# 8.7. Priority Queues with Binary HeapsΒΆ

In earlier sections you learned about the first-in first-out data
structure called a queue. One important variation of a queue is called a
**priority queue**. A priority queue acts like a queue in that you
dequeue an item by removing it from the front. However, in a priority
queue the logical order of items inside a queue is determined by their
priority. The highest priority items are at the front of the queue and
the lowest priority items are at the back. Thus when you enqueue an item
on a priority queue, the new item may move all the way to the front. We
will see that the priority queue is a useful data structure for some of
the graph algorithms we will study in the next chapter.

You can probably think of a couple of easy ways to implement a priority
queue using sorting functions and vectors. However, inserting into a vector
is \(O(n)\) and sorting a vector is \(O(n \log{n})\). We can
do better. The classic way to implement a priority queue is using a data
structure called a **binary heap**. A binary heap will allow us both
enqueue and dequeue items in \(O(\log{n})\).

The binary heap is interesting to study because when we diagram the heap
it looks a lot like a tree, but when we implement it we use only a
single vector as an internal representation. The binary heap has two
common variations: the **min heap**, in which the smallest key is always
at the front, and the **max heap**, in which the largest key value is
always at the front. In this section we will implement the min heap. We
leave a max heap implementation as an exercise.