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Heaps & priority queues

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Pedro Hugo Valencia Morales
Pedro Hugo Valencia Morales
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Heaps & Priority Queues
Overview ,[object Object],[object Object],[object Object],[object Object],[object Object]
Complete Binary Tree ,[object Object],[object Object],[object Object]
Array-Based Representation of a Binary Tree Bob Mike Sam Tom Pam Joe Sue Ann Jane Mary root 0 free 10
Left Child & Right Child in a Complete, Array-Based Binary Tree ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Left Child & Right Child in a Complete, Array-Based Binary Tree ,[object Object],[object Object],Bob Mike Sam Tom Pam Joe Sue Ann Jane Mary
Parent in a Complete, Array-Based Binary Tree ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Parent in a Complete , Array-Based Binary Tree ,[object Object],Bob Mike Sam Tom Pam Joe Sue Ann Jane Mary
Array-Based Representation of a Complete Binary Tree ,[object Object],[object Object]
Array-Based Representation of a Complete Binary Tree Bob Mike Sam Tom Pam Joe Sue Ann Jane Mary
Heaps ,[object Object],[object Object],[object Object],[object Object],[object Object]
Array-Based Representation of a Heap Mary Mike Sam Ann Tom Pam Sue Joe Bob Jane 3 2 1 5 4 7 6 9 8 10
Array-Based Representation of a Heap ,[object Object],[object Object],[object Object],[object Object]
The ADT Priority Queue ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
PriorityQ: Array-Based Implementation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
PriorityQ: Array-Based Implementation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
PriorityQ: Public Member Function Definitions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
PriorityQ: Retrieve & Delete ,[object Object],[object Object],[object Object],[object Object],[object Object]
PriorityQ: Public Member Function Definition ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Semiheap ,[object Object],10 5 10 40 30 15 50 35 20 55 60 45 25 55 45 25
Rebuilding a Heap: Basic Idea ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Retrieve & Delete: Example ,[object Object],[object Object],5 10 40 30 15 50 35 20 55 45 60 25 move 25 to here
Rebuilding a Heap: Example (Cont’d.) ,[object Object],[object Object],15 50 35 20 5 10 40 30 55 45 25 swap 25 with the item in this node
Rebuilding a Heap: Example (Cont’d.) ,[object Object],[object Object],15 50 35 20 5 10 40 30 25 45 55 swap 25 with the item in this node
Rebuilding a Heap: Example (Cont’d.) ,[object Object],[object Object],15 25 35 20 5 10 40 30 50 45 55 swap 25 with the item in this node
Rebuilding a Heap: Example (Cont’d.) ,[object Object],[object Object],[object Object],15 40 35 20 5 10 25 30 50 45 55
PriorityQ: Private Member Function Definition ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
PriorityQ Insert : Basic Idea ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
PriorityQ Insert: Example ,[object Object],[object Object],15 40 35 20 5 10 25 30 50 45 55 47
PriorityQ Insert: Example (Cont’d.) ,[object Object],15 40 35 20 5 10 25 30 50 45 55 47
PriorityQ Insert: Example (Cont’d.) ,[object Object],15 40 47 20 5 10 25 30 50 45 55 35
PriorityQ Insert: Example (Cont’d.) ,[object Object],[object Object],15 40 45 20 5 10 25 30 50 47 55 35
PriorityQ: Public Member Function Definition ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Heap-Based PriorityQ: Efficiency ,[object Object],[object Object],[object Object],[object Object],[object Object]
Heapsort: Basic Idea ,[object Object],[object Object],[object Object],[object Object]
Transform an Array Into a Heap: Basic Idea ,[object Object],[object Object],[object Object],[object Object]
Transform an Array Into a Heap: Example ,[object Object],[object Object],[object Object],7 2 4 10 9 3 5 6 5 4 3 2 1 0 4 2 7 5 3 6 7 6 9 10 rebuildHeap
Transform an Array Into a Heap: Example ,[object Object],[object Object],rebuildHeap 9 2 4 10 7 3 5 6 5 4 3 2 1 0 4 2 9 5 3 6 7 6 7 10
Transform an Array Into a Heap: Example ,[object Object],[object Object],rebuildHeap 9 2 4 5 7 3 10 6 5 4 3 2 1 0 4 2 9 10 3 6 7 6 7 5
Transform an Array Into a Heap: Example ,[object Object],[object Object],rebuildHeap 3 2 4 5 7 9 10 6 5 4 3 2 1 0 4 2 3 10 9 6 7 6 7 5
Transform an Array Into a Heap: Example ,[object Object],[object Object],[object Object],rebuildHeap 7 2 4 5 3 9 10 6 5 4 3 2 1 0 4 2 7 10 9 6 7 6 3 5
Transform an Array Into a Heap: Example ,[object Object],[object Object],7 2 4 5 3 9 6 10 5 4 3 2 1 0 4 2 7 6 9 10 7 6 3 5
Transform an Array Into a Heap (Cont’d.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Transform an Array Into a Heap: C++ ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Rebuild a Heap: C++ ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Transform a Heap Into a Sorted Array ,[object Object],[object Object],[object Object]
Transform a Heap Into a Sorted Array: Basic Idea ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],[object Object],7 2 4 5 3 9 6 10 4 2 7 6 9 10 5 4 3 2 1 0 7 6 3 5 a[ ]: Heap
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],[object Object],7 2 4 5 9 6 3 4 2 7 6 9 3 5 4 3 2 1 0 7 6 10 5 a[ ]: Semiheap Sorted rebuildHeap
Transform a Heap Into a Sorted Array: Example ,[object Object],7 2 4 5 3 6 9 4 2 7 6 3 9 5 4 3 2 1 0 7 6 10 5 a[ ]: Becoming a Heap Sorted rebuildHeap
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],3 2 4 5 7 6 9 4 2 3 6 7 9 5 4 3 2 1 0 7 6 10 5 a[ ]: Heap Sorted
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],[object Object],3 2 4 7 6 5 4 2 3 6 7 5 5 4 3 2 1 0 7 6 10 9 a[ ]: Semiheap Sorted rebuildHeap
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],[object Object],3 2 4 5 6 7 4 2 3 6 5 7 5 4 3 2 1 0 7 6 10 9 a[ ]: Heap Sorted rebuildHeap
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],[object Object],3 2 5 6 4 7 2 3 6 5 4 5 4 3 2 1 0 7 6 10 9 a[ ]: Semiheap Sorted rebuildHeap
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],3 2 5 4 6 7 2 3 4 5 6 5 4 3 2 1 0 7 6 10 9 a[ ]: Heap Sorted
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],[object Object],3 5 4 2 7 6 3 4 5 2 5 4 3 2 1 0 7 6 10 9 a[ ]: Semiheap Sorted rebuildHeap
Transform a Heap Into a Sorted Array: Example ,[object Object],Becoming a Heap 3 2 4 5 7 6 3 4 2 5 5 4 3 2 1 0 7 6 10 9 a[ ]: Sorted rebuildHeap
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],2 3 4 5 7 6 2 4 3 5 5 4 3 2 1 0 7 6 10 9 a[ ]: Heap Sorted
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],[object Object],3 4 2 7 6 5 4 3 2 5 4 3 2 1 0 7 6 10 9 a[ ]: Semiheap Sorted rebuildHeap
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],3 2 4 7 6 5 2 3 4 5 4 3 2 1 0 7 6 10 9 a[ ]: Heap Sorted
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],[object Object],3 2 7 6 5 4 3 2 5 4 3 2 1 0 7 6 10 9 a[ ]: Semiheap Sorted rebuildHeap
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],2 3 7 6 5 4 2 3 5 4 3 2 1 0 7 6 10 9 a[ ]: Heap Sorted
Transform a Heap Into a Sorted Array: Example ,[object Object],[object Object],[object Object],2 7 6 5 4 3 2 5 4 3 2 1 0 7 6 10 9 a[ ]: Heap Sorted
Transform a Heap Into a Sorted Array: Example ,[object Object],7 6 5 4 3 2 5 4 3 2 1 0 7 6 10 9 a[ ]: Sorted
Heapsort: C++ ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Heapsort: Efficiency ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Growth Rates for Selected Sorting Algorithms † According to Knuth, the average growth rate of Insertion sort is about 0.9 times that of Selection sort and about 0.4 times that of Bubble Sort. Also, the average growth rate of Quicksort is about 0.74 times that of Mergesort and about 0.5 times that of Heapsort.

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Heaps & priority queues

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  • 4. Array-Based Representation of a Binary Tree Bob Mike Sam Tom Pam Joe Sue Ann Jane Mary root 0 free 10
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  • 10. Array-Based Representation of a Complete Binary Tree Bob Mike Sam Tom Pam Joe Sue Ann Jane Mary
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  • 12. Array-Based Representation of a Heap Mary Mike Sam Ann Tom Pam Sue Joe Bob Jane 3 2 1 5 4 7 6 9 8 10
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  • 67. Growth Rates for Selected Sorting Algorithms † According to Knuth, the average growth rate of Insertion sort is about 0.9 times that of Selection sort and about 0.4 times that of Bubble Sort. Also, the average growth rate of Quicksort is about 0.74 times that of Mergesort and about 0.5 times that of Heapsort.