1 HEAPS & PRIORITY QUEUES Array and Tree implementations.

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HEAPS & PRIORITY QUEUES

Array and Tree implementations

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Priority queuePriority 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

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The definition of “smallest” is up to the programmer (for example,

you might define it by implementing Comparator or Comparable)

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Priority queuePriority queue

If there are several “smallest” elements,

the implementer must decide which to remove first

Remove any “smallest” element (don’t care which)

Remove the first one added

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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

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A priority queue ADTA priority queue ADT

PriorityQueue(): Methods

void add(Comparable o): void add(Comparable o): inserts o into the priority queue

Comparable removeLeast(): Comparable removeLeast(): removes and returns the least element

Comparable getLeast(): Comparable getLeast(): returns (but does not remove) the least element

boolean isEmpty():isEmpty(): returns true iff empty

int size(): int size(): returns the number of elements

void clear():void clear(): discards all elements

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Evaluating implementationsEvaluating implementations

When we choose a data structure,

it is important to look at usage patternsusage 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

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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 searchesfew searches,

we probably don’t want to spend time sorting the array

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Evaluating implementationsEvaluating implementations

For almost all uses of a queue (including a uses of a queue (including a priority queue),priority queue),

we eventually remove everything that we remove everything that we addadd

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TIMINGTIMINGHence,

when we analyze a priority queue, analyze a priority queue,

neither “add” nor “remove” is more important—

we need to look at the timing for timing for (“add + remove”)

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Array implementationsArray 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 inefficient representation – a BigO(N)

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Array implementationsArray 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

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Linked list implementationsLinked list implementations

A priority queue could be implemented as an

unsorted linked list

Adding an element would take O(1)O(1) time (why?)

Removing an element would take (N) time (why?)

Again, inefficient!Again, inefficient!

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SORTED LINKED LISTSORTED LINKED LIST

A priority queue could be implemented as a

sorted linked list

Adding an element Adding an element would take O(n) time (why?)

Removing an element Removing an element would take O(1) time (why?)

Again, an inefficient algorithminefficient algorithm

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Binary tree implementationsBinary tree implementations

A priority queue could be represented as a A priority queue could be represented as a

balancedbalanced binary search tree binary search tree

Insertion and removal could destroy the balanceInsertion and removal could destroy the balance

We would need an algorithm to rebalance the binary tree

Good rebalancing algorithms Good rebalancing algorithms require O(log n)O(log n) time time, but are

complicatedcomplicated

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Heap ImplementatonHeap 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

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Heap is a complete binary tree 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)

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Therefore we would like to find a better algorithm. We want to:

Insert and Remove with a Big O of oneInsert and Remove with a Big O of one

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HEAPS & Priority QueuesHEAPS & 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

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HeapsHeapsHeapsHeaps

A A heapheap is a certain is a certain kind of complete kind of complete binary tree.binary tree.

When a completebinary tree is built,

its first node must bethe root.

When a completebinary tree is built,

its first node must bethe root.

Root

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Complete binary tree.

Right childRight child

The Second and thirdNodes are always the left and right childleft and right childof the root.of the root.

The Second and thirdNodes are always the left and right childleft and right childof the root.of the root.

Left Left childchild

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Complete binary tree.

The next nodesalways fill the next

level from left-to-right.left-to-right.

The next nodesalways fill the next

level from left-to-right.left-to-right.

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Complete binary Complete binary tree.tree.

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A heap is a A heap is a certaincertain kind of kind of complete binary complete binary tree.tree.

Each node in a heapEach node in a heapcontains a key thatcontains a key thatcan be compared tocan be compared toother nodes' keys.other nodes' keys.

Each node in a heapEach node in a heapcontains a key thatcontains a key thatcan be compared tocan be compared toother nodes' keys.other nodes' keys.

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To implement a heap as a priority queueTo implement a heap as a priority queue

A node has the A node has the heap property heap property if it is if it is Equal to or greater Equal to or greater than than as its children as its children

ORORif it is smaller than or equal to its children ( (since smaller since smaller numbers represent higher prioritiesnumbers represent higher priorities))

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Heap: Yellow node has the heap property

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Priority queue: Yellow node has the Priority queue: Yellow node has the heap property - less than its childrenheap property - less than its children

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HeapsHeaps

A heap is a certain kind of complete binary tree.

This example is a max heap.

The "heap property"requires that each

node's key is >= to thekeys of its children

The "heap property"requires that each

node's key is >= to thekeys of its children

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Priority QueuesPriority Queues

A heap can easily implement a priority queue

but what happens with we remove the element remove the element of highest priority? of highest priority?

How do we re-arrange the tree?

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Removing from a heap 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!ALL HEAPS ARE COMPLETE TREES!

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To remove an element at top of treeTo remove an element at top of tree::

Remove the element at location 0

Move the element in the lastIndex to location to location 00, , decrement decrement lastIndex

(lastIndex is at the end of the heap)(lastIndex is at the end of the heap)

Reheap the new root node Reheap the new root node (the one now at location 0)

This is called down-heap bubbling or down-heap bubbling or percolating down percolating down

Down-heap bubblingDown-heap bubbling requires O(log n) time

to remove an element

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Removing the Top of a HeapRemoving the Top of a Heap(max heap)(max heap)


Move the last node onto the root.

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Move the value in the last node into the root.

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Removing the Top of a HeapRemoving the Top of a HeapRemoving the Top of a HeapRemoving the Top of a Heap


Push the out-of-Push the out-of-place node place node downwarddownward, ,

swapping with its swapping with its larger child larger child

until the new node until the new node reaches an reaches an acceptable location.acceptable location.

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Move the last node onto the

root.

Push the out-of-Push the out-of-place node place node downward, downward, swapping with its larger child until the new node reaches an acceptable location.acceptable location.

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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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reheapificationreheapification downwarddownward..reheapificationreheapification downwarddownward..

We stop when:

The children all have keys <= to the out-of-keys <= to the out-of-place node, orplace node, or

The node reaches the The node reaches the leaf.leaf.

The process of pushing the node downward is called: reheapification downward.

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Priority QueuesPriority 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

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An Example of a MAX HeapAn Example of a MAX Heap

The root has largest valueThe root has largest value

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Deleting a node from the heapDeleting a node from the heapWe use a heap a heap to implement a priority queue priority queue

the "next" element ( the root next" element ( the root ) in the queue was removed (pop operation).

We end up with a root with no value

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Root has no valueRoot has no value

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Deleting a node from the heapDeleting a node from the heap

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The obvious questionis: which node can we

use to replace this one?

The obvious questionis: which node can we

use to replace this one?

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If we want this tree to stay complete

the rightmost element in thebottommost level is the

obvious choice

If we want this tree to stay complete

the rightmost element in thebottommost level is the

obvious choice

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We now have aproblem: We have a complete tree butthis is not a heap!

WHY???

This can be solved by applying systematic swap of nodes.

We now have aproblem: We have a complete tree butthis is not a heap!

WHY???

This can be solved by applying systematic swap of nodes.

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1 3We systematically swapthe root with the largerof the children nodes

until no more swaps canbe done

We systematically swapthe root with the largerof the children nodes

until no more swaps canbe done

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The tree has restored itsheap property!

The tree has restored itsheap property!

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Insert operationInsert operation

Another example - the heap below

Recall that while delete causes pairwise swaps from the root to the Recall that while delete causes pairwise swaps from the root to the bottom, bottom,

insert causes pairwise swaps from the bottom to the top. insert causes pairwise swaps from the bottom to the top.

Suppose that we want to insert an element with value 20insert an element with value 2022

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insert (20)insert (20)

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Implementing a HeapImplementing a HeapImplementing a HeapImplementing a Heap

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

An array of dataAn array of data

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


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


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Points about the ImplementationPoints about the ImplementationPoints about the ImplementationPoints 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.manipulate the data.


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Array representation of a heapArray representation of a heap

Left child of node i is 2*i + 1, right child is 2*i + 2

Unless the value is larger than lastIndex lastIndex -therefore no such child

Parent of node i is (i – 1)/2 unless i == 0

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3 12 6 18 14 8

0 1 2 3 4 5 6 7 8 9 10 11 12

lastIndex = location 5

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algorithm for the array implementationalgorithm for the array implementation

Increase lastIndex and put the new value where lastIndex now new value where lastIndex now references -- references --

that is add to the end of the arraythat is add to the end of the array

ReheapReheap the newly added node

This is called up-heap bubbling or percolating up-heap bubbling or percolating up

Up-heap bubbling requires O(log n) time

Adding to heap Adding to heap - - - Implementation- Implementation

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Points about the Points about the ImplementationImplementation

Points about the Points about the ImplementationImplementation

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

[0][0] [1] [2] [3] [4]

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

[0][0] [1] [2] [3] [4] [5]

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If we add 45 to the tree and the array, 45’s parent is at what index?

(i-1)/2 which is 2.

We do a We do a comparison of 45 with its comparison of 45 with its parent and parent and

swap if it is greater than the swap if it is greater than the parent - it is greater than itparent - it is greater than it

[0][0] [1] [2] [3] [4] [5]

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If we add 45 to the tree and to the tree and the array,

45’s parent is at what index? (i-1)/2 which is 2.

We do a comparison and swap.

It is still out of place.It is still out of place.

[0][0] [1] [2] [3] [4] [5]

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45 is still out of place,

so we do another comparison

and swap with 45’s parent which is 42. So what is the terminating condition)(s)?

[0][0] [1] [2] [3] [4] [5]

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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 Summary

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Summary Adding/Deleting

To add an entry to a heap, To add an entry to a heap,

place the new entry at the next available spot, place the new entry at the next available spot,

and perform a reheapification upward.and perform a reheapification upward.

To remove the biggest entryTo remove the biggest entry, ,

move the value in the last node onto the root, and move the value in the last node onto the root, and

perform a reheapification downward.perform a reheapification downward.

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Reheapifying a TreeReheapifying a Tree

Fine,

we know how to delete one element and we know how to delete one element and

restore the heap,restore the heap,

but there are several other operations but there are several other operations involving heapsinvolving heaps

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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 TreeReheapifying a Tree

we can heapify a tree using a quite simple heapify a tree using a quite simple idea.idea.

Given a binary tree T (not necessarily a heap) with root R Given a binary tree T (not necessarily a heap) with root R and left subtree Tand left subtree T

leftleft and T and Trightright,,

if these two subtrees are heaps, if these two subtrees are heaps,

we can make T a heap using the process described in the we can make T a heap using the process described in the previous slide (of applying systematic swaps)previous slide (of applying systematic swaps)

We start the process of heapifying a tree We start the process of heapifying a tree from the lower levelsfrom the lower levels

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Algorithm for Reheapifying TreeAlgorithm for Reheapifying Tree

The algorithm aims at heapifying a tree by The algorithm aims at heapifying a tree by always comparing always comparing three nodes at a time.three nodes at a time.

To To start from the lower levels start from the lower levels

we choose to look at the internal nodes in we choose to look at the internal nodes in

reverse level orderreverse level order

External nodExternal nodes have no children so they es have no children so they are (individually) are (individually) heapsheaps

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HEAPIFYHEAPIFY

void heapify (Heap h) {void heapify (Heap h) {

NodeType n;NodeType n;

for (n = for (n = the internal nodes of h in reverse level-the internal nodes of h in reverse level-orderorder) {) {

Reheapify the heap h Reheapify the heap h starting at node nstarting at node n

}}

}}

void heapify (Heap h) {void heapify (Heap h) {

NodeType n;NodeType n;

for (n = for (n = the internal nodes of h in reverse level-the internal nodes of h in reverse level-orderorder) {) {

Reheapify the heap h Reheapify the heap h starting at node nstarting at node n

}}

}}

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Let change this tree into a heap - A MAX HEAP

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An exampleAn example6

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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 swapped3 and 1 have swappedInternal Nodes in Reverse Level-Order: 3, 12, 17, 4, 6

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An exampleAn example

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Compare 12 with its children and swap 12 and 22 :Compare 12 with its children and swap 12 and 22 :

Internal Nodes in Reverse Level-Order: 3, 22, 17, 4, 6

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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 Compare 22 with 4 22 with 4 and swap : and swap : 4 IS NOW OUT OF PLACE4 IS NOW OUT OF PLACE

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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, 6

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Reheap down 6 : Compare 6 and 11Reheap down 6 : Compare 6 and 11

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6 is in place and the tree is reheaped:

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heapsortheapsort

Heapsort is another algorithm for sorting lists

The idea behind heap sort is that

to sort a list

we can insert the elements of a list into a heap using heapify

Then remove all the elements using remove.

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The idea behind heap sort is that

to sort a list we can insert the elements of a list into a heap using heapify

Then remove all the elements using remove.

It is guaranteed that the largest element is removed first, largest element is removed first,

So by systematically removing the elements

we get the list ordered ( in reverse order)we get the list ordered ( in reverse order)

1 HEAPS & PRIORITY QUEUES Array and Tree implementations.

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