Heaps are based on the notion of a complete tree, for which we gave an informal definition earlier.
Formally:
A binary tree is completely full if it is of height, h, and has 2h+1-1 nodes.
A binary tree of height, h, is complete iff
- it is empty or
- its left subtree is complete of height h-1 and its right subtree is completely full of height h-2 or
- its left subtree is completely full of height h-1 and its right subtree is complete of height h-1.
- all the leaves are on
- the same level or
- two adjacent ones and
- all nodes at the lowest level are as far to the left as possible.
Heaps
A binary tree has the heap property iff
- it is empty or
- the key in the root is larger than that in either child and both subtrees have the heap property.
A heap can be used as a priority queue: the highest priority item is at the root and is trivially extracted. But if the root is deleted, we are left with two sub-trees and we must efficiently re-create a single tree with the heap property.
The value of the heap structure is that we can both extract the highest priority item and insert a new one in O(logn) time.
How do we do this?
| Let's start with this heap. A deletion will remove the T at the root. |  |
| To work out how we're going to maintain the heap property, use the fact that a complete tree is filled from the left. So that the position which must become empty is the one occupied by the M. Put it in the vacant root position. |  |
| This has violated the condition that the root must be greater than each of its children. So interchange the M with the larger of its children. |  |
| The left subtree has now lost the heap property. So again interchange the M with the larger of its children. |  |
This tree is now a heap again, so we're finished. We need to make at most h interchanges of a root of a subtree with one of its children to fully restore the heap property. Thus deletion from a heap is O(h) or O(logn).
Addition to a heap
| To add an item to a heap, we follow the reverse procedure. Place it in the next leaf position and move it up. Again, we require O(h) or O(logn) exchanges. |  |
Storage of complete trees
The properties of a complete tree lead to a very efficient storage mechanism using n sequential locations in an array.
 | If we number the nodes from 1 at the root and place:
|
| Viewed as an array, we can see that the nth node is always in index position n. |  |
The code for extracting the highest priority item from a heap is, naturally, recursive. Once we've extracted the root (highest priority) item and swapped the last item into its place, we simply call MoveDown recursively until we get to the bottom of the tree.See code of delete heap below:
/* Extract the highest priority from the heap */
#define LEFT(k) (2*k)
#define RIGHT(k) (2*k+1)
#define EMPTY(c,k) (k>=c->item_cnt)
#define SWAP(i,j) { void *x = c->items[i]; \
c->items[i] = c->items[j]; \
c->items[j] = x; }
void MoveDown( Collection c, int k )
{
int larger, right, left;
left = LEFT(k);
right = RIGHT(k);
if ( !EMPTY(c,k) ) /* Termination condition! */
{
larger=left;
if ( !EMPTY(c,right) )
{
if ( ItemCmp( c->items[right], c->items[larger] ) > 0 )
larger = right;
}
if ( ItemCmp( c->items[k], c->items[larger] ) )
{
SWAP( k, larger );
MoveDown( c, larger );
}
}
}
void *HighestPriority( Collection c )
/* Return the highest priority item
Pre-condition: (c is a collection created by a call to
ConsCollection) &&
(existing item count >= 1) &&
(item != NULL)
Post-condition: item has been deleted from c
*/
{
int i, cnt;
void *save;
assert( c != NULL );
assert( c->item_cnt >= 1 );
/* Save the root */
save = c->items[0];
/* Put the last item in the root */
cnt = c->item_cnt;
c->items[0] = c->items[cnt-1];
/* Adjust the count */
c->item_cnt--;
/* Move the new root item down if necessary */
MoveDown( c, 1 );
return save;
}
Note the macros LEFT and RIGHT which simply encode the relation between the index of a node and its left and right children. Similarly the EMPTY macro encodes the rule for determining whether a sub-tree is empty or not.
Inserting into a heap follows a similar strategy, except that we use a MoveUp function to move the newly added item to its correct place. (For the MoveUp function, a further macro which defines the PARENT of a node would normally be added.)
Heaps provide us with a method of sorting, known as heapsort. However, we will examine and analyse the simplest method of sorting first.
Animation
In the animation, note that both the array representation (used in the implementation of the algorithm) and the (logical) tree representation are shown. This is to demonstrate how the tree is restructured to make a heap again after every insertion or deletion.
