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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: One of the cutest little data structures that was ever invented is called the heap. And we're going to use the heap as an example implementation of a priority queue. And we'll also use heaps to build a sorting algorithm, called heap sort, that is very, very different from either insertion sort or merge sort. And it has some nice properties that neither insertions sort nor merge sort have. But what I want to do is get started with motivating the heap data structure, regardless of whether you're interested in sorting or not. So the notion of a priority queue, I think, makes intuitive sense to all of you. It's essentially a structure that implements a set S of elements. And each of these elements is associated with the key. And as you can imagine, a priority queue is something where you queue up for something, you want to buy something, you want to sell something. You have certain priorities assigned to you, and you want to pick the maximum priority or the min priority. You want to be able to delete it from the queue. You want to be able to insert things into this queue. You want to be able to change priorities in the queue. So all of these operations are interesting operations that should run fast, and for some definition of fast. Obviously we are interested in the asymptotic complexity definition of fast. In that case, we'll be saying does this operation run an order n time, order log n time, et cetera. So in general, I think for the next few lectures, you're going to see a specification of data structure in terms of the operations that the data structure should perform. And those of you who have taken six double O five, you'll see that it's basically an abstract data type that's associated with these operations. So it's a spec for the abstract data type. In six double O five, you had really spent a lot of time on asymptotic complexity, or the efficiency of operations on the abstract data type. Here, in double O six, you'll specify this ADT, and specify the set of operations or methods in the ADT. And we'll talk about whether these are order end complexity log end complexity, and compare and contrast different ADTs. So today's ADT is a heap. And what is the set of operations that we'd like to perform on a priority queue? So we can use that to motivate the development of the heap. And those are, insert s x. So you have a set of elements s, and you want to be able to insert element x into set s. You want to be able to do max of...