1. Priority Queues and Heaps

2. Outline and Reading PriorityQueue ADT (§8.1)
Total order relation (§8.1.1)
Comparator ADT (§8.1.2)
Sorting with a Priority Queue (§8.1.5)
Implementing a PQ with a list (§8.2)
Selection-sort and Insertion Sort (§8.2.2)
Heaps (§8.3)
Complete Binary Trees (§8.3.2)
Implementing a PQ with a heap (§8.3.3)
Heapsort (§8.3.5)

3. Priority Queue ADT A priority queue stores a collection of items
An item is a pair (key, element)
Main methods of the Priority Queue ADT
insertItem(k, o) inserts an item with key k and element o
removeMin() removes the item with the smallest key
Additional methods
minKey() returns, but does not remove, the smallest key of an item
minElement() returns, but does not remove, the element of an item with smallest key
size(), isEmpty()
Applications:
Standby flyers
Auctions

4. Comparator ADT
A comparator encapsulates the action of comparing two objects according to a given total order relation
A generic priority queue uses a comparator as a template argument, to define the comparison function (<,=,>)
The comparator is external to the keys being compared. Thus, the same objects can be sorted in different ways by using different comparators.
When the priority queue needs to compare two keys, it uses its comparator

5. Using Comparators in C++
A comparator class overloads the “()” operator with a comparison function.
Example: Compare two points in the plane lexicographically.
class LexCompare { public: int operator()(Point a, Point b) { if (a.x < b.x) return –1; else if (a.x > b.x) return +1; else if (a.y < b.y) return –1; else if (a.y > b.y) return +1; else return 0; } };

6. Total Order Relation Mathematical concept of total order relation 
Keys in a priority queue can be arbitrary objects on which an order is defined
Two distinct items in a priority queue can have the same key
Mathematical concept of total order relation 
Reflexive property: x  x
Antisymmetric property: x  y  y  x  x = y
Transitive property: x  y  y  z  x  z

7. PQ-Sort: Sorting with a Priority Queue
We can use a priority queue to sort a set of comparable elements
Insert the elements one by one with a series of insertItem(e, e) operations
Remove the elements in sorted order with a series of removeMin() operations
Running time depends on the PQ implementation
Algorithm PQ-Sort(S, C)
Input sequence S, comparator C for the elements of S
Output sequence S sorted in increasing order according to C
P  priority queue with comparator C
while !S.isEmpty ()
e  S.remove (S.first())
P.insertItem(e, e)
while !P.isEmpty()
e  P.minElement() P.removeMin()
S.insertLast(e)

8. List-based Priority Queue
Unsorted list implementation
Store the items of the priority queue in a list-based sequence, in arbitrary order
Performance:
insertItem takes O(1) time since we can insert the item at the beginning or end of the sequence
removeMin, minKey and minElement take O(n) time since we have to traverse the entire sequence to find the smallest key
sorted list implementation
Store the items of the priority queue in a sequence, sorted by key
Performance:
insertItem takes O(n) time since we have to find the place where to insert the item
removeMin, minKey and minElement take O(1) time since the smallest key is at the beginning of the sequence 4 5 2 3 1 1 2 3 4 5

9. Selection-Sort
Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted sequence
Running time of Selection-sort:
Inserting the elements into the priority queue with n insertItem operations takes O(n) time
Removing the elements in sorted order from the priority queue with n removeMin operations takes time proportional to …+ n
Selection-sort runs in O(n2) time 4 5 2 3 1

10. Exercise: Selection-Sort
Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted sequence (first n insertItems, then n removeMins)
Illustrate the performance of selection-sort on the following input sequence:
(22, 15, 36, 44, 10, 3, 9, 13, 29, 25) 4 5 2 3 1

11. Insertion-Sort
Insertion-sort is the variation of PQ-sort where the priority queue is implemented with a sorted sequence
Running time of Insertion-sort:
Inserting the elements into the priority queue with n insertItem operations takes time proportional to …+ n
Removing the elements in sorted order from the priority queue with a series of n removeMin operations takes O(n) time
Insertion-sort runs in O(n2) time 1 2 3 4 5

12. Exercise: Insertion-Sort
Insertion-sort is the variation of PQ-sort where the priority queue is implemented with a sorted sequence (first n insertItems, then n removeMins)
Illustrate the performance of insertion-sort on the following input sequence:
(22, 15, 36, 44, 10, 3, 9, 13, 29, 25) 1 2 3 4 5

13. In-place Insertion-sort
Instead of using an external data structure, we can implement selection-sort and insertion-sort in-place (only O(1) extra storage)
A portion of the input sequence itself serves as the priority queue
For in-place insertion-sort
We keep sorted the initial portion of the sequence
We can use swapElements instead of modifying the sequence 5 4 2 3 1 5 4 2 3 1 4 5 2 3 1 2 4 5 3 1 2 3 4 5 1 1 2 3 4 5 1 2 3 4 5

14. Heaps and Priority Queues
Trees
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Heaps and Priority Queues 2 6 5 7 9

15. Trees
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What is a heap?
A heap is a binary tree storing keys at its internal nodes and satisfying the following properties:
Heap-Order: for every internal node v other than the root, key(v)  key(parent(v))
Complete Binary Tree: let h be the height of the heap
for i = 0, … , h - 1, there are 2i nodes of depth i
at depth h - 1, the internal nodes are to the left of the leaf nodes
The last node of a heap is the rightmost internal node of depth h - 1 2 5 6 9 7
last node

16. Height of a Heap Theorem: A heap storing n keys has height O(log n)
Proof: (we apply the complete binary tree property)
Let h be the height of a heap storing n keys
Since there are 2i keys at depth i = 0, … , h - 2 and at least one key at depth h - 1, we have n  … + 2h
Thus, n  2h-1 , i.e., h  log n + 1
depth
keys 1 1 2
h-2
2h-2
h-1 1

17. Exercise: Heaps
Where may an item with the largest key be stored in a heap?
True or False:
A pre-order traversal of a heap will list out its keys in sorted order. Prove it is true or provide a counter example.
Let H be a heap with 7 distinct elements (1,2,3,4,5,6, and 7). Is it possible that a preorder traversal visits the elements in sorted order? What about an inorder traversal or a postorder traversal? In each case, either show such a heap or prove that none exists.

18. Heaps and Priority Queues
We can use a heap to implement a priority queue
We store a (key, element) item at each internal node
We keep track of the position of the last node
For simplicity, we show only the keys in the pictures
(2, Sue)
(5, Pat)
(6, Mark)
(9, Jeff)
(7, Anna)

19. Insertion into a Heap
Method insertItem of the priority queue ADT corresponds to the insertion of a key k to the heap
The insertion algorithm consists of three steps
Find the insertion node z (the new last node)
Store k at z and expand z into an internal node
Restore the heap-order property (discussed next) 2 6 5 7 9 z
insertion node 2 5 6 z 9 7 1

20. Upheap
After the insertion of a new key k, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 2 1 5 1 5 2 z z 9 7 6 9 7 6

21. Removal from a Heap
Method removeMin of the priority queue ADT corresponds to the removal of the root key from the heap
The removal algorithm consists of three steps
Replace the root key with the key of the last node w
Compress w and its children into a leaf
Restore the heap-order property (discussed next) 2 6 5 7 9 w
last node 7 5 6 w 9

22. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap property by swapping key k with the child with the smallest key along a downward path from the root
Downheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 7 5 6 7 9 w 5 6 w 9

23. Updating the Last Node
The insertion node can be found by traversing a path of O(log n) nodes
Go up until a left child or the root is reached
If a left child is reached, go to the right child
Go down left until a leaf is reached
Similar algorithm for updating the last node after a removal

24. Heap-Sort
Consider a priority queue with n items implemented by means of a heap
the space used is O(n)
methods insertItem and removeMin take O(log n) time
methods size, isEmpty, minKey, and minElement take time O(1) time
Using a heap-based priority queue, we can sort a sequence of n elements in O(n log n) time
The resulting algorithm is called heap-sort
Heap-sort is much faster than quadratic sorting algorithms, such as insertion-sort and selection-sort

25. Exercise: Heap-Sort
Heap-sort is the variation of PQ-sort where the priority queue is implemented with a heap (first n insertItems, then n removeMins)
Illustrate the performance of heap-sort on the following input sequence:
(22, 15, 36, 44, 10, 3, 9, 13, 29, 25) 4 5 2 3 1

26. Vector-based Heap Implementation
We can represent a heap with n keys by means of a vector of length n
For the node at rank i
the left child is at rank 2i+1
the right child is at rank 2i +2
parent is
Links between nodes are not explicitly stored
The leaves are not represented
Operation insertItem corresponds to inserting at rank n
Operation removeMin corresponds to removing at rank n-1
Yields in-place heap-sort 2 6 5 7 9 2 5 6 9 7 1 2 3 4

27. Priority Queue Sort Summary
PQ-Sort consists of n insertions followed by n removeMin ops
Insert
RemoveMin
PQ-Sort Total
Insertion Sort (ordered sequence)
O(n)
O(1)
O(n2)
Selection Sort (unordered sequence)
Heap Sort (binary heap, vector-based implementation)
O(logn)
O(n logn)

28. Merging Two Heaps We are given two two heaps and a key k3 5 8 2 6 4
We are given two two heaps and a key k
We create a new heap with the root node storing k and with the two heaps as subtrees
We perform downheap to restore the heap-order property 7 3 2 8 5 4 6 2 3 4 8 5 7 6

29. Bottom-up Heap Construction
We can construct a heap storing n given keys in using a bottom-up construction with log n phases
In phase i, pairs of heaps with 2i -1 keys are merged into heaps with 2i+1-1 keys
2i -1
2i+1-1

30. Example 16 15 4 12 6 9 23 20

31. Example 25 15 16 5 12 4 11 9 6 27 20 23

32. Example 15 25 16 4 12 5 6 9 11 20 27 23

33. Example 7 8 15 4 6 20 16 25 5 12 11 9 23 27

34. Example 4 6 15 5 8 20 16 25 7 12 11 9 23 27

35. Example 10 4 6 15 5 8 20 16 25 7 12 11 9 23 27

36. Example 4 5 6 15 7 8 20 16 25 10 12 11 9 23 27

37. Analysis At each level we insert nodes
Each node can generate h-i swaps

38. Analysis At each level we insert nodes
Each node can generate h-i swaps

39. Analysis At each level we insert nodes
Each node can generate h-i swaps

40. Analysis At each level we insert nodes
Each node can generate h-i swaps

41. Analysis At each level we insert nodes
Each node can generate h-i swaps

42. Analysis At each level we insert nodes
Each node can generate h-i swaps

43. Analysis At each level we insert nodes
Each node can generate h-i swaps
Thus, bottom-up heap construction runs in O(n) time
Bottom-up heap construction is faster than n successive insertions and speeds up the first phase of heap-sort

44. Exercise: Bottom-up Heap
Build a heap from the bottom up on the following input sequence:
(22, 15, 36, 44, 10, 3, 9, 13, 29, 25, 2, 11, 7, 1, 17)

45. Trees
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Locators 3 a 1 g 4 e

46. Trees
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Locators
A locator identifies and tracks an item (key, element) within a data structure
A locator sticks with a specific item, even if that element changes its position in the data structure
Intuitive notion:
claim check
reservation number
Methods of the locator ADT:
key(): returns the key of the item associated with the locator
element(): returns the element of the item associated with the locator

47. Locator-based Methods
Locator-based priority queue methods:
insert(k, o): inserts the item (k, o) and returns a locator for it
min(): returns the locator of an item with smallest key
remove(l): remove the item with locator l
replaceKey(l, k): replaces the key of the item with locator l
replaceElement(l, o): replaces with o the element of the item with locator l
locators(): returns an iterator over the locators of the items in the priority queue
Locator-based dictionary methods:
insert(k, o): inserts the item (k, o) and returns its locator
find(k): if the dictionary contains an item with key k, returns its locator, else return a special null locator
remove(l): removes the item with locator l and returns its element
locators(), replaceKey(l, k), replaceElement(l, o)

48. Possible Implementation
The locator is an object storing
key
element
position (or rank) of the item in the underlying structure
In turn, the position (or array cell) stores the locator
Example:
binary search tree with locators 6 d 3 a 9 b 1 g 4 e 8 c

49. Positions vs. Locators Position
represents a “place” in a data structure
related to other positions in the data structure (e.g., previous/next or parent/child)
often implemented as a pointer to a node or the index of an array cell
Position-based ADTs (e.g., sequence and tree) are fundamental data storage schemes
Locator
identifies and tracks a (key, element) item
unrelated to other locators in the data structure
often implemented as an object storing the item and its position in the underlying structure
Key-based ADTs (e.g., priority queue and dictionary) can be augmented with locator-based methods