1. DATA STRUCTURES AND ALGORITHMS
LAB 10
Bianca Tesila
FILS, April 2014

2. OBJECTIVES
 Heap
 Heap Sort

3. HEAP: INTRODUCTION
 What is a heap - a data structure that:
 is stored as an array in memory
 can be represented/viewed as an almost complete
binary tree
 respects the heap property

4. HEAP: INTRODUCTION
 What is an almost complete binary tree - a
binary tree such that:
 all the leaves are at the bottom level or the bottom 2
levels
 all the leaves are in the leftmost possible positions
 all the levels are completely filled with nodes (except
for the bottom level)

5. HEAP: INTRODUCTION
The heap property:
 Max heaps
 For each node (except for the root): A[i] <= A[parent(i)]
 Conclusion: The root (A[1]) contains the largest element
 Min heaps
 For each node (except the root): A[i] >= A[parent(i)]
 Conclusion: The root (A[1]) contains the smallest element
!! Heap Sort uses max heaps

6. HEAP: DATA STRUCTURE
 Store the elements in an array - the advantage
of this method over using the usual pointers and
nodes is that there is no wasting of space due to
storing two pointer fields in each node.
 Start by taking the nodes level by level from the
top down, left to right. Then, store them in an
array.

7. HEAP: DATA STRUCTURE
 An array A[0..n] can be represented as a binary
tree:
 A[0] –root of the tree
 Parent of A[i] = parent(i) = A[(i-1)/2]
 Left child of A[i] = left(i) = A[2*i+1]
 Right child of A[i] = right(i) = A[2*(i+ 1)]
 Height of the heap: Θ(log n)
 Number of nodes from the root to the farthest leaf

8. HEAP: BASIC OPERATIONS
Insert a new element:
 add it as the last element in the heap
 however, the heap property may be broken
 if the added element is larger than its parent, you
need to find its correct position in the heap => filter
up
Filter up an element:
 compare it with its parent
 if the parent is smaller than it, stop
 else, swap it with the parent and continue

9. HEAP: BASIC OPERATIONS
Insert E in this heap
1. Place E in the next available position:
2. Filter up E:

10. HEAP: BASIC OPERATIONS
Delete ( extract min/max):
 simply remove A[0] (the root element)
 however, the heap is broken as it has no root element
 need to find a new root
 move the last element in the heap as the new root
 now, the heap property is lost (almost certainly) =>
need to filter down
Filter down an element:
 compare it to its children
 if it is larger than both children, stop
 else, swap it with the largest child and continue

11. HEAP: BASIC OPERATIONS
Remove C from this heap:

12. HEAP SORT
 convert the array you want to sort into a heap
 remove the root item (the smallest), readjust the
remaining items into a heap, and place the
removed item at the end of the heap (array)
 remove the new item in the root (the second
smallest), readjust the heap, and place the
removed item in the next to the last position and
so on

13. HEAP SORT
Example: HeapSort.pdf

14. HEAP
!! Exercise:
Implement the functions for returning the parent and the childs of a
given node, using the following signatures:
template <typename T>
int Heap<T>::parent(int index)
{
// TODO
}
template <typename T>
int Heap<T>::leftSubtree(int index)
{
// TODO
}
template <typename T>
int Heap<T>::rightSubtree(int index)
{
// TODO
}
!! Use heap.h

15. HEAP SORT
!! Exercise:
Implement the heapsort and test it for the
following array:
25, 17, 36, 2, 3, 100, 1, 19, 17
!! Use heap.h

16. HOMEWORK
Finish all the lab assignments.