HEAPS

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HEAPS

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Title: HEAPS

1
HEAPS PRIORITY QUEUES

Array and Tree implementations

2
Priority queue

A stack is first in, last out
A queue is first in, first out
A priority queue is least-first-out
The smallest element is the first one removed

The definition of smallest is up to the
programmer (for example,
you might define it by implementing Comparator or
Comparable)

4
Priority queue

If there are several smallest elements,
the implementer must decide which to remove first
Remove any smallest element (dont care which)
Remove the first one added

A priority queue is an
ADT where items are removed based on their
priority
(highest-to-lowest), regardless of the order they
ARE inserted in the structure

6
A priority queue ADT

PriorityQueue() Methods
void add(Comparable o) inserts o into the
priority queue
Comparable removeLeast() removes and returns the
least element
Comparable getLeast() returns (but does not
remove) the least element
boolean isEmpty() returns true iff empty
int size() returns the number of elements
void clear() discards all elements

7
Evaluating implementations

When we choose a data structure,
it is important to look at usage patterns
If we load an array once and do thousands of
searches on it,
we want to make searching fast
so we would probably sort the array

When we choose a data structure,
it is important to look at usage patterns
If we load a huge array and expect to do only a
few searches,
we probably dont want to spend time sorting the
array

9
Evaluating implementations

For almost all uses of a queue (including a
priority queue),
we eventually remove everything that we add

10
TIMING

Hence,
when we analyze a priority queue,
neither add nor remove is more important
we need to look at the timing for (add remove)

11
Array implementations

A priority queue could be implemented as an
unsorted array (with a count of elements)
Adding an element would take O(1) time (why?)
Removing an element would take O(n) time (why?)
Hence, (adding and removing) an element takes
O(n) time
This is an inefficient representation a BigO(N)

12
Array implementations

A priority queue could be implemented as a
sorted array (again, with a counter of
elements)
Adding an element would take O(n) time (why?)
Removing an element would take O(1) time (why?)
So adding and removing) an element takes
O(n) time
Again, this is very inefficient

13
Linked list implementations

A priority queue could be implemented as an
unsorted linked list
Adding an element would take O(1) time (why?)
Removing an element would take (N) time (why?)
Again, inefficient!

14
SORTED LINKED LIST

A priority queue could be implemented as a
sorted linked list
Adding an element would take O(n) time (why?)
Removing an element would take O(1) time (why?)
Again, an inefficient algorithm

15
Binary tree implementations

A priority queue could be represented as a
balanced binary search tree
Insertion and removal could destroy the balance
We would need an algorithm to rebalance the
binary tree
Good rebalancing algorithms require O(log n)
time, but are complicated

16
Heap Implementaton

The concepts of heaps is used to refer to memory
in a computer,
E.g. the part of the memory that provides dynamic
memory allocation ( get it as you need it)
Heap is also used as a name for a special kind
of binary tree

17
Heap is a complete binary tree

A heap is a complete binary tree
with values stored in its nodes such
that
no child has a value bigger than the
value of its parent
(though it can be equal)

Therefore we would like to find a better
algorithm. We want to
Insert and Remove with a Big O of one

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HEAPS Priority Queues

A binary tree represented as a heap facilitates
the operation
to find the maximum element
it is always in the root of the tree
Because finding the node with the maximum value
in a heap is a
BigO(I)
A HEAP is the most efficient structure for
implementing
priority queues

20
Heaps

A heap is a certain kind of complete binary tree.

Root
When a complete binary tree is built, its first
node must be the root.
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Heaps

Complete binary tree.

Right child
Left child
The Second and third Nodes are always the left
and right child of the root.
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Heaps

Complete binary tree.

The next nodes always fill the next level from
left-to-right.
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Heaps

Complete binary tree.

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Heaps
45

A heap is a certain kind of complete binary tree.

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Each node in a heap contains a key that can be
compared to other nodes' keys.
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To implement a heap as a priority queue

A node has the heap property if it is
Equal to or greater than
as its children
OR
if it is smaller than or equal to its children
(since smaller numbers represent higher
priorities)

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Heaps

A heap is a certain kind of complete binary tree.
This example is a max heap.

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The "heap property" requires that each node's key
is gt to the keys of its children
27
Priority Queues

A heap can easily implement a priority queue
but what happens with we remove the element of
highest priority?
How do we re-arrange the tree?

28
Removing from a heap

There are two things we need to consider when
rearranging
We want to end up with a heap which means that
The tree has to be complete
ALL HEAPS ARE COMPLETE TREES!

29
To remove an element at top of tree

Remove the element at location 0
Move the element in the lastIndex to location 0,
decrement
lastIndex
(lastIndex is at the end of the heap)
Reheap the new root node (the one now at location
0)
This is called down-heap bubbling or
percolating down
Down-heap bubbling requires O(log n) time
to remove an element

30
Removing the Top of a Heap(max heap)

Move the last node onto the root.

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Removing the Top of a Heap(max heap)
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Move the value in the last node into the root.

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Removing the Top of a Heap
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Move the last node onto the root.
Push the out-of-place node downward,
swapping with its larger child
until the new node reaches an acceptable location.

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Removing the Top of a Heap
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Move the last node onto the root.
Push the out-of-place node downward, swapping
with its larger child until the new node reaches
an acceptable location.

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Removing the Top of a Heap

Move the last node onto the root.
Push the out-of-place node downward, swapping
with its larger child until the new node reaches
an acceptable location.
Note that we discard half of the tree with each
move downward

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reheapification downward.
42

We stop when
The children all have keys lt to the out-of-place
node, or
The node reaches the leaf.
The process of pushing the node downward is
called reheapification
downward.

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Priority Queues

Another Example
How to delete a node from a Priority Queue
Since it is queue - first in first out,
we remove the first one
The Root

37
An Example of a MAX HeapThe root has largest
value
38
Deleting a node from the heap

We use a heap to implement a priority queue
the "next" element ( the root ) in the queue was
removed (pop operation).
We end up with a root with no value

Root has no value
39
Deleting a node from the heap
The obvious question is which node can we use to
replace this one?
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Deleting a node from the heap
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If we want this tree to stay complete the
rightmost element in the bottommost level is the
obvious choice
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Deleting a node from the heap
We now have a problem We have a complete tree
but this is not a heap! WHY??? This can be
solved by applying systematic swap of nodes.
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Deleting a node from the heap
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We systematically swap the root with the
larger of the children nodes until no more swaps
can be done
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Deleting a node from the heap
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Deleting a node from the heap
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Deleting a node from the heap
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The tree has restored its heap property!
46
Insert operation

Another example - the heap below
Recall that while delete causes pairwise swaps
from the root to the bottom,
insert causes pairwise swaps from the bottom to
the top.
Suppose that we want to insert an element with
value 20

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insert (20)
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insert (20)
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insert (20)
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insert (20)
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Implementing a Heap

We will store the data from the nodes in a
partially-filled array.

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An array of data
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Implementing a Heap

Data from the root goes in the first
location of the
array.

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An array of data
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Implementing a Heap

Data from the next row goes in the next two array
locations.

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An array of data
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Implementing a Heap

Data from the next row goes in the next two array
locations.

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An array of data
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Implementing a Heap

Data from the next row goes in the next two array
locations.

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An array of data
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Points about the Implementation

The links between the tree's nodes are not
actually stored as pointers, or in any other way.
The only way we "know" that "the array is a tree"
is from the way we manipulate the data.

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An array of data
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Array representation of a heap
lastIndex location 5

Left child of node i is 2i 1, right child is
2i 2
Unless the value is larger than lastIndex
-therefore no such child
Parent of node i is (i 1)/2 unless i 0

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Adding to heap - - Implementation

algorithm for the array implementation
Increase lastIndex and put the new value where
lastIndex now references --
that is add to the end of the array
Reheap the newly added node
This is called up-heap bubbling or percolating up
Up-heap bubbling requires O(log n) time

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Points about the Implementation
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If you know the index of a node, then it is easy
to figure out the
indexes of that node's parent and children.

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Points about the Implementation
42

If we add 45 to the tree and to the array,
45s parent is at what index? i ?
(i-1)/2 2

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Points about the Implementation

If we add 45 to the tree and the array, 45s
parent is at what index?
(i-1)/2 which is 2.
We do a comparison of 45 with its parent and
swap if it is greater than the parent - it is
greater than it

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Points about the Implementation

If we add 45 to the tree and to the tree and the
array,
45s parent is at what index?
(i-1)/2 which is 2.
We do a comparison and swap.
It is still out of place.

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Points about the Implementation
45

45 is still out of place,
so we do another comparison
and swap with 45s parent which is 42.
So what is the terminating condition)(s)?

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Summary

A heap is a complete binary tree, where
the entry at each node is greater than or equal
to the entries in its children.
OR
the entry at each node is less than or equal to
the entries in its children.

Summary Adding/Deleting
To add an entry to a heap,
place the new entry at the next available spot,
and perform a reheapification upward.
To remove the biggest entry,
move the value in the last node onto the root,
and
perform a reheapification downward.

66
Reheapifying a Tree

Fine,
we know how to delete one element and
restore the heap,
but there are several other operations involving
heaps

If we find a systematic way of heapifying a tree,
we could perform any operation
because we can execute the procedure heapify

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Reheapifying a Tree

we can heapify a tree using a quite simple idea.
Given a binary tree T (not necessarily a heap)
with root R and left subtree Tleft and Tright,
if these two subtrees are heaps,
we can make T a heap using the process described
in the previous slide (of applying systematic
swaps)
We start the process of heapifying a tree from
the lower levels

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Algorithm for Reheapifying Tree

The algorithm aims at heapifying a tree by
always comparing three nodes at a time.
To start from the lower levels
we choose to look at the internal nodes in
reverse level order
External nodes have no children so they are
(individually) heaps

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HEAPIFY

void heapify (Heap h)
NodeType n
for (n the internal nodes of h in reverse
level-order)
Reheapify the heap h starting at node n

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Let change this tree into a heap - A MAX
HEAP
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An example
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Internal Nodes in Reverse Level-Order are 1,
12, 17, 4, 6 , start with the leftmost leaf
compare 3 To 1
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3 and 1 have
swapped Internal Nodes in Reverse Level-Order 3,
12, 17, 4, 6
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An example
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Compare 12 with its children and swap 12 and
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Internal Nodes in Reverse Level-Order 3, 22, 17,
4, 6

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An example
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Compare 17 with its children and no changes
necessary 3, 22, 17, 4, 6 ---
NEXT COMPARE 4 AND 22
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Compare 22 with 4 and swap 4 IS NOW OUT
OF PLACE
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An example
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Compare 12 and 4 and swap, NEXT compare 22 to 6
3, 12, 17, 22, 6
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Heapify 6 downwards until it finds its proper
location 3, 12, 17,
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Reheap down 6 Compare 6 and 11
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An example
6 is in place and the tree is
reheaped
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An example
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