1. Priority Queues and Heaps

2. Priority Queues We already know a ”standard” queue:
Elements can only be added to the start of the queue
Elements can only be removed from the end of the queue
Order of removal is FIFO (first in, first out)
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3. Priority Queues
In a Priority Queue, each element is assigned a priority as well
A priority is a numeric value – the lower the number, the higher the priority
Elements are removed in order of their priority – no longer FIFO
Can still add elements in any order
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4. Priority Queues public interface PriorityQueue<T> {
void add(T element);
T remove();
T peek(); }
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5. Priority Queues
The interface to a priority queue is almost the same as to a regular queue
However, elements in the queue must now be of the type Comparable (i.e. implement the interface)
The remove method always returns the element with highest priority (lowest numeric value…)
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6. Priority Queues
A Priority Queue is an abstract data structure, which can be implemented in various ways
Linked list, removal will be O(n)
Binary search tree, removal will usually be O(log(n)), but not always
Heap, removal will always be O(log(n))
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7. Heaps A Heap is a Binary Tree, but not a Binary Search Tree
In order for a Binary Tree to be a Heap, two conditions must be fulfilled:
A heap is almost complete; only nodes missing in bottom layer
All nodes store values which are at most as large as the values stored in any descendant
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8. Heaps
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9. Heaps On a heap, only two operations are of interest:
Insert a new element
Remove the element with lowest value, i.e. the root element
However, both of these operations must preserve the heap property of the tree
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10. Heaps - insertion 32 23 29 42 37 56 Find an empty slot in the tree
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11. Heaps - insertion 32 23 29 42
If the value in the parent is larger than the new value, swap the parent and the new slot (repeat this) 37 56
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12. Heaps - insertion 32 23 29 32 37 56 42 Now insert the new value
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13. Heaps - insertion
Since the heap is ”almost complete”, the number of layers in a tree with n nodes is at most (log(n) + 1)
Each swap operation can be done in constant time, so insertion of an element has the run-time complexity O(log(n))
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14. Heaps - removal
Remove the root node (always smallest) 23 29 32 37 56 42
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15. Heaps - removal 29 32
Move the last element into the root position 37 56 42
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16. Heaps - removal
If any child has a lower value, swap position with child with lowest value
(repeat this) 42 29 32 37 56
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17. Heaps - removal
If any child has a lower value, swap position with child with lowest value
(repeat this) 29 42 32 37 56
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18. Heaps - removal
The heap property has now been reestablished
This is known as ”fixing the heap” 29 37 32 42 56
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19. Heaps - removal
Since the heap is ”almost complete”, the number of layers in a tree with n nodes is at most (log(n) + 1)
Each swap operation can be done in constant time, so deletion of an element has the run-time complexity O(log(n))
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20. Heaps - removal
The important point is that for a heap, we are guaranteed O(log(n)) run time for insertion and deletion
This cannot be guaranteed for a binary search tree
A binary search tree can ”degenerate” into a linked list; a heap cannot
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21. Heaps - representation
Due to the regularity of a heap, it can efficiently be stored in an array
Root node is stored in position 1 (not 0)
A node in position i has its children stored in position 2i and (2i + 1)
A node in position i has its parent stored in position i/2 (integer division)
When running out of space, double the size
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22. Heapsort
A heap can be used for a quite efficient way of sorting an array of n objects
Run-time complexity of O(nlog(n))
Does not use extra space
Main steps
Turn the array into a heap
Repeatedly remove the root element (which has the smallest value)
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23. Heapsort
In order to turn the array into a heap, we could just insert all the elements into a new heap
However, we can do this without using an extra heap!
We use the ”fix heap” procedure from the bottom in the tree and upwards
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24. Heapsort Why will this work…?
Remember that the ”fix heap” procedure takes two ”subheaps” as input, plus a root node with a ”wrong” value
If we work from the bottom and up, the input will always be like above
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25. Heapsort Fix heap here Fix heaps here Fix heaps here Trivially a heap
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26. Heapsort Now the tree is a heap
Repeatedly remove the root from the heap, and fix the remaining heap
We ”remove” the root by placing it at the end of the array, beyond the last element in the remaining heap
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27. Heapsort 23 29 32 37 56 42 23 29 32 37 56 42
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28. Heapsort 29 32 37 56 42 29 32 37 56 42 23
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29. Heapsort 29 37 32 42 56 29 37 32 42 56 23
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30. Heapsort 37 32 42 56 37 32 42 56 29 23
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31. Heapsort 32 37 56 42 32 37 56 42 29 23
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32. Heapsort 37 56 42 37 56 42 32 29 23
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33. Heapsort 37 42 56 37 42 56 32 29 23
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34. Heapsort 42 56 42 56 37 32 29 23
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35. Heapsort 42 56 42 56 37 32 29 23
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36. Heapsort 56 56 42 37 32 29 23
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37. Heapsort 56 56 42 37 32 29 23
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38. Heapsort 56 42 37 32 29 23
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39. Heapsort One minor issue – numbers are sorted in wrong order
Could just reverse order, takes O(n)
Or use max-heap
Min-heap: All nodes store values at most as large as the values stored in any descendant
Max-heap: All nodes store values at least as large as the values stored in any descendant
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