Heaps and Priority Queues

• C++ Reading: Chapter 8 Heaps and Priority Queues
• Java Reading: Chapter 9 Priority Queues

Priority Queues

A priority queue is a queue where removal depends not upon order of insertion but upon importance.

The key gives an element’s importance. Keys must have a comparison operation that is reflexive, antisymmetric, and transitive.

Such a queue is straightforwardly implemented using a list. There are two strategies: sorted and unsorted.

A priority queue gives a straightforward scheme for sorting: Insert all elements into the priority queue and simply read them back out.

The priority queue sorting scheme, implemented using sorted and unsorted lists, gives rise to the insertion-sort and selection-sort algorithms, respectively. Both are $$O(n^2)$$.

If we can improve the performance of the priority queue, we can naturally improve the performance of sorting.

Heaps

A heap:

• Is a binary tree
• Is complete: Level $$i$$ has $$2^i$$ nodes and the last level fills from the left.
• This means that the height $$h=\lfloor\log_2 n\rfloor$$.
• Has the heap order property: A node $$v$$ has a key greater than or equal to the key of its parent.
• This means that the minimum key is always found at the root.
• Has an efficient insertion algorithm:
• Place the new node in the last position.
• While the heap order property does not hold:
• Swap the node with its parent.
• This is $$O(\log n)$$ because the height of the tree is at most $$\log n$$
• Has an efficient root removal algorithm:
• Copy the node in the last position to the root.
• While the heap order property does not hold:
• Swap the node with the smaller of its children.
• This is $$O(\log n)$$ because the height of the tree is at most $$\log n$$
• Has an elegant zero-based array representation:
• Let $$x(v)$$ be the array index of node $$v$$, defined as follows:
• If $$v$$ is the root then $$x(v)=0$$.
• If $$u$$ is the left child of $$v$$ then $$x(u)=2\,x(v)+1$$.
• If $$u$$ is the right child of $$v$$ then $$x(u)=2\,x(v)+2$$.
• If $$p$$ is the parent of $$v$$ then $$x(p)=\lfloor\frac{x(v)-1}{2}\rfloor$$.

Applying the priority queue sorting scheme to a heap results in an $$O(n\log n)$$ sort.

The array representation of the heap also gives rise to an extremely efficient in-place sorting algorithm.