Welcome

Hello, and welcome to competitive programming. Today we are going to start talking about more serious data structures: in this case balanced trees and heaps.

Objectives

We are, again, going to assume you know the basics of the balanced binary tree, either the AVL tree or the Red-Black tree. We also are going to assume you know about heaps. We have two goals. First, we want you to learn the STL interfaces to these structures and become proficient at determining which one you need to use. But you will not be able to rely on these forever. More advanced problems will require you to augment these data structures in ways the STL cannot accommodate. For those problems, you will need to be able to write a customized version of these quickly and accurately.

BSTs

A rotation

As before, we are going to advocate for the use of library versions of trees and sets. But…. the is a major disclaimer. Some of the more advanced problems we will get to later in the course will require customized versions of these libraries. So you are going to have to be able to write bug free versions of red-black or AVL trees sooner or later. You should practice writing an implementation on a periodic basis so that you gain speed during the contests.

As a reminder, here is the code in C++ to do a left rotation. You might recognize it from CS 225. The right rotation is the dual of this, and there are left-right and right-left rotations that you make by composing these single rotations. My advice is to give yourself a speed drill… how quickly can you code a tree or other data structure that uses rotations?

The STL

But for now, know that C++ STL uses red-black trees to implement the map and set libraries. The map library stores both key and value, whereas set only stores the keys. The time complexity for lookup is $\log(n)$, which is not as good as a hash table, but good enough for most ICPC type problems where the input size is less than one million.

There is another advantage of using map and set rather than hashing: you can get an iterator to the ordered contents.

You’ll want to look at the C++ reference guide for these to see what other operations they offer, but here are some highlights.

There is insert, which inserts an element. If the key is already part of the data structure, then it will not be overwritten. Use replace to replace an existing key. There is also erase to remove a single element, and clear to erase the whole structure. To access the values, use the array access operator, or else the at method.

Hashing

There is one kind of hashing that may be of use though. This is when you can ``bucket sort’’ your elements. In other words, you have an array into which you can locate a key without needing to hash or sort it. For example, if your keys were integers from 1 to 1024, or perhaps single characters, then a simple array does what we need very quickly.

Here’s a simple example: suppose you needed to implement a Ceaser Cipher, where each letter of the alphabet was transposed to make a secret code. You might make a direct lookup table like this one. You only need 26 letters, but since the total amount of data is small and the keys are unique, we can just use an array.

Priority Queues

The priority_queue is a max heap, and has $O(1)$ access to the maximum member, and $O(\log n)$ insertion and removal.

A max heap has two very useful functions. First, it can serve as a priority queue. Second, it can partially sort data for you very efficiently. If you have a million elements and want the top 10, this is a good choice. If you want a much larger fraction of these elements, then a modified version of quicksort may be better.