Heaps Chapter 6 1 Objectives Understand what a priority queue is - - PowerPoint PPT Presentation

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Heaps Chapter 6 1 Objectives Understand what a priority queue is - - PowerPoint PPT Presentation

Jan 31, 2023 143 likes •540 views

Heaps Chapter 6 1 Objectives Understand what a priority queue is Identify the applications where a priority queue can be used Analyze the running time and space overhead of the heap data structure Use heaps to solve the top-k problem 2

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SLIDE 1

Heaps

Chapter 6

1

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SLIDE 2

Objectives

Understand what a priority queue is Identify the applications where a priority queue can be used Analyze the running time and space

  • verhead of the heap data structure

Use heaps to solve the top-k problem

2

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SLIDE 3

Top-K

Given n numbers, find the k largest numbers k=1 k=2 k=n/100 (Top 1%) Sort and select Use a BST (or AVL tree) to keep track of the top-k items Is there something more efficient?

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SLIDE 4

Top-K

Given n numbers, find the k largest numbers k=1 k=2 k=n/100 (Top 1%) Sort and select O(n log n) Use a BST (or AVL tree) to keep track of the top-k items O(n log k) Is there something more efficient?

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SLIDE 5

Regular Queues

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HPC

Jobs

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SLIDE 6

Priority Queue

Airport check-in

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HPC

! Jobs

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SLIDE 7

Priority Queue ADT

Queue

Enqueue Dequeue Front Empty?

Priority Queue

Insert DeleteMin/Max PeekMin/Max Empty? IncreaseKey DecreaseKey Remove

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SLIDE 8

Priority Queue Implementation

Straight-forward implementations? Unsorted list Sorted list BST/AVL

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Unsorted list Sorted list BST/AVL Insert PeekMin DeleteMin IncreaseKey DecreaseKey

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SLIDE 9

Priority Queue Implementation

Straight-forward implementations Unsorted list Sorted list BST/AVL

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Unsorted list Sorted list BST/AVL Insert O(1) O(n) O(log n) PeekMin O(n) O(1) O(log n) DeleteMin O(n) O(1) O(log n) IncreaseKey O(1)* O(n) O(log n) DecreaseKey O(1)* O(n) O(log n)

* Assuming the record is already located

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SLIDE 10

Heap Implementation

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13 21 16 Example of a MinHeap 24 31 19 68 65 26 32

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SLIDE 11

Heap Implementation

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68 32 65 Example of a MaxHeap 24 31 19 16 13 21 26

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SLIDE 12

Heap Properties

A complete binary tree For any internal node n

n.Key ≤ n.child.Key (Min Heap) n.Key ≥ n.child.Key (Max Heap)

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SLIDE 13

MinHeap Insertion

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13 21 16 24 31 19 68 65 26 32 Insert 15 15

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SLIDE 14

MinHeap Insertion

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13 21 16 24 15 19 68 65 26 32 Insert 15 31

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SLIDE 15

MinHeap Insertion

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13 15 16 24 21 19 68 65 26 32 Insert 15 31

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SLIDE 16

MinHeap Insertion

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13 15 16 24 21 19 68 65 26 32 Insert 15 31

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SLIDE 17

MinHeap Insertion

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13 15 16 24 21 19 68 65 26 32 Insert 25 31 25

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SLIDE 18

MinHeap Insertion

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13 15 16 24 21 19 68 65 26 32 Insert 10 31 25 10

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SLIDE 19

MinHeap Insertion

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13 15 16 24 21 10 68 65 26 32 Insert 10 31 25 19

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SLIDE 20

MinHeap Insertion

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13 15 10 24 21 16 68 65 26 32 Insert 10 31 25 19

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SLIDE 21

MinHeap Insertion

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10 15 13 24 21 16 68 65 26 32 Insert 10 31 25 19

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SLIDE 22

MinHeap Insertion

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10 15 13 24 21 16 68 65 26 32 Insert 10 31 25 19

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SLIDE 23

MinHeap Deletion

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10 15 13 24 21 16 68 65 26 32 DeleteMin 31 25 19

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SLIDE 24

MinHeap Deletion

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15 13 24 21 16 68 65 26 32 DeleteMin 31 25 19

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SLIDE 25

MinHeap Deletion

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19 15 13 24 21 16 68 65 26 32 DeleteMin 31 25

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SLIDE 26

MinHeap Deletion

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13 15 19 24 21 16 68 65 26 32 DeleteMin 31 25

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SLIDE 27

MinHeap Deletion

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13 15 16 24 21 19 68 65 26 32 DeleteMin 31 25

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SLIDE 28

MinHeap Deletion

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13 15 16 24 21 19 68 65 26 32 DeleteMin 31 25

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SLIDE 29

MinHeap Deletion

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15 16 24 21 19 68 65 26 32 DeleteMin 31 25

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SLIDE 30

MinHeap Deletion

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25 15 16 24 21 19 68 65 26 32 DeleteMin 31

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SLIDE 31

MinHeap Deletion

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15 25 16 24 21 19 68 65 26 32 DeleteMin 31

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SLIDE 32

MinHeap Deletion

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15 21 16 24 25 19 68 65 26 32 DeleteMin 31

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SLIDE 33

MinHeap Deletion

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15 21 16 24 25 19 68 65 26 32 DeleteMin 31

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SLIDE 34

Array Representation

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15 21 16 24 25 19 68 65 26 32 31 Level-order Traversal 15 21 16 24 25 19 68 65 26 32 31

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SLIDE 35

Array Representation

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15 21 16 24 25 19 68 65 26 32 31 Level-order Traversal 15 21 16 24 25 19 68 65 26 32 31 1 2 3 4 5 6 7 8 9 10 11

1 2 3 4 5 6 7 8 9 10 11

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SLIDE 36

Array Implementation

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template <typename T> class MinHeap { T[] keys; int size; int capacity; }; #define ROOT 1 #define LEFT_CHILD(p) p*2 #define RIGHT_CHILD(p) p*2+1 #define PARENT(p) p/2 int treePointer = ROOT; Do you spot any mistakes or bad practices?

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SLIDE 37

Array Implementation

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template <typename T> class MinHeap { T[] keys; int size; int capacity; }; #define ROOT 1 #define LEFT_CHILD(p) ((p)*2) #define RIGHT_CHILD(p) ((p)*2+1) #define PARENT(p) ((p)/2) int treePointer = ROOT;

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SLIDE 38

Top-K Revisit

Given an unordered list of n values, and an integer K, find the K largest elements

Initialize a MinHeap of size K Insert the first K values in the list into the Heap For each additional value x

If x is smaller than the top of the heap (PeekMin)

Discard x

Else

DeleteMin Insert(x)

Return all the values in the MinHeap

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O(n log k)