1. CS210- Lecture 14 July 5, 2005 Agenda Inserting into Heap
Removing from Heap
Heap Sort
Heap Construction
Top down Heap Construction
Bottom up Heap Construction
Adaptable PQ
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CS210-Summer 2005, Lecture 14

2. Heaps
An efficient realization of a priority queue uses a data structure called a heap.
Heap allows us to perform both insertions and removals in logarithmic time.
Heap stores entries in a binary tree rather than in a list.
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CS210-Summer 2005, Lecture 14

3. Heaps and Priority Queues
We can use a heap to implement a priority queue
We store a (key, element) item at each node.
We keep track of the position of the last node
(2, Sue)
(5, Pat)
(6, Mark)
(9, Jeff)
(7, Anna)
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CS210-Summer 2005, Lecture 14

4. Insertion into a Heap
(2, C)
Method insert of the priority queue ADT corresponds to the insertion of an entry (k,x) to the heap
The insertion algorithm consists of three steps
Add a new node z (the new last node)
Store e at z
Restore the heap-order property (discussed next)
(5, A)
(6, Z) z
(15,K)
(9, F)
insertion node
(2, C)
(5, A)
(6, Z) z
(15,K)
(9, F)
(1, D)
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CS210-Summer 2005, Lecture 14

5. Upheap
After the insertion of a new entry e, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping e along an upward path from the insertion node
Upheap terminates when the entry e reaches the root or a node whose parent has a key smaller than or equal to key k of e.
Since a heap has height O(log n), upheap runs in O(log n) time
(2, C)
(1, D)
(5, A)
(1,D)
(5, A)
(2, C) z z
(15,K)
(9, F)
(6, Z)
(9, F)
(6, Z)
(15,K)
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CS210-Summer 2005, Lecture 14

6. Removal from a Heap
Method removeMin of the priority queue ADT corresponds to the removal of the root entry from the heap
The removal algorithm consists of three steps
Replace the root entry with the entry of the last node w
Remove w
Restore the heap-order property (discussed next)
(2, C)
(5, A)
(6, Z) w
(15,K)
(9, F)
last node
(9, F)
(5, A)
(6, Z) w
(15,K)
new last node
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CS210-Summer 2005, Lecture 14

7. Downheap
After replacing the root entry with the entry e of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping entry e along a downward path from the root
Upheap terminates when key k of entry e 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
(9, F)
(5, A)
(5, A)
(6, Z)
(9, F)
(6, Z) w w
(15,K)
(15,K)
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CS210-Summer 2005, Lecture 14

8. Heap-Sort
Consider a priority queue with n items implemented by means of a heap
methods insert and removeMin take O(log n) time
methods size, isEmpty, 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
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9. Heap Construction
We can construct a heap storing n entries in O(nlogn) time, by means of n successive insert operations.
However if all the n key-value pairs to be stored in the heap are given in advance, there is an alternative bottom up construction method that runs in O(n) time.
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10. Bottom up Heap Construction16 15 4 12 6 7 23 20 25 5 11 27 9 8 14 16 4 15 12 6 7 23 20 25 5 11 27 9 8 14
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11. Bottom up Heap Construction16 4 15 12 6 7 8 20 25 5 11 27 9 23 14 16 15 4 12 6 7 8 20 25 5 11 27 9 23 14
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12. Bottom up Heap Construction16 4 15 12 6 7 8 20 25 5 11 27 9 23 14 16 15 4 12 6 7 8 20 25 5 11 27 9 23 14
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13. Bottom up Heap Construction16 4 15 12 5 7 8 20 25 6 11 27 9 23 14 16 15 4 12 5 7 8 20 25 6 11 27 9 23 14
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14. Bottom up Heap Construction16 4 15 12 5 7 8 20 25 6 11 27 9 23 14 16 15 4 12 5 7 8 20 25 6 11 27 9 23 14
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15. Bottom up Heap Construction16 4 15 12 5 7 8 20 25 6 11 27 9 23 14 16 15 4 12 5 7 8 20 25 6 11 27 9 23 14
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16. Bottom up Heap Construction16 4 5 12 6 7 8 20 25 15 11 27 9 23 14 16 5 4 12 6 7 8 20 25 15 11 27 9 23 14
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17. Bottom up Heap Construction4 7 5 12 6 9 8 20 25 15 11 27 16 23 14 16 5 4 12 6 7 8 20 25 15 11 27 9 23 14
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18. Adaptable Priority Queues
Suppose we have an online trading system where orders to purchase and sell a given 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
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?
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CS210-Summer 2005, Lecture 14

19. Methods of the Adaptable PQ ADT
remove(e): Remove and return entry e.
replaceKey(e,k): Replace with k and return the key of entry e of P; an error condition occurs if k is invalid (that is, k cannot be compared with other keys).
replaceValue(e,x): Replace with x and return the value of entry e of P.
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20. Example Operation Output P insert(5,A) e1 (5,A)
insert(3,B) e2 (3,B),(5,A)
insert(7,C) e3 (3,B),(5,A),(7,C)
min() e2 (3,B),(5,A),(7,C)
key(e2) 3 (3,B),(5,A),(7,C)
remove(e1) e1 (3,B),(7,C)
replaceKey(e2,9) 3 (7,C),(9,B)
replaceValue(e3,D) C (7,D),(9,B)
remove(e2) e2 (7,D)
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21. Locating Entries
In order to implement the operations remove(k), replaceKey(e), and replaceValue(k), we need fast ways of locating an entry e in a priority queue.
We can always just search the entire data structure to find an entry e, but there are better ways for locating entries.
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22. Location-Aware Entries
A locator-aware entry identifies and tracks the location of its (key, value) object within a data structure
Main idea:
Since entries are created and returned from the data structure itself, it can return location-aware entries, thereby making future updates easier
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23. List Implementation A location-aware list entry 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
trailer
header
nodes/positions 2 c 4 c 5 c 8 c
entries
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24. Heap Implementation A location-aware heap entry is an object storing
key
value
position of the entry in the underlying heap
In turn, each heap position stores an entry
Back pointers are updated during entry swaps 2 d 4 a 6 b 8 g 5 e 9 c
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25. Performance
Using location-aware entries we can achieve the following running times:
Method Unsorted List Sorted List Heap
size, isEmpty O(1) O(1) O(1)
insert O(1) O(n) O(log n)
min O(n) O(1) O(1)
removeMin O(n) O(1) O(log n)
remove O(1) O(1) O(log n)
replaceKey O(1) O(n) O(log n)
replaceValue O(1) O(1) O(1)
7/28/2019
CS210-Summer 2005, Lecture 14