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Chapter 8: Priority Queues and Heaps
Nancy AmatoParasol Lab, Dept. CSE, Texas A&M University
Acknowledgement: These slides are adapted from slides provided with Data Structures and Algorithms in C++, Goodrich, Tamassia and Mount (Wiley 2004)
http://parasol.tamu.edu
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Priority Queues 04/19/23 00:21Trees 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)
Priority Queues 04/19/23 00:21Trees 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• Stock market
Priority Queues 04/19/23 00:21Trees 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
Priority Queues 04/19/23 00:21Trees 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; }};
Priority Queues 04/19/23 00:21Trees 6
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
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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 SOutput sequence S sorted in increasing order according to CP priority queue with
comparator Cwhile !S.isEmpty ()
e S.remove (S.first())
P.insertItem(e, e)while !P.isEmpty()
e P.minElement()P.removeMin()
S.insertLast(e)
Priority Queues 04/19/23 00:21Trees 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
Priority Queues 04/19/23 00:21Trees 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
1 2 …n• Selection-sort runs in O(n2) time
4 5 2 3 1
Priority Queues 04/19/23 00:21Trees 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
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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 1 2 …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
Priority Queues 04/19/23 00:21Trees 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)
4 5 2 3 1
Priority Queues 04/19/23 00:21Trees 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
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Heaps and Priority Queues
2
65
79
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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 external nodes
2
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79
• The last node of a heap is the rightmost internal node of depth h 1
last node
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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 1 2 4 … 2h2 1
• Thus, n 2h1 , i.e., h log n 1
1
2
2h2
1
keys
0
1
h2
h1
depth
Priority Queues 04/19/23 00:21Trees 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.
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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)
(6, Mark)(5, Pat)
(9, Jeff) (7, Anna)
Priority Queues 04/19/23 00:21Trees 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)
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insertion node
2
65
79 1
z
z
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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
15
79 6z
1
25
79 6z
Priority Queues 04/19/23 00:21Trees 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)
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last node
w
7
65
9
w
Priority Queues 04/19/23 00:21Trees 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-order property by swapping key k along a downward path from the root
• Upheap 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
65
9
w
5
67
9
w
Priority Queues 04/19/23 00:21Trees 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
Priority Queues 04/19/23 00:21Trees 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
Priority Queues 04/19/23 00:21Trees 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
Priority Queues 04/19/23 00:21Trees 26
Vector-based Heap Implementation
• We can represent a heap with n keys by means of a vector of length n 1
• For the node at rank i• the left child is at rank 2i• the right child is at rank 2i 1
• Links between nodes are not explicitly stored
• The leaves are not represented• The cell of at rank 0 is not used• Operation insertItem corresponds
to inserting at rank n 1• Operation removeMin corresponds
to removing at rank n• Yields in-place heap-sort
2
65
79
2 5 6 9 7
1 2 3 4 50
Priority Queue Sort Summary
• PQ-Sort consists of n insertions followed by n removeMin ops
04/19/23 00:21Vectors 27
Insert RemoveMin
PQ-Sort total
Insertion Sort (ordered sequence)
O(n) O(1) O(n2)
Selection Sort (unordered sequence)
O(1) O(n) O(n2)
Heap Sort (binary heap, vector-based implementation)
O(logn) O(logn) O(nlogn)
Priority Queues 04/19/23 00:21Trees 28
Merging Two Heaps
• 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
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Priority Queues 04/19/23 00:21Trees 29
• 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 2i11 keys
Bottom-up Heap Construction
2i 1 2i 1
2i11
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Example
1516 124 76 2023
25
1516
5
124
11
76
27
2023
Priority Queues 04/19/23 00:21Trees 31
Example (cont’d)
25
1516
5
124
11
96
27
2023
15
2516
4
125
6
911
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2027
Priority Queues 04/19/23 00:21Trees 32
Example (cont’d)
7
15
2516
4
125
8
6
911
23
2027
4
15
2516
5
127
6
8
911
23
2027
Priority Queues 04/19/23 00:21Trees 33
Example (end)
4
15
2516
5
127
10
6
8
911
23
2027
5
15
2516
7
1210
4
6
8
911
23
2027
Priority Queues 04/19/23 00:21Trees 34
Analysis
• We visualize the worst-case time of a downheap with a proxy path that goes first right and then repeatedly goes left until the bottom of the heap (this path may differ from the actual downheap path)
• Since each node is traversed by at most two proxy paths, the total number of nodes of the proxy paths is O(n)
• 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
Priority Queues 04/19/23 00:21Trees 35
Locators
3 a
1 g 4 e
Priority Queues 04/19/23 00:21Trees 36
Locators
• A locator identifies and tracks a (key, element) item 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
• Application example:• Orders to purchase and sell a
given stock are stored in two priority queues (sell orders and buy orders)
• the key of an order is the price
• the element is the number of shares
• When an order is placed, a locator to it is returned
• Given a locator, an order can be canceled or modified
Priority Queues 04/19/23 00:21Trees 37
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)
Priority Queues 04/19/23 00:21Trees 38
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
3 a
6 d
9 b
1 g 4 e 8 c
Priority Queues 04/19/23 00:21Trees 39
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