logarithmic in nature?
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5.03
Binary.
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0.62
So binary search, as Derek proposed, once we knew the list was sorted
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was an alternative to that.
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And constant time.
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What about this?
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What's an algorithm, maybe independent of today,
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that allows you constant time, running time?
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It could be anything.
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How about hello world?
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The [? say ?] block, or printf like that.
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If you can just say a whole phrase at once.
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Or maybe just adding two numbers together is like one step.
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No matter how big the numbers are, it still just takes one step
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to add them all together.
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It depends on some of the lower-level implementation details.
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And even printf or the [? say ?] block has multiple characters.
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So we'll actually come back to this in the future
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as to what it really means to be a step.
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But anything that takes a constant number of steps.
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Checking an "if" condition, returning a value.
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Statements, generally, might be broadly speaking just constant time,
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but we'll see contradictions to that before long.
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So let's take a look at two other symbols.
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This one being a capital omega symbol.
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And this omega symbol is the opposite, really, of big O. Big O
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3.64
is an upper bound on the running time.
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1.61
In the worst case, how many steps might it
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take to find an element in a list of n elements?
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n steps.
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So big O of n.
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Bubble sort, selection sort, insertion sort?
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They're on the order of n squared.
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1.69
But what about the lower bound?
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Especially when you get lucky in the best case so to speak,
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how few steps might you solve problems with?
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So for instance, if we consider the same formulas,
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but this time with capital omega, what's an algorithm
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that no matter what takes n squared steps,
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even if, for instance, the list is sorted?
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So suppose our 8 volunteers here were already 1, 2, 3, 4, 5, 6, 7, 8.
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Which of those algorithms would still have required n squared steps?
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So not linear search or binary search because those are much lower.
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But insertion, bubble sort, selection sort.
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Well, what about selection sort?
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Even if the list is already sorted, how do I find the smallest element?
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I'm lucky and it's right here, but I don't know that yet, right?
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To know that, I have to confidently walk the whole list and realize,
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well, that was a waste of time.
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1 is indeed the smallest element.
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It's sorted.
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How do I find the next smallest element?
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Start at the left.
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But you can ignore number 1.
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So it's a minor savings.
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OK, 2.
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Feel's pretty small, but got to check.
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Dammit, that was the smallest.
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And so selection sort, even in the best case when the list is already sorted,
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is going to take at least on the order of n squared
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steps because of that naivete of it.
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Because of that underlying design of just looking
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for the smallest element with no optimizations means
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the algorithm's always going to take us that many steps.
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But it doesn't have to be that way.
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What about bubble sort?
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Suppose that all 8 humans are sorted, 1 all the way through 8.
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And here I am comparing the first two.
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1 and 2?
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Not out of order.
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2 and 3, 3 and 4, 4 and 5, 5 and 6, 6 and 7, 7 and 8.
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OK, everything looks good.
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And in particular, what did I not do while walking from left to right?
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I didn't do any swaps.
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So it would be foolish of me algorithmically to do any of that
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again because the answer is not going to change.
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And so done.
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It took me n steps to sort n elements with bubble sort
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if the list is already sorted.
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So it would be omega of n because in the best case, it takes you n steps.
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Could a sorting algorithm be constant time
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or logarithmic, log n or order of 1?
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Or omega of 1 rather?
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Can you sort n elements in logarithmic time or constant time?
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What's the intuition there?
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These are kind of like now the fundamental meaning of computing
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and what you can actually do.
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What does it mean for me to ask, can you sort n elements
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in log n time or constant time?
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Well, that effectively is like saying, can you
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sort n elements without even looking at some of those elements?
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Because both log n and 1-- or mathematically,
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and if you don't recall, take it on faith today-- less than omega of n
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in general.
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So can you possibly, intuitively, in the real world,
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sort n elements in some random order in fewer than n steps?
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Yeah.
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AUDIENCE: [INAUDIBLE].
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