1. 09 Priority Queues
Hongfei Yan
Apr. 27, 2016

2. Contents 01 Python Primer (P2-51)
02 Object-Oriented Programming (P57-103)
03 Algorithm Analysis (P )
04 Recursion (P )
05 Array-Based Sequences (P )
06 Stacks, Queues, and Deques (P )
07 Linked Lists (P )
08 Trees (P )
09 Priority Queues (P )
10 Maps, Hash Tables, and Skip Lists (P )
11 Search Trees (P )
12 Sorting and Selection (P )
13 Text Processing (P )
14 Graph Algorithms (P )
15 Memory Management and B-Trees (P )
ISLR_Print6

3. Contents
9.1 The Priority Queue 9.2 Implementing a Priority Queue 9.3 Heaps 9.4 Sorting with a Priority Queue 9.5 Adaptable Priority Queues

4. Preview

5. 9.1 The Priority Queue Abstract Data Type
9.1.1 Priorities The Priority Queue ADT

6. E.g., An air-traffic control center that has to decide which flight to clear for landing
This choice may be influenced by factors such as each plane’s
distance from the runway,
time spent waiting in a holding pattern,
or amount of remaining fuel.
It is unlikely that the landing decisions are based purely on a FIFO policy.

7. E.g., To use another airline analogy
suppose a certain flight is fully booked an hour prior to departure.
Because of the possibility of cancellations, the airline maintains a queue of standby passengers hoping to get a seat.
Although the priority of a standby passenger is influenced by the check-in time of that passenger, other considerations include the fare paid and frequent-flyer status.
So it may be that an available seat is given to a passenger who has arrived later than another, if such a passenger is assigned a better priority by the airline agent.

8. Priority Queue A new abstract data type known as a priority queue.
This is a collection of prioritized elements that allows
arbitrary element insertion, and allows
the removal of the element that has first priority.
When an element is added to a priority queue,
the user designates its priority by providing an associated key.
The element with the minimum key will be the next to be removed from the queue

9. Total Order Relations
Keys in a priority queue can be arbitrary objects on which an order is defined
Two distinct entries 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

10. Priority Queue ADT Additional methods Applications:
A priority queue stores a collection of items
Each item is a pair (key, value)
Main methods of the Priority Queue ADT
add (k, x) inserts an item with key k and value x
remove_min() removes and returns the item with smallest key
Additional methods
min() returns, but does not remove, an item with smallest key
len(P), is_empty()
Applications:
Standby flyers
Auctions
Stock market

11. Priority Queue Example

12. 9.2 Implementing a Priority Queue
9.2.1 The Composition Design Pattern Implementation with an Unsorted List Implementation with a Sorted List
In this section, we show how to implement a priority queue by storing its entries in a positional list L. (See Section 7.4.) We provide two realizations, depending on whether or not we keep the entries in L sorted by key.

13. 9.2.1 Composition Design Pattern
An item in a priority queue is simply a key-value pair
Priority queues store items to allow for efficient insertion and removal based on keys
reminiscent
英 [ ˌremɪˈnɪsnt ] 美 [ ˌrɛməˈnɪsənt ]
adj.怀旧的; 回忆往事的; 使人联想…的
n.回忆录作者

14. Sequence-based Priority Queue
Implementation with an unsorted list
Performance:
add takes O(1) time since we can insert the item at the beginning or end of the sequence
remove_min and min take O(n) time since we have to traverse the entire sequence to find the smallest key
Implementation with a sorted list
Performance:
add takes O(n) time since we have to find the place where to insert the item
remove_min and min take O(1) time, since the smallest key is at the beginning 4 5 2 3 1 1 2 3 4 5

15. 9.2.2 Unsorted List Implementation

17. 9.2.3 Sorted List Implementation

18. 2 6 5 7 9
9.3 Heaps
9.3.1 The Heap Data Structure Implementing a Priority Queue with a Heap Array-Based Representation of a Complete Binary Tree Python Heap Implementation Analysis of a Heap-Based Priority Queue Python’s heapq Module

20. 9.3.1 The Heaps Data Structure
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9.3.1 The Heaps Data Structure
A heap is a binary tree storing keys at its nodes and satisfying the following properties:
Heap-Order: for every position 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
and the remaining nodes at level h reside in the leftmost possible positions at that level.
The last node of a heap is the rightmost node of maximum depth 2 5 6 9 7
last node

21. Height of a Heap Let h be the height of a heap storing n keys
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 - 1 and at least one key at depth h, we have n  … + 2h
Thus, n  2h , i.e., h  log n
depth
keys 1
Theorem
英 [ ˈθɪərəm ] 美 [ ˈθiərəm ]
n.[数] 定理; （能证明的）一般原理，公理，定律，法则 1 2
h-1
2h-1 h 1

22. 9.3.2 Implementing a Priority Queues with a Heap
Use a heap to implement a priority queue
Store a (key, element) item at each internal node
Keep track of the position of the last node
(2, Sue)
(5, Pat)
(6, Mark)
(9, Jeff)
(7, Anna)

23. Adding an Item to the Heap
Method add 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
Restore the heap-order property (discussed next) 2 6 5 7 9
insertion node 1 z

24. Up-Heap Bubbling After an Insertion
After the insertion of a new key k, the heap-order property may be violated
Algorithm up-heap restores the heap-order property by swapping k along an upward path from the insertion node
Up-heap 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), up-heap runs in O(log n) time 2 1 5 1 5 2 z z 9 7 6 9 7 6

25. Removing the Item with Minimum Key
Method remove_min 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
Remove w
Restore the heap-order property (discussed next) 2 5 6 w 9 7
last node 7 5 6 w 9
new last node

26. Down-Heap Bubbling After a Removal
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm down-heap restores the heap-order property by swapping key k along a downward path from the root
Down-heap 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), down-heap runs in O(log n) time 7 5 5 6 7 6 w w 9 9

27. 9.3.3 Array-based Heap Implementation
We can represent a heap with n keys by means of an array 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
Links between nodes are not explicitly stored
Operation add corresponds to inserting at rank n + 1
Operation remove_min corresponds to removing at rank n
Yields in-place heap-sort 2 5 6 9 7 2 5 6 9 7 1 3 4

28. 9.3.4 Python Heap Implementation

29. Merging Two Heaps We are given two heaps and a key k3 2
We are given 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 down-heap to restore the heap-order property 8 5 4 6 7 3 2 8 5 4 6 2 3 4 8 5 7 6

30. 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

31. Example 16 15 4 12 6 7 23 20 25 5 11 27 16 15 4 12 6 7 23 20

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

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

34. Example (end) 10 4 6 15 5 8 23 16 25 7 12 11 9 27 20 4 5 6 15 7 8 23 16 25 10 12 11 9 27 20

35. 9.3.5 Analysis of Heap Construction
We visualize the worst-case time of a down-heap 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 down-heap 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

36. 9.4 Sorting with a Priority Queue
9.4.1 Selection-Sort and Insertion-Sort Heap-Sort

37. Priority Queues
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9.4.1 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 insert 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
the bottleneck computation
is the repeated “selection” of the minimum element in Phase 2. For this
reason, this algorithm is better known as selection-sort. (See Figure 9.7.)

38. Selection-Sort Example
Sequence S Priority Queue P
Input: (7,4,8,2,5,3,9) ()
Phase 1
(a) (4,8,2,5,3,9) (7)
(b) (8,2,5,3,9) (7,4)
(g) () (7,4,8,2,5,3,9)
Phase 2
(a) (2) (7,4,8,5,3,9)
(b) (2,3) (7,4,8,5,9)
(c) (2,3,4) (7,8,5,9)
(d) (2,3,4,5) (7,8,9)
(e) (2,3,4,5,7) (8,9)
(f) (2,3,4,5,7,8) (9)
(g) (2,3,4,5,7,8,9) ()

39. Priority Queues
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9.4.1 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 insert 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

40. Insertion-Sort Example
Sequence S Priority queue P
Input: (7,4,8,2,5,3,9) ()
Phase 1
(a) (4,8,2,5,3,9) (7)
(b) (8,2,5,3,9) (4,7)
(c) (2,5,3,9) (4,7,8)
(d) (5,3,9) (2,4,7,8)
(e) (3,9) (2,4,5,7,8)
(f) (9) (2,3,4,5,7,8)
(g) () (2,3,4,5,7,8,9)
Phase 2
(a) (2) (3,4,5,7,8,9)
(b) (2,3) (4,5,7,8,9)
(g) (2,3,4,5,7,8,9) ()

41. In-place Insertion-Sort
Instead of using an external data structure, we can implement selection-sort and insertion-sort in-place
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 swaps 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

42. 9.4.2 Heap-based Priority Queue Sorting
Use a priority queue to sort a set of comparable elements
Insert the elements one by one with a series of add operations
Remove the elements in sorted order with a series of remove_min operations
The running time depends on the priority queue implementation:
Unsorted sequence gives selection-sort: O(n2) time
Sorted sequence gives insertion-sort: O(n2) time
Can we do better?
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.is_empty ()
e  S.remove_first ()
P.add (e, )
while P.is_empty()
e  P.removeMin().key()
S.add_last(e)

43. Heap-Sort Performance
Consider a priority queue with n items implemented by means of a heap
the space used is O(n)
methods add and remove_min take O(log n) time
methods len, is_empty, and min 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

44. 9.5 Adaptable Priority Queues3 a 5 g 4 e
9.5.1 Locators Implementing an Adaptable Priority Queue

45. Items and Priority Queues
Locators
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Items and Priority Queues
An item stores a (key, value) pair
Item fields:
_key: the key associated with this item
_value: the value paired with the key associated with this item
Priority Queue ADT:
add(k, x) inserts an item with key k and value x
remove_min() removes and returns the item with smallest key
min() returns, but does not remove, an item with smallest key
len(P),
is_empty()

46. E.g.: Standby airline passenger application
A standby passenger with a pessimistic attitude may become tired of waiting and decide to leave ahead of the boarding time
Operation remove min does not suffice since the passenger leaving does not necessarily have first priority.
Instead, we want a new operation, remove, that removes an arbitrary entry.
Another standby passenger finds her gold frequent-flyer card and shows it to the agent
Thus, her priority has to be modified accordingly.
To achieve this change of priority, we would like to have a new operation update allowing us to replace the key of an existing entry with a new key.

47. Locators
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Example
Online trading system where orders to purchase and sell a stock are stored in two priority queues (one for sell orders and one for buy orders) as (p,s) entries:
The key, p, of an order is the price
The value, s, for an entry is the number of shares
A buy order (p,s) is executed when a sell order (p’,s’) with price p’<p is added (the execution is complete if s’>s)
A sell order (p,s) is executed when a buy order (p’,s’) with price p’>p is added (the execution is complete if s’>s)
What if someone wishes to cancel their order before it executes?
What if someone wishes to update the price or number of shares for their order?

48. Methods of the Adaptable Priority Queue ADT
P.remove(loc):
Remove from P and return item e for locator loc.
P.update(loc,k,v):
Replace the key-value pair for locator, loc, with (k,v).
In order to implement methods update and remove efficiently, we need a mechanism
for finding a user’s element within a priority queue that avoids performing a
linear search through the entire collection. To support our goal, when a new element
is added to the priority queue, we return a special object known as a locator to
the caller. We then require the user to provide an appropriate locator as a parameter
when invoking the update or remove method, as follows, for a priority queue P:
Adaptable
英 [ əˈdæptəbl ] 美 [ əˈdæptəbəl ]
adj.可适应的; 有适应能力的; 适合的; 可改编的

49. Locators
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Locators
A locator-aware item identifies and tracks the location of its (key, value) object within a data structure
Intuitive notion:
Coat claim check
Valet claim ticket
Reservation number
Main idea:
Since items are created and returned from the data structure itself, it can return location-aware items, thereby making future updates easier
locator
英 [ ləʊˈkeɪtə(r) ] 美 [ loʊˈkeɪtər ]
n.表示位置之物，土地
Coat claim check 外套要求检查 
Valet claim ticket 代客索票 

50. List Implementation A location-aware list item is an object storing
key
value
position (or rank) of the item in the list
In turn, the position (or array cell) stores the entry
Back pointers (or ranks) are updated during swaps
trailer
header
nodes/positions
in turn
英 [ in tə:n ] 美 [ ɪn tɚn ]
释义依次; 轮流地; 相应地; 转而 2 c 4 c 5 c 8 c
items

51. Heap Implementation A location-aware heap item is an object storing
key
value
position of the item in the underlying heap
In turn, each heap position stores an item
Back pointers are updated during item swaps 4 a 2 d 6 b 8 g 5 e 9 c

54. Performance
Improved times thanks to location-aware items are highlighted in red
Method Unsorted List Sorted List Heap
len, is_empty O(1) O(1) O(1)
add O(1) O(n) O(log n)
min O(n) O(1) O(1)
remove_min O(n) O(1) O(log n)
remove O(1) O(1) O(log n)
update O(1) O(n) O(log n)

55. Summary