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© 2010 Pearson Addison-Wesley. All rights reserved. Addison Wesley is an imprint of CHAPTER 11: Priority Queues and Heaps Java Software Structures: Designing and Using Data Structures Third Edition John Lewis & Joseph Chase

1-2 © 2010 Pearson Addison-Wesley. All rights reserved. 1-2 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

1-3 © 2010 Pearson Addison-Wesley. All rights reserved. 1-3 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

1-7 © 2010 Pearson Addison-Wesley. All rights reserved. 1-7 The HeapADT (continued) /** * Removes element with the lowest value from this heap. * * @return 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. * * @return a reference to the element with the lowest value in this heap */ public T findMin(); } /** * Removes element with the lowest value from this heap. * * @return 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. * * @return a reference to the element with the lowest value in this heap */ public T findMin(); }

1-10 © 2010 Pearson Addison-Wesley. All rights reserved. 1-10 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

1-11 © 2010 Pearson Addison-Wesley. All rights reserved. 1-11 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

1-15 © 2010 Pearson Addison-Wesley. All rights reserved. 1-15 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

1-16 © 2010 Pearson Addison-Wesley. All rights reserved. 1-16 The removeMin Operation The replacement element is always the last leaf The last leaf is always the last element at level h

1-18 © 2010 Pearson Addison-Wesley. All rights reserved. 1-18 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

1-20 © 2010 Pearson Addison-Wesley. All rights reserved. 1-20 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

1-21 © 2010 Pearson Addison-Wesley. All rights reserved. 1-21 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

1-22 © 2010 Pearson Addison-Wesley. All rights reserved. 1-22 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

1-23 © 2010 Pearson Addison-Wesley. All rights reserved. 1-23 The PriorityQueueNode class /** * PriorityQueueNode represents a node in a priority queue containing a * comparable object, order, and a priority value. * * @author Dr. Lewis * @author Dr. Chase * @version 1.0, 8/19/08 */ public class PriorityQueueNode implements Comparable { private static int nextorder = 0; private int priority; private int order; private T element; /** * Creates a new PriorityQueueNode with the specified data. * * @param obj the element of the new priority queue node * @param prio the integer priority of the new queue node */ /** * PriorityQueueNode represents a node in a priority queue containing a * comparable object, order, and a priority value. * * @author Dr. Lewis * @author Dr. Chase * @version 1.0, 8/19/08 */ public class PriorityQueueNode implements Comparable { private static int nextorder = 0; private int priority; private int order; private T element; /** * Creates a new PriorityQueueNode with the specified data. * * @param obj the element of the new priority queue node * @param prio the integer priority of the new queue node */

1-24 © 2010 Pearson Addison-Wesley. All rights reserved. 1-24 The PriorityQueueNode class (cont.) public PriorityQueueNode (T obj, int prio) { element = obj; priority = prio; order = nextorder; nextorder++; } /** * Returns the element in this node. * * @return the element contained within this node */ public T getElement() { return element; } /** * Returns the priority value for this node. * * @return the integer priority for this node */ public PriorityQueueNode (T obj, int prio) { element = obj; priority = prio; order = nextorder; nextorder++; } /** * Returns the element in this node. * * @return the element contained within this node */ public T getElement() { return element; } /** * Returns the priority value for this node. * * @return the integer priority for this node */

1-25 © 2010 Pearson Addison-Wesley. All rights reserved. 1-25 The PriorityQueueNode class (cont.) public int getPriority() { return priority; } /** * Returns the order for this node. * * @return 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; } public int getPriority() { return priority; } /** * Returns the order for this node. * * @return 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; }

1-26 © 2010 Pearson Addison-Wesley. All rights reserved. 1-26 The PriorityQueueNode class (cont.) /** * Returns the 1 if the current node has higher priority than * the given node and -1 otherwise. * * @param obj the node to compare to this node * @return the integer result of the comparison of the obj node and this * this one */ public int compareTo(PriorityQueueNode obj) { int result; PriorityQueueNode temp = obj; if (priority > temp.getPriority()) result = 1; else if (priority < temp.getPriority()) result = -1; else if (order > temp.getOrder()) result = 1; else result = -1; return result; } /** * Returns the 1 if the current node has higher priority than * the given node and -1 otherwise. * * @param obj the node to compare to this node * @return the integer result of the comparison of the obj node and this * this one */ public int compareTo(PriorityQueueNode obj) { int result; PriorityQueueNode temp = obj; if (priority > temp.getPriority()) result = 1; else if (priority < temp.getPriority()) result = -1; else if (order > temp.getOrder()) result = 1; else result = -1; return result; }

1-27 © 2010 Pearson Addison-Wesley. All rights reserved. 1-27 The PriorityQueue class /** * PriorityQueue demonstrates a priority queue using a Heap. * * @author Dr. Lewis * @author Dr. Chase * @version 1.0, 8/19/08 */ import jss2.*; public class PriorityQueue extends ArrayHeap > { /** * Creates an empty priority queue. */ public PriorityQueue() { super(); } /** * PriorityQueue demonstrates a priority queue using a Heap. * * @author Dr. Lewis * @author Dr. Chase * @version 1.0, 8/19/08 */ import jss2.*; public class PriorityQueue extends ArrayHeap > { /** * Creates an empty priority queue. */ public PriorityQueue() { super(); }

1-29 © 2010 Pearson Addison-Wesley. All rights reserved. 1-29 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

1-30 © 2010 Pearson Addison-Wesley. All rights reserved. 1-30 The HeapNode class /** * HeapNode creates a binary tree node with a parent pointer for use * in heaps. * * @author Dr. Lewis * @author Dr. Chase * @version 1.0, 9/9/2008 */ package jss2; public class HeapNode extends BinaryTreeNode { protected HeapNode parent; /** * Creates a new heap node with the specified data. * * @param obj the data to be contained within the new heap nodes */ HeapNode (T obj) { super(obj); parent = null; } /** * HeapNode creates a binary tree node with a parent pointer for use * in heaps. * * @author Dr. Lewis * @author Dr. Chase * @version 1.0, 9/9/2008 */ package jss2; public class HeapNode extends BinaryTreeNode { protected HeapNode parent; /** * Creates a new heap node with the specified data. * * @param obj the data to be contained within the new heap nodes */ HeapNode (T obj) { super(obj); parent = null; }

1-35 © 2010 Pearson Addison-Wesley. All rights reserved. 1-35 LinkedHeap - getNextParentAdd /** * Returns the node that will be the parent of the new node * * @return the node that will be a parent of the new node */ private HeapNode getNextParentAdd() { HeapNode result = lastNode; while ((result != root) && (result.parent.left != result)) result = result.parent; if (result != root) if (result.parent.right == null) result = result.parent; else { result = (HeapNode )result.parent.right; while (result.left != null) result = (HeapNode )result.left; } else while (result.left != null) result = (HeapNode )result.left; return result; } /** * Returns the node that will be the parent of the new node * * @return the node that will be a parent of the new node */ private HeapNode getNextParentAdd() { HeapNode result = lastNode; while ((result != root) && (result.parent.left != result)) result = result.parent; if (result != root) if (result.parent.right == null) result = result.parent; else { result = (HeapNode )result.parent.right; while (result.left != null) result = (HeapNode )result.left; } else while (result.left != null) result = (HeapNode )result.left; return result; }

1-37 © 2010 Pearson Addison-Wesley. All rights reserved. 1-37 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

1-38 © 2010 Pearson Addison-Wesley. All rights reserved. 1-38 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. * * @return the element with the lowest value in * this heap * @throws 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 { /** * Remove the element with the lowest value in this heap and * returns a reference to it. Throws an EmptyCollectionException * if the heap is empty. * * @return the element with the lowest value in * this heap * @throws 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 {

1-39 © 2010 Pearson Addison-Wesley. All rights reserved. 1-39 LinkedHeap – removeMin (cont.) HeapNode 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; } HeapNode 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; }

1-40 © 2010 Pearson Addison-Wesley. All rights reserved. 1-40 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

1-41 © 2010 Pearson Addison-Wesley. All rights reserved. 1-41 LinkedHeap – getNewLastNode /** * Returns the node that will be the new last node after a remove. * * @return the node that willbe the new last node after a remove */ private HeapNode getNewLastNode() { HeapNode result = lastNode; while ((result != root) && (result.parent.left == result)) result = result.parent; if (result != root) result = (HeapNode )result.parent.left; while (result.right != null) result = (HeapNode )result.right; return result; } /** * Returns the node that will be the new last node after a remove. * * @return the node that willbe the new last node after a remove */ private HeapNode getNewLastNode() { HeapNode result = lastNode; while ((result != root) && (result.parent.left == result)) result = result.parent; if (result != root) result = (HeapNode )result.parent.left; while (result.right != null) result = (HeapNode )result.right; return result; }

1-42 © 2010 Pearson Addison-Wesley. All rights reserved. 1-42 LinkedHeap – heapifyRemove /** * Reorders this heap after removing the root element. */ private void heapifyRemove() { T temp; HeapNode node = (HeapNode )root; HeapNode left = (HeapNode )node.left; HeapNode right = (HeapNode )node.right; HeapNode 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) next = left; else next = right; /** * Reorders this heap after removing the root element. */ private void heapifyRemove() { T temp; HeapNode node = (HeapNode )root; HeapNode left = (HeapNode )node.left; HeapNode right = (HeapNode )node.right; HeapNode 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) next = left; else next = right;

1-43 © 2010 Pearson Addison-Wesley. All rights reserved. 1-43 LinkedHeap – heapifyRemove (cont.) temp = node.element; while ((next != null) && (((Comparable)next.element).compareTo (temp) < 0)) { node.element = next.element; node = next; left = (HeapNode )node.left; right = (HeapNode )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) next = left; else next = right; } node.element = temp; } temp = node.element; while ((next != null) && (((Comparable)next.element).compareTo (temp) < 0)) { node.element = next.element; node = next; left = (HeapNode )node.left; right = (HeapNode )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) next = left; else next = right; } node.element = temp; }

1-44 © 2010 Pearson Addison-Wesley. All rights reserved. 1-44 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

1-45 © 2010 Pearson Addison-Wesley. All rights reserved. 1-45 ArrayHeap /** * ArrayHeap provides an array implementation of a minheap. * * @author Dr. Lewis * @author Dr. Chase * @version 1.0, 9/9/2008 */ package jss2; import jss2.exceptions.*; public class ArrayHeap extends ArrayBinaryTree implements HeapADT { public ArrayHeap() { super(); } /** * ArrayHeap provides an array implementation of a minheap. * * @author Dr. Lewis * @author Dr. Chase * @version 1.0, 9/9/2008 */ package jss2; import jss2.exceptions.*; public class ArrayHeap extends ArrayBinaryTree implements HeapADT { public ArrayHeap() { super(); }

1-46 © 2010 Pearson Addison-Wesley. All rights reserved. 1-46 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

1-48 © 2010 Pearson Addison-Wesley. All rights reserved. 1-48 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; } /** * 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; }

1-49 © 2010 Pearson Addison-Wesley. All rights reserved. 1-49 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

1-50 © 2010 Pearson Addison-Wesley. All rights reserved. 1-50 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. * * @return a reference to the element with the * lowest value in this head * @throws 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; } /** * Remove the element with the lowest value in this heap and * returns a reference to it. Throws an EmptyCollectionException if * the heap is empty. * * @return a reference to the element with the * lowest value in this head * @throws 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; }

1-51 © 2010 Pearson Addison-Wesley. All rights reserved. 1-51 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) next = left; else next = right; temp = tree[node]; /** * 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) next = left; else next = right; temp = tree[node];

1-52 © 2010 Pearson Addison-Wesley. All rights reserved. 1-52 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) next = left; else next = right; } tree[node] = temp; } 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) next = left; else next = right; } tree[node] = temp; }

1-53 © 2010 Pearson Addison-Wesley. All rights reserved. 1-53 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

1-54 © 2010 Pearson Addison-Wesley. All rights reserved. 1-54 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

1-55 © 2010 Pearson Addison-Wesley. All rights reserved. 1-55 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

1-56 © 2010 Pearson Addison-Wesley. All rights reserved. 1-56 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