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Priority Queues and Heaps
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 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
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
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; }};
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
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)
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
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
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
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
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
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
Heaps and Priority Queues
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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
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• The last node of a heap is the rightmost internal node of depth h 1
last node
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
keys0
1
h2
h1
depth
• 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.
Exercise: Heaps
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)
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
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79 1
z
z
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
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
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9
w
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
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9
w
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9
w
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
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
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
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
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0 1 2 3 4
2
1i
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)
O(1) O(n) O(n2)
Heap Sort (binary heap, vector-based implementation)
O(logn) O(logn) O(n logn)
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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• 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
Example
1516 124 96 2023
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2023
Example
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Example
Example
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Example
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Example10
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Example4
Analysis
• At each level we insert nodes• Each node can generate h-i swaps
i2
Analysis
i2
• At each level we insert nodes• Each node can generate h-i swaps
i2
Analysis
ih
• At each level we insert nodes• Each node can generate h-i swaps
i2
Analysis
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0
)(2h
i
i ih
• At each level we insert nodes• Each node can generate h-i swaps
i2
Analysis
)2(2)(2 11
0
hih hh
i
i
• At each level we insert nodes• Each node can generate h-i swaps
i2
Analysis
)2)(log(2)(21
0
nnihh
i
i
• At each level we insert nodes• Each node can generate h-i swaps
i2
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
i2
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)
Locators 3 a
1 g 4 e
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
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)
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
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
