High Probability Bounds for Treaps (and Quicksort)

So far, we have argued about the expected work of quick sort and the expected depth of a node in a treap, but is this good enough? Let’s say my company DontCrash is selling you a new air traffic control system and I say that in expectation, no two planes will get closer than 500 meters of each other — would you be satisfied? More relevant to this class, let’s say you wanted to run 1000 jobs on 1000 processors and I told you that in expectation each finishes in an hour — would you be happy? How long might you have to wait? There are two problems with expectations, at least on their own. Firstly, they tell us very little if anything about the variance. And secondly, as mentioned in an earlier lecture, the expectation of a maximum can be much higher than the maximum of expectations. The first has implications in real time systems where we need to get things done in time, and the second in getting efficient parallel algorithms (e.g., span is the max span of the two parallel calls). Fortunately, in many cases we can use expectations to prove much stronger bounds at least when we have independent events. These bounds are what we call high probability bounds, again as mentioned in an earlier lecture. Basically, it says that we guarantee some property with probability very close to 1 — in fact, so close that the difference is only inverse polynomially small in the problem size. To get the high probability bound, we will again use the Chernoff bound that we saw in some previous lecture: