1. CHAPTER 11: Priority Queues and Heaps
Java Software Structures:
Designing and Using Data Structures
Third Edition
John Lewis & Joseph Chase

2. Chapter Objectives Define a heap abstract data structure
Demonstrate how a heap can be used to solve problems
Examine various heap impmentations
Compare heap implementations

3. Heaps A heap is a binary tree with two added properties:
It is a complete tree
For each node, the node is less than or equal to both the left child and the right child
This definition describes a minheap

4. Heaps
In addition to the operations inherited from a binary tree, a heap has the following additional operations:
addElement
removeMin
findMin

5. The operations on a heap

6. The HeapADT /** * HeapADT defines the interface to a Heap. *
Dr. Lewis
Dr. Chase
1.0, 9/9/2008 */
package jss2;
public interface HeapADT<T> extends BinaryTreeADT<T> {
* Adds the specified object to this heap.
obj the element to added to this head
public void addElement (T obj);

7. The HeapADT (continued)
/**
* Removes element with the lowest value from this heap. *
the element with the lowest value from this heap */
public T removeMin();
* Returns a reference to the element with the lowest value in
* this heap.
a reference to the element with the lowest value in this heap
public T findMin(); }

8. UML description of the HeapADT

9. The addElement Operation
The addElement method adds a given element to the appropriate location in the heap
The proper location is the one location that will maintain the completeness of the tree

10. The addElement Operation
There is only one correct location for the insertion of a new node
Either the next open position from the left at level h
Or the first position in level h+1 if level h is full

11. The addElement Operation
Once we have located the new node in the proper position, then we must account for the ordering property
We simply compare the new node to its parent value and swap the values if necessary
We continue this process up the tree until either the new value is greater than its parent or the new value becomes the root of the heap

12. Two minheaps containing the same data

13. Insertion points for a heap

14. Insertion and reordering in a heap

15. The removeMin Operation
The removeMin method removes the minimum element from the heap
The minimum element is always stored at the root
Thus we have to return the root element and replace it with another element

16. The removeMin Operation
The replacement element is always the last leaf
The last leaf is always the last element at level h

17. Examples of the last leaf in a heap

18. The removeMin Operation
Once the element stored in the last leaf has been moved to the root, the heap will have to reordered
This is accomplished by comparing the new root element to the smaller of its children and swapping them if necessary
This process is repeated down the tree until the element is either in a leaf or is less than both of its children

19. Removal and reordering in a heap

20. Using Heaps: Priority Queue
A priority queue is a collection that follows two ordering rules:
Items which have higher priority go first
Items with the same priority use a first in, first out method to determine their ordering
A priority queue could be implemented using a list of queues where each queue represents items of a given priority

21. Using Heaps: Priority Queue
Another solution is to use a minheap
Sorting the heap by priority accomplishes the first ordering
However, the first in, first out ordering for items with the same priority has to be manipulated

22. Using Heaps: Priority Queue
The solution is to create a PriorityQueueNode object that stores the element to be placed on the queue, the priority of the element and the arrival order of the element
Then we simply define a compareTo method for the PriorityQueueNode class that first compares priority then arrival time
The PriorityQueue class then extends the Heap class and stores PriorityQueueNodes

23. The PriorityQueueNode class
/**
* PriorityQueueNode represents a node in a priority queue containing a
* comparable object, order, and a priority value. *
Dr. Lewis
Dr. Chase
1.0, 8/19/08 */
public class PriorityQueueNode<T> implements Comparable<PriorityQueueNode> {
private static int nextorder = 0;
private int priority;
private int order;
private T element;
* Creates a new PriorityQueueNode with the specified data.
obj the element of the new priority queue node
prio the integer priority of the new queue node

24. The PriorityQueueNode class (cont.)
public PriorityQueueNode (T obj, int prio) {
element = obj;
priority = prio;
order = nextorder;
nextorder++; }
/**
* Returns the element in this node. *
the element contained within this node */
public T getElement()
return element;
* Returns the priority value for this node.
the integer priority for this node

25. The PriorityQueueNode class (cont.)
public int getPriority() {
return priority; }
/**
* Returns the order for this node. *
the integer order for this node */
public int getOrder()
return order;
* Returns a string representation for this node.
public String toString()
String temp = (element.toString() + priority + order);
return temp;

26. The PriorityQueueNode class (cont.)
/**
* Returns the 1 if the current node has higher priority than
* the given node and -1 otherwise. *
obj the node to compare to this node
the integer result of the comparison of the obj node and this
* this one */
public int compareTo(PriorityQueueNode obj) {
int result;
PriorityQueueNode<T> temp = obj;
if (priority > temp.getPriority())
result = 1;
else if (priority < temp.getPriority())
result = -1;
else if (order > temp.getOrder())
else
return result; }

27. The PriorityQueue class
/**
* PriorityQueue demonstrates a priority queue using a Heap. *
Dr. Lewis
Dr. Chase
1.0, 8/19/08 */
import jss2.*;
public class PriorityQueue<T> extends ArrayHeap<PriorityQueueNode<T>> {
* Creates an empty priority queue.
public PriorityQueue()
super(); }

28. The PriorityQueue class (continued)
/**
* Adds the given element to this PriorityQueue. *
object the element to be added to the priority queue
priority the integer priority of the element to be added */
public void addElement (T object, int priority) {
PriorityQueueNode<T> node = new PriorityQueueNode<T> (object, priority);
super.addElement(node); }
* Removes the next highest priority element from this priority
* queue and returns a reference to it.
a reference to the next highest priority element in this queue
public T removeNext()
PriorityQueueNode<T> temp = (PriorityQueueNode<T>)super.removeMin();
return temp.getElement();

29. Implementing Heaps with Links
A linked implementation of a minheap would simply be an extension of our LinkedBinaryTree class
However, since we need each node to have a parent pointer, we will create a HeapNode class to extend our BinaryTreeNode class we used earlier

30. The HeapNode class
/**
* HeapNode creates a binary tree node with a parent pointer for use
* in heaps. *
Dr. Lewis
Dr. Chase
1.0, 9/9/2008 */
package jss2;
public class HeapNode<T> extends BinaryTreeNode<T> {
protected HeapNode<T> parent;
* Creates a new heap node with the specified data.
obj the data to be contained within the new heap nodes
HeapNode (T obj)
super(obj);
parent = null; }

31. Implementing Heaps with Links
The addElement method must accomplish three tasks:
Add the new node at the appropriate location
Reorder the heap
Reset the lastNode pointer to point to the new last node

32. LinkedHeap /** * Heap implements a heap. * * @author Dr. Lewis
Dr. Chase
1.0, 9/9/2008 */
package jss2;
import jss2.exceptions.*;
public class LinkedHeap<T> extends LinkedBinaryTree<T> implements HeapADT<T> {
public HeapNode<T> lastNode;
public LinkedHeap()
super(); }

33. LinkedHeap - addElement
/**
* Adds the specified element to this heap in the appropriate
* position according to its key value. Note that equal elements
* are added to the right. *
obj the element to be added to this head */
public void addElement (T obj) {
HeapNode<T> node = new HeapNode<T>(obj);
if (root == null)
root=node;
else
HeapNode<T> next_parent = getNextParentAdd();
if (next_parent.left == null)
next_parent.left = node;
next_parent.right = node;
node.parent = next_parent; }
lastNode = node;
count++;
if (count>1)
heapifyAdd();

34. LinkedHeap - the addElement Operation
The addElement operation makes use of two private methods:
getNextParentAdd that returns a reference to the node that will be the parent of the new node
heapifyAdd that reorders the heap after the insertion

35. LinkedHeap - getNextParentAdd
/**
* Returns the node that will be the parent of the new node *
the node that will be a parent of the new node */
private HeapNode<T> getNextParentAdd() {
HeapNode<T> result = lastNode;
while ((result != root) && (result.parent.left != result))
result = result.parent;
if (result != root)
if (result.parent.right == null)
else
result = (HeapNode<T>)result.parent.right;
while (result.left != null)
result = (HeapNode<T>)result.left; }
return result;

36. LinkedHeap - heapifyAdd
/**
* Reorders this heap after adding a node. */
private void heapifyAdd() {
T temp;
HeapNode<T> next = lastNode;
temp = next.element;
while ((next != root) && (((Comparable)temp).compareTo
(next.parent.element) < 0))
next.element = next.parent.element;
next = next.parent; }
next.element = temp;

37. LinkedHeap - the removeMin Operation
The removeMin operation must accomplish three tasks:
Replace the element stored in the root with the element stored in the last leaf
Reorder the heap if necessary
Return the original root element

38. LinkedHeap - removeMin
/**
* Remove the element with the lowest value in this heap and
* returns a reference to it. Throws an EmptyCollectionException
* if the heap is empty. *
the element with the lowest value in
* this heap
EmptyCollectionException if an empty collection exception occurs */
public T removeMin() throws EmptyCollectionException {
if (isEmpty())
throw new EmptyCollectionException ("Empty Heap");
T minElement = root.element;
if (count == 1)
root = null;
lastNode = null; }
else

39. LinkedHeap – removeMin (cont.)
HeapNode<T> next_last = getNewLastNode();
if (lastNode.parent.left == lastNode)
lastNode.parent.left = null;
else
lastNode.parent.right = null;
root.element = lastNode.element;
lastNode = next_last;
heapifyRemove(); }
count--;
return minElement;

40. LinkedHeap - the removeMin Operation
Like the addElement operation, the removeMin operation makes use of two private methods
getNewLastNode that returns a reference to the new last node in the heap
heapifyRemove that reorders the heap after the removal

41. LinkedHeap – getNewLastNode
/**
* Returns the node that will be the new last node after a remove. *
the node that willbe the new last node after a remove */
private HeapNode<T> getNewLastNode() {
HeapNode<T> result = lastNode;
while ((result != root) && (result.parent.left == result))
result = result.parent;
if (result != root)
result = (HeapNode<T>)result.parent.left;
while (result.right != null)
result = (HeapNode<T>)result.right;
return result; }

42. LinkedHeap – heapifyRemove
/**
* Reorders this heap after removing the root element. */
private void heapifyRemove() {
T temp;
HeapNode<T> node = (HeapNode<T>)root;
HeapNode<T> left = (HeapNode<T>)node.left;
HeapNode<T> right = (HeapNode<T>)node.right;
HeapNode<T> next;
if ((left == null) && (right == null))
next = null;
else if (left == null)
next = right;
else if (right == null)
next = left;
else if (((Comparable)left.element).compareTo(right.element) < 0)
else

43. LinkedHeap – heapifyRemove (cont.)
temp = node.element;
while ((next != null) && (((Comparable)next.element).compareTo
(temp) < 0)) {
node.element = next.element;
node = next;
left = (HeapNode<T>)node.left;
right = (HeapNode<T>)node.right;
if ((left == null) && (right == null))
next = null;
else if (left == null)
next = right;
else if (right == null)
next = left;
else if (((Comparable)left.element).compareTo(right.element) < 0)
else }
node.element = temp;

44. Implementing Heaps with Arrays
An array implementation of a heap may provide a simpler alternative
In an array implementation, the location of parent and child can always be calculated
Given that the root is in position 0, then for any given node stored in position n of the array, its left child is in position 2n + 1 and its right child is in position 2(n+1)
This means that its parent is in position (n-1)/2

45. ArrayHeap
/**
* ArrayHeap provides an array implementation of a minheap. *
Dr. Lewis
Dr. Chase
1.0, 9/9/2008 */
package jss2;
import jss2.exceptions.*;
public class ArrayHeap<T> extends ArrayBinaryTree<T> implements HeapADT<T> {
public ArrayHeap()
super(); }

46. Implementing Heaps with Arrays
Like the linked version, the addElement operation for an array implementation of a heap must accomplish three tasks:
Add the new node,
Reorder the heap,
Increment the count by one
The ArrayHeap version of this method only requires one private method, heapifyAdd which reorders the heap after the insertion

47. ArrayHeap - addElement
/**
* Adds the specified element to this heap in the appropriate
* position according to its key value. Note that equal elements
* are added to the right. *
obj the element to be added to this heap */
public void addElement (T obj) {
if (count==tree.length)
expandCapacity();
tree[count] =obj;
count++;
if (count>1)
heapifyAdd(); }

48. ArrayHeap - heapifyAdd
/**
* Reorders this heap to maintain the ordering property after
* adding a node. */
private void heapifyAdd() {
T temp;
int next = count - 1;
temp = tree[next];
while ((next != 0) && (((Comparable)temp).compareTo
(tree[(next-1)/2]) < 0))
tree[next] = tree[(next-1)/2];
next = (next-1)/2; }
tree[next] = temp;

49. ArrayHeap - the removeMin Operation
The removeMin operation must accomplish three tasks:
Replace the element stored at the root with the element stored in the last leaf
Reorder the heap as necessary
Return the original root element
Like the addElement operation, the removeMin operation makes use of a private method, heapifyRemove to reorder the heap

50. ArrayHeap - removeMin /**
* Remove the element with the lowest value in this heap and
* returns a reference to it. Throws an EmptyCollectionException if
* the heap is empty. *
a reference to the element with the
* lowest value in this head
EmptyCollectionException if an empty collection exception occurs */
public T removeMin() throws EmptyCollectionException {
if (isEmpty())
throw new EmptyCollectionException ("Empty Heap");
T minElement = tree[0];
tree[0] = tree[count-1];
heapifyRemove();
count--;
return minElement; }

51. ArrayHeap - heapifyRemove
/**
* Reorders this heap to maintain the ordering property. */
private void heapifyRemove() {
T temp;
int node = 0;
int left = 1;
int right = 2;
int next;
if ((tree[left] == null) && (tree[right] == null))
next = count;
else if (tree[left] == null)
next = right;
else if (tree[right] == null)
next = left;
else if (((Comparable)tree[left]).compareTo(tree[right]) < 0)
else
temp = tree[node];

52. ArrayHeap – heapifyRemove (cont.)
while ((next < count) && (((Comparable)tree[next]).compareTo
(temp) < 0)) {
tree[node] = tree[next];
node = next;
left = 2*node+1;
right = 2*(node+1);
if ((tree[left] == null) && (tree[right] == null))
next = count;
else if (tree[left] == null)
next = right;
else if (tree[right] == null)
next = left;
else if (((Comparable)tree[left]).compareTo(tree[right]) < 0)
else }
tree[node] = temp;

53. Analysis of Heap Implementations
The addElement operation is O(log n) for both implementations
The removeMin operation is O(log n) for both implementations
The findMin operation is O(1) for both implementations

54. Using Heaps: Heap Sort
Given the ordering property of a heap, it is natural to think of using a heap to sort a list of objects
One approach would be to simply add all of the objects to a heap and then remove them one at a time in ascending order

55. Using Heaps: Heap Sort
Insertion into a heap is O(log n) for any given node and thus would be O(n log n) for n nodes to build the heap
However, it is also possible to build a heap in place using an array
Since we know the relative position of each parent and its children in the array, we simply start with the first non-leaf node in the array, compare it to its children and swap if necessary

56. Using Heaps: Heap Sort
We then work backward in the array until we reach the root
Since at most, this will require us to make two comparisons for each non-leaf node, this approach is O(n) to build the heap