1. Priority Queues (Chapter 6.6):
Abstract model of Priority Queues
Priority Queue
Get the one with the highest priority
insert
the minimum one
(or equivalently, maximum, or other requirement)
a black box
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2. Let’s cleaning up some confusion.
Binary Search Tree 17
Get the one with the highest priority
insert 14 19 12 15 18
the minimum one
(or maximum,)
Is Binary Search Tree a priority queue?
No! It is not, but it can be used (in the black box) to implement a priority queue.
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3. Items with higher priority will be treated first.
Stacks (FILO), queues (FIFO) are Priority Queues
Arrays are not, since we can randomly access the items in an array
The priority of the item may be fixed or changed over the time.
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4. Using a Binary Search Tree is overkilled. Why?17
Get the one with the highest priority
insert 14 19 12 15 18 11 13
the minimum one
(or maximum)
Items are totally ordered, but we don’t
really this information (nothing comes free)
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5. Our goals: 1. Can get the minimum one efficiently
2. and low maintenance 11 9 16 12 11 12
Get the minimum one 9 13
insert 13 12 14 17 15 19 18 13 14 17 15 19 18 16 14 17 15 19 18
Binary Heap
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6. Binary Heap Using (abstractly) binary tree structure
For every node in the tree, its key is smaller than (or equal to) the keys of its two children.
Therefore, for every node in the tree, its data is the smallest one among data stored in the sub-trees (the two children).
Different from BST, not every key in the left-sub-tree is less than every key in the right-sub-tree.
Loosing this requirement, we gain some benefits.
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7. A Binary Heap but we can organize it in a better way
without too much extra cost. 1 4 6 5 9 23 10 25 8 14 11 25 12 29 28 12 17 20 15 22 29 30 19 36 13 29 23 30 37
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8. A Binary Heap in complete binary tree10 12 13 15 14 16 17 23 22 21 18 19 24 25 26 27 32 29 30 25 26 37
pack
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9. A Binary Heap implemented in an array10 1 12 13 2 3 15 14 16 17 4 5 6 7 23 22 21 28 39 24 25 42 10 11 15 8 9 12 13 14 27 32 29 40 35 36 37 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
1st child (left-child) of i is *i
2nd child (right-child) of i is 2*i+1
The parent of i is i / 2
double the size of the array if the space ran out
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10. Heap operation: insert an item11 10 12 13 15 14 16 17 23 22 21 28 39 24 25 42 27 32 29 40 35 36 37 11
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11. Heap operation: insert an item11 10 12 13 15 14 16 17 23 22 21 39 24 25 42 11 27 32 29 40 35 36 37 28
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12. Heap operation: insert an item11 10 12 13 15 11 16 17 23 22 21 39 24 25 42 14 27 32 29 40 35 36 37 28
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13. Heap operation: insert an item11 10 11 13 15 12 16 17 23 22 21 39 24 25 42 14 27 32 29 40 35 36 37 28
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14. Heap operation: delete the minimum10 12 13 15 14 16 17 23 22 21 28 39 24 25 42 27 32 29 40 35 36 37
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15. Heap operation: delete the minimum10 12 13 15 14 16 17 23 22 21 28 39 24 25 42 27 32 29 40 35 36 37
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16. Heap operation: delete the minimum10 37 12 13 15 14 16 17 23 22 21 28 39 24 25 42 27 32 29 40 35 36
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17. Heap operation: delete the minimum10 12 37 13 15 14 16 17 23 22 21 28 39 24 25 42 27 32 29 40 35 36
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18. Heap operation: delete the minimum10 12 14 13 15 37 16 17 23 22 21 28 39 24 25 42 27 32 29 40 35 36
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19. Heap operation: delete the minimum10 12 14 13 15 16 21 17 23 22 37 28 39 24 25 42 27 32 29 40 35 36
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20. Heap operation: delete the minimum10 12 14 13 15 16 21 17 23 22 35 28 39 24 25 42 27 32 29 40 37 36
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21. Time complexity for insertion
best case: O(1)
worst case: O(log n) 14
log n 15 21 23 22 37 25 27 32 29 40 35 36 30 27
average case:
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22. Time complexity of Building a heap
Insert them one by one into the empty heap
What is the cost?
best case:
O(n)
average case:
O(n)
worst case:
O(n log n)
Can we improve the worst case? 34 15 30 2 23 9 25 6 32 5 40 11 25 21 7
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23. Percolating a non-heap
Worst case: O(n) 34 2 15 2 5 30 7 7 2 15 6 5 23 9 25 7 30 21 6 23 9 21 15 6 32 5 23 40 11 31 30 21 7 25 15 32 34 40 11 31 30 25
Binary Heap
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24. Percolating a non-heap
Worst case: O(S)
S = Sum of heights of all nodes h
O(S) = O(n)
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