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
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
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