ADSA : Heaps/11 1

241-423 Advanced Data Structures and Algorithms

Objectives– implement heaps (a kind of array-based complete

binary tree), heap sort, priority queues

Semester 2, 2012-2013

11. Heaps

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Contents1. Tree Terminology

2. Binary Trees

3. The Comparator Interface

4. Generalized Array Sorting

5. Heaps

6. Sorting with a Max Heap

7. API for the Heaps Class

8. Priority Queue Collection

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1. Tree Terminology

A

JI

HGFE

DCB

ro o t

parent

ch ild

subtree

leaf no de

sib ling

in ter io r (in ternal) no de

continued

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• A path between a parent node X0 anda subtree node N is a sequence of nodes P = X0, X1, . . ., (Xk is N)

– k is the length of the path

– each node Xi is the parent of Xi+1 for 0 i k-1

continued

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• The level of a node is the length of the path from root to that node.

• The height of a tree is the maximum level in the tree.

continued

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2. Binary Trees

• In a binary tree, each node has at most two children.

continued

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• Each node of a binary tree defines a left and a Each node of a binary tree defines a left and a right subtree. Each subtree is a tree.right subtree. Each subtree is a tree.

Left child of T

Right child of T

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Height of a Binary Tree Nodeo Node N starts a subtree TN , with TL and TR the

left and right subtrees. Then:

-1 if TN is empty1+max( height(TL), height(TR)) if TN not empty

height(N) = height(TN) ={

continued

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degenerate binary tree (a list)

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A Complete Binary Tree• A complete binary tree of height h has all its

nodes filled through level h-1, and the nodes at depth h run left to right with no gaps.

continued

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• An array arr[] can be viewed as a complete binary tree if:– the root is stored in arr[0]– the level 1 children are in arr[1], arr[2]– the level 2 children are in arr[3], arr[4], arr[5],

arr[6]– etc.

An Array-based Complete Binary Tree

continued

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Integer[] arr = {5, 1, 3, 9, 6, 2, 4, 7, 0, 8};

continued

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o For arr[i] in an n-element array‑based complete binary tree:

Left child of arr[i] is arr[2*i + 1]; orundefined if (2*i + 1) n

Right child of arr[i] is arr[2*i + 2]; orundefined if (2*i + 2) n

Parent of arr[i] is arr[(i-1)/2]; orundefined if i = 0

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interface COMPARATOR<T> java.util

Methods

int compare(T x, T y) Compares its two arguments for order. Returns a negative integer, zero, or a positive integer to specify that x is less than, equal to, or greater than y.

boolean equals(Object obj) Returns true if this object equals obj and false otherwise.

3. The Comparator Interface

continued

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3.1. Comparator vs. Comparable

• Comparator.compare(T x, T y)– comparison is done between two objects, x and y– the method is implemented outside the T class

• Comparable.compareTo(T y)– comparison is done between 'this' object and

object y– the method is implemented by the T class

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3.2. Benefits of Comparator

The comparison code is in a separate place from the other code for a class.

1. This means that the same objects can be compared in different ways by defining different Comparators– see the two comparators for circles

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• 2. A Comparator can be passed to a sort method as an argument to control the ordering of objects.• this means that the same sort method can sort in

different ways depending on its Comparator argument.

• e.g. sorting an array into ascending or descending order

• e.g. sorting data into max heaps or min heaps

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3.3. Comparing Circlespublic class Circle{ private double xCenter, yCenter; private double radius;

public Circle(double x, double y, double r) { xCenter = x; yCenter = y; radius = r; }

public double getX() { return xCenter; }

:

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public double getY()

{ return yCenter; }

public double getRadius()

{ return radius; }

public double getArea()

{ return Math.PI * radius * radius; }

: // more methods, but no comparison function

} // end of Circle class

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Comparing Circle Radiipublic class RadiiComp implements Comparator<Circle>{ public int compare(Circle c1, Circle c2) { double radC1 = c1.getRadius(); double radC2 = c2.getRadius();

// returns < 0 if radC1 < radC2, // 0 if radC1 == radC2, // > 0 if radC1 > radC2 return (int)(radC1 – radC2); } // end of compare()

// equals() is inherited from Object superclass

} // end of RadiiComp class

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Comparing Circle Positionspublic class PosComp implements Comparator<Circle>{ public int compare(Circle c1, Circle c2) { double c1Dist = (c1.getX() * c1.getX()) + (c1.getY() * c1.getY()); double c2Dist = (c2.getX() * c2.getX()) + (c2.getY() * c2.getY());

// returns < 0 if c1Dist < c2Dist, // 0 if c1Dist == c2Dist, // > 0 if c1Dist > c2Dist return (int)(c1Dist - c2Dist); } // end of compare()

// equals() is inherited from Object superclass} // end of PosComp class

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Comparing Circles in Two Ways

Circle c1 = new Circle(0, 0, 5);Circle c2 = new Circle(3, 2, 7);

RadiiComp rComp = new RadiiComp();if (rComp.compare(c1, c2) < 0) System.out.println("Circle 1 is smaller than circle 2");

PosComp pComp = new PosComp();if (pComp.compare(c1, c2) < 0) System.out.println("Circle 1 is nearer to origin than circle 2");

Circle 1 is smaller than circle 2Circle 1 is nearer to origin than circle 2

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3.4. Comparators for Sorting

• The Less and Greater Comparator classes make sorting and searching functions more flexible– see selectionSort() in the next few slides

• Less and Greater will also be used to create min and max heaps.

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The Less Comparator

import java.util.Comparator;

// the < Comparatorpublic class Less<T> implements Comparator<T>{ public int compare(T x, T y) { return ((Comparable<T>)x).compareTo(y); }}

uses T's compareTo() to compare x and y

x Less y== x.compareTo y== x < y

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Using Less

Comparator<Integer> lessInt = new Less<Integer>();

Integer a = 3, b = 5;

if (lessInt.compare(a, b) < 0)

System.out.println(a + " < " + b); 3 < 5

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The Greater Comparator

// the > Comparatorpublic class Greater<T> implements Comparator<T>{ public int compare(T x, T y) {

return -((Comparable<T>)x).compareTo(y); }}

uses T's compareTo() to compare x and yx Greater y

== -(x.compareTo y)== -(x < y)== x > y

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Using Greater

Comparator<Integer> greaterInt = new Greater<Integer>();

Integer a = 9, b = 5;

if (greaterInt.compare(a, b) < 0)

System.out.println(a + " > " + b); 9 > 5

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4. Generalized Array Sorting

• A single selectionSort() method can sort array in different ways by being passed different Comparator arguments.

• The Less or Greater Comparators let the method sort an array of objects into ascending or descending order.

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Selection Sort with Comparator

// new version adds a Comparator parameterpublic static <T> void selectionSort(T[] arr, Comparator<? super T> comp){ int smallIndex;

int n = arr.length; for (int pass = 0; pass < n-1; pass++) { // scan the sublist starting at index smallIndex = pass; :

Compare with selectionSort() in Part 2which uses Comparable.

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// j traverses the sublist arr[pass+1] to arr[n-1] for (int j = pass+1; j < n; j++) // if smaller element found, assign posn to smallIndex if (comp.compare(arr[j], arr[smallIndex]) < 0) smallIndex = j;

// swap smallest element into arr[pass] T temp = arr[pass]; arr[pass] = arr[smallIndex]; arr[smallIndex] = temp; }

} // end of selectionSort()

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Example

String[] arr = {"red", "green", "blue", "yellow", "teal", "orange"};

Less<String> lessComp = new Less<String>();

Greater<String> gtComp = new Greater<String>();

Arrays.selectionSort(arr, lessComp);

System.out.println("Less sort: " + Arrays.toString(arr));

Arrays.selectionSort(arr, gtComp);

System.out.println("Greater sort: " + Arrays.toString(arr));

Less sort: [blue, green, orange, red, teal, yellow]

Greater sort: [yellow, teal, red, orange, green, blue]

// ascending

// descending

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5. Heaps• A maximum heap is an array‑based complete

binary tree in which the value of a parent is ≥ the value of both its children.

continued

0

555052 251011 5 2022

1 2 3 4 5 6 7 8

lvl 0 1 2 3

there’s no orderingwithin a level

(between siblings)

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0

401530 10

1 2 3

lvl 0 1 2

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• A minimum heap uses the relation ≤.

continued

0

5 1050 112052552522

1 2 3 4 5 6 7 8

lvl 0 1 2 3

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0

101530 40

1 2 3

lvl 0 1 2

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A max heap will be ordered using the Greater comparator.– I’ll focus on max heaps in these notes

A min heap will be ordered using the Less comparator.

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Heap Uses• Heapsort

– one of the best sorting methods• in-place; no quadratic worst-case

• Selection algorithms– finding the min, max, median, k-th element in

sublinear time

• Graph algorithms– Prim's minimal spanning tree; – Dijkstra's shortest path

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Max Heap Operations

o Inserting an element: pushHeap()

o Deleting an element: popHeap()o most of the work is done by calling adjustHeap()

o Array --> heap conversion: makeHeap()o most of the work is done by calling adjustHeap()

o Heap sorting: heapSort()o utilizes makeHeap() then popHeap()

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5.1. Inserting into a Max Heap

• Assume that the array is a maximum heap.– a new item will enter the array at index last with

the heap expanding by one element

continued

pushHeap()

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• Insert an item by moving the nodes on the path of parents down one level until the item is correctly placed as a parent in the heap.

path of parents

insert 50

continued

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• At each step, compare item with parent– if item is larger, move parent down one level

• arr[currPos] = parent;• currPos = the parent index;

• Stop when parent is larger than item– assign item to the currPos position

• arr[currPos] = item;

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pushHeap()

public static <T> void pushHeap(T[] arr, int last, T item, Comparator<? super T> comp){ // insert item into the heap arr[] // assume that arr[0] to arr[last-1] // are in heap order

// currPos is the index that moves up the // path of parents int currPos = last; int parentPos = (currPos-1)/2; // see slide 16 :

Depending on the comparator,pushHeap() can work with maxor min heaps.

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// move up parents path to the root while (currPos != 0) { // compare target and parent value if (comp.compare(item,arr[parentPos]) < 0) { arr[currPos] = arr[parentPos]; // move data from parent --> current currPos = parentPos; parentPos = (currPos-1)/2; // get next parent } else // heap condition is ok break; }

arr[currPos] = item; // put item in right location} // end of pushHeap()

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5.2. Deleting from a Heap

• Deletion is normally restricted to the root only– remove the maximum element (in a max heap)

• To erase the root of an n‑element heap:– exchange the root with the last element (the one at

index n‑1); delete the moved root– filter (sift) the new root down to its correct

position in the heap

popHeap()

call adjustHeap()

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Deletion Example• Delete 63

– exchange with 18; remove 63– filter down 18 to correct position

continued

for a Max Heap

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1 8

35

3 882 51 0

4 03 0

a r r [ 0 ]

a r r [ 1 ] a r r [ 2 ]

a r r [ 4 ]

a r r [ 8 ]a r r [ 7 ]

a r r [ 5 ]a r r [ 3 ] a r r [ 6 ]

F ilte r d o w n 1 8

continued(63) removed

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• Move 18 down:– smaller than 30 and 40; swap with 40

18

18

38

continued

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• Move 18 down:– smaller than 38; swap with 38

18

3818

• Stop since 18 is now a leaf node.

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Filter (Sift) Down a Max Heap

• Move root value down the tree:– compare value with its two children– if value < a child then heap order is wrong– select largest child and swap with value

– repeat algorithm but with new child

– continue until value ≥ both children or at a leaf

adjustHeap()

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// filter a value down the heappublic static <T> void adjustHeap(T[] arr, int first, int last, Comparator<? super T> comp){ int currentPos = first; // start at first T value = arr[first]; // filter value down the heap

// compute the left child index int childPos = 2*currentPos + 1; // see slide 16 :

adjustHeap() Depending on the comparator,adjustHeap() can work with maxor min heaps.

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// scan down path of children while (childPos < last) { // compare the two children; select bigger if ((childPos+1 < last) && comp.compare(arr[childPos+1], arr[childPos]) < 0) childPos = childPos + 1; // since cp+1 < cp

// compare selected child with value if (comp.compare(arr[childPos],value) < 0) { // swap child and value arr[currentPos] = arr[childPos]; arr[childPos] = value; :

index of right child is childPos+1

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// update indices to continue the scan currentPos = childPos; childPos = 2*currentPos + 1; // left child index } else // value in right position break; }} // end of adjustHeap()

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Deletion method: popHeap()

• Exchange the root with the last value in the heap (arr[last-1])

• Call adjustHeap() to filter value down heap– heap now has index range [0, last-1)

• Return the original root value

see slide 47 for more details

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// delete the maximum (or minimum) element in the heap // and return its valuepublic static <T> T popHeap(T[] arr, int last, Comparator<? super T> comp){ // value that is removed from the heap T temp = arr[0];

// exchange last element in heap with the root value arr[0] = arr[last-1]; arr[last-1] = temp;

// filter the value down over the range (0, last-1) adjustHeap(arr, 0, last-1, comp); return temp;}

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5.3. Using Heapsimport ds.util.Heaps;import ds.util.Greater;import ds.util.Less;

public class UsingHeaps{ public static void main(String[] args) { // integer array, and arrA and arrB heaps Integer[] intArr = {15, 29, 52, 17, 21, 39, 8}; Integer[] heapArrA = new Integer[intArr.length], Integer[] heapArrB = new Integer[intArr.length];

// comparators for maximum and minimum heaps Greater<Integer> gtComp = new Greater<Integer>(); Less<Integer> lessComp = new Less<Integer>(); :

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// load intArr into heapArrA to form a maximum heap // and into heapArrB to form a minimum heap for (i = 0; i < intArr.length; i++) { Heaps.pushHeap(heapArrA, i, intArr[i], grComp); //max Heaps.pushHeap(heapArrB, i, intArr[i], lessComp);//min }

// print heapArrA System.out.println("Display maximum heap:"); System.out.println( Heaps.displayHeap(heapArrA, heapArrA.length, 2));

// draw heapArrB Heaps.drawHeap(heapArrB, heapArrB.length, 2); :

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// pop minimum value Integer minObj = Heaps.popHeap(heapArrB, heapArrB.length, less); System.out.println("\nMinimum value is " + minObj);

// draw heapArrB before and after popHeap() // the index range is 0 to heapArrB.length-1 Heaps.drawHeap(heapArrB, heapArrB.length-1, 2); } // end of main()

} // end of UsingHeaps class

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remove 8

minimum heap, heapArrB

max heap, heapArrA

Execution

continued

displayHeap() drawHeap()

ADSA : Heaps/11 62minimum heap, heapArrB

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5.4. Complexity of Heap Operations

• A heap stores elements in an array-based complete tree.

• pushHeap() reorders elements in the tree by moving up the path of parents.

• adjustHeap() reorders elements in the tree by moving down the path of the children.– their cost depends on path length

continued

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• Assuming the heap has n elements, the maximum length for a path between a leaf node and the root is log2n

• since the tree is balanced

• The runtime efficiency of the algorithms is O(log2 n)

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5.5. Heapifying O(n)!

• Transforming an array into a heap is called "heapifying the array".

• Turn an n‑element array into a heap by filtering down each parent in the tree – begin with the last parent at index (n-2)/2– end with the root node at index 0

continued

makeHeap()

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Integer[] arr = {9, 12, 17, 30, 50, 20, 60, 65, 4, 19};

The grey nodesare the parents.

Adjust in order: 50, 30, 17, 12, 9

continued

Max Heap Creation

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// arrange array elements into a heappublic static <T> void makeHeap(T[] arr, Comparator<? super T> comp){ int lastPos = arr.length; // heap size int heapPos = (lastPos - 2)/2; // see slide 16 // the index of the last parent

// filter down every parent in order // from last parent up to root while (heapPos >= 0) { adjustHeap(arr, heapPos, lastPos, comp); heapPos--; }} // end of makeHeap()

Depending on the comparator,makeHeap() can create maxor min heaps.

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6. Sorting with a Max Heap

• If the array is a maximum heap, it has an efficient sorting algorithm:– For each iteration i, the largest element is arr[0].– Exchange arr[0] with arr[i] and then reorder the

array so that elements in the index range [0, i) are a heap.

– This is done by popHeap(), which is O(log2n)

heapSort()

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Max Heap Sort

continued

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the max heap sort is into ascending order

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Heapsort

• Heap sort is a modified version of selection sort for an array that is a heap.– for each i = n, n-1, ..., 2, call popHeap() which

pops arr[0] from the heap and assign it at index i-1.

• A maximum heap is sorted into ascending order

• A minimum heap is sorted into descending order.

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public static <T> void heapSort(T[] arr, Comparator<? super T> comp){ Heaps.makeHeap(arr, comp); // "heapify" arr[]

int n = arr.length; // iteration through arr[n-1] ... arr[1] for (int i = n; i > 1; i--) { // popHeap() moves largest (smallest) to arr[n-1] Heaps.popHeap(arr, i, comp); }} // end of heapSort()

Depending on the comparator,heapSort() can work with maxor min heaps.

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ExampleInteger[] arr = {7,1,9,0,8,2,4,3,6,5};

// make a max heap

Arrays.heapSort(arr, new Greater<Integer>() );

System.out.println("Sort ascending: " +

Arrays.toString(arr));

// make a min heap

Arrays.heapSort(arr, new Less<Integer>() );

System.out.println("Sort decending: " +

Arrays.toString(arr));

Sort ascending: [0,1,2,3,4,5,6,7,8,9]

Sort descending: [9,8,7,6,5,4,3,2,1,0]

// max heap

// min heap

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• The worst case running time of makeHeap() is closer to O(n), not O(n log2 n).

• During the second phase of the heap sort, popHeap() executes n-1 times. Each operation has efficiency O(log2 n).

• The worst-case complexity of the heap sort is O(n) + O(n log2 n) = O(n log2 n).

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7. API for the Heaps Classclass Heaps ds.util

Static Methods

static <T> void adjustHeap(T[] arr, int first, int last, Comparator<? super T> comp) Filters the array element arr[first] down the heap. Called by makeHeap() to convert an array to a heap.

static String displayHeap(Object[] arr, int n, int maxCharacters) Returns a string that presents the array elements in the index range [0, n) as a complete binary tree.

static void drawHeap(Object[] arr, int n, int maxCharacters) Provides a graphical display of a heap as a complete binary tree

static void drawHeaps (Object[] arr, int n, int maxCharacters) Provides a graphical display of a heap as a complete binary tree. Keeps the window open for new frames created by subsequent calls to drawHeap() or drawHeaps().

continued

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class Heaps (continued) ds.util

Static Methods

static <T> void heapSort(T[] arr, Comparator<? super T> comp) Sorts the array in the ordered specified by the comparator; if comp is Greater, the array is sorted in ascending order; if comp is Less, the array is sorted in descending order.

static <T> void makeHeap(T[] arr, Comparator<? super T> comp) The method that is responsible for converting an array to a heap. At each element that may not satisfy the heap property, the algorithm reorders elements in the subtree by calling adjustHeap()

static <T> T popHeap(T[] arr, int last, Comparator<? super T> comp) The index range [0, last) is a heap. Deletes the optimum element from the heap, stores the deleted value at index last-1, and returns the value. The index range [0,last-1) is a heap with one less element.

static <T> void pushHeap(T[] arr, int last, T item, Comparator<? super T> comp) Inserts item into a heap that consists of the array elements in the index range [0,last-1). The elements in the index range [0, last) become a heap.

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8. Priority Queue Collectiono In a priority queue, all the elements have priority

values.o A deletion always removes the element with the

highest priority.

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o Two types of priority queues:– maximum priority queue

• remove the largest value first• what I’ll be using

– minimum priority queue• remove the smallest value first

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PQueue Interface

o The generic PQueue resembles a queue with the same method names.

interface PQueue<T> ds.util boolean isEmpty()

Returns true if the priority queue is empty and false otherwise.

T peek() Returns the value of the highest-priority item. If empty, throws a NoSuchElementException.

continuedcontinued

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interface PQueue<T> ds.util T pop()

Removes the highest priority item from the queue and returns its value. If it is empty, throws a NoSuchElementException.

void push(T item) Inserts item into the priority queue.

int size() Returns the number of elements in this priority queue

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HeapPQueue Class Example// create a max priority queue of StringsHeapPQueue<String> pq = new HeapPQueue<String>();pq.push("green");pq.push("red");pq.push("blue");

// output the size, and element with highest prioritySystem.out.println(pq.size() + ", " + pq.peek());

// use pop() to empty the pqueue and list elements // in decreasing priority orderwhile ( !pq.isEmpty() ) System.out.print( pq.pop() + " ");

3, redred green blue

The max priority for Stringsis z --> a order

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Implementing PQueue

• We can use a heap (the HeapPQueue class) to implement the PQueue interface.

• The user can specify either a Greater (max heap) or Less (min heap) comparator • this dictates whether deletion removes the maximum or

the minimum element from the collection• max heap --> maximum priority queue

• min heap --> minimum priority queue

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The HeapPQueue Classpublic class HeapPQueue<T> implements PQueue<T>{ private T[] heapElt; // the heap of queue elems private int numElts; // number of elements in queue private Comparator<T> comp;

public HeapPQueue() // create an empty maximum priority queue { comp = new Greater<T>(); // so ordered big small numElts = 0; heapElt = (T[]) new Object[10]; }

// more methods. . .

} // end of HeapPQueue class

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public T peek()// return the highest priority item; O(1){ // check for an empty heap if (numElts == 0) throw new NoSuchElementException( "HeapPQueue peek(): empty queue"); return heapElt[0]; // return heap root} // end of peek()

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// erase the highest priority item and return itpublic T pop() // O(log n){ // check for an empty priority queue if (numElts == 0) throw new NoSuchElementException( "HeapPQueue pop(): empty queue");

// pop heap and save return value in top T top = Heaps.popHeap(heapElt, numElts, comp);

numElts--; // heap has one less element return top;} // end of pop()

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public void push(T item)// insert item into the priority queue; O(log n){ // if full then double the capacity if (numElts == heapElt.length) enlargeCapacity();

// insert item into the heap Heaps.pushHeap(heapElt, numElts, item, comp); numElts++;} // end of push()

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Heapify AnalysisHeapify Analysis

• Heapify performs better than O(n log n) Heapify performs better than O(n log n) because most of the node adjusting is done because most of the node adjusting is done over heights less than log n.over heights less than log n.

level height0

1

2

3

3 = log n

2

1

0

Not examinable

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Cost of Heap BuildingCost of Heap Building

less than O(log n)most of the time

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The summation converges to 2. Why?

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Proof of ConvergenceProof of Convergence

• The sum of a geometric series:The sum of a geometric series:

• Take the derivatives of both sides:Take the derivatives of both sides:

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