1. Lecture 5 HeapSort Priority queue and Heapsort
ACKNOWLEDGEMENTS: Some contents in this lecture source from COS226 of Princeton University by Kevin Wayne and Bob Sedgewick

2. Priority queues
A priority queue is a data structure for maintaining a set S of elements, each with an associated value called a key.
A max-priority queue supports the following operations:
insert(S, x)
max(S)
delete-max(S)
decrease-key(x, key)

3. Array and sorted array Maximum Insert DeleteMax Array O(n) O(1)
Heap
O(log n)

4. Data structure matters.

5. Almost complete binary tree
Array representation
complete tree with N = 16 nodes (height = 4)
Property. Height of an almost complete tree with N nodes is ⎣log N⎦.

6. Binary tree

7. Binary Heap
A max-heap (min-heap) is an almost complete binary tree with vertices satisfying the max-heap (min-heap) property:
If v and p(v) are a vertex and its parent respectively, then p(v) ≥ v (p(v) ≤ v).

8. Insert: SiftUp 16 14 10 8 7 9 3 2 18

9. Insert: SiftUp 16 14 10 8 7 9 3 2 18

10. Insert: SiftUp 16 14 10 18 7 9 3 2 8

11. Insert: SiftUp 16 18 10 14 7 9 3 2 8

12. Insert: SiftUp 18 16 10 14 7 9 3 2 8
O(logn)

13. Insert: SiftUp 16 14 10
SiftUp(H,i) 8 7 9 3 2
O(logn) 18

14. Maximum 16 14 10 8 7 9 3 2 4
Θ(1)
return H[1];

15. Delete-maximum: SiftDown16
O(2logn) 14 10 8 7 9 3
SiftDown(H,i) 2 4

16. Heap and Array Maximum Insert DeleteMax Array O(n) O(1) Sorted array
O(log n)
d-heap
O(logd n)
O(dlogd n)

17. Heap sort Question: How to make a heap?
Create max-heap with all N keys.
Repeatedly remove the maximum key.
Question: How to make a heap?

18. How to make a heap

19. Heap sort

20. Quiz
Give the array that results after heap construction on the following array of 10 keys:
X U M V Y T A J I N
Give the array that results after performing 3 successive delete-the-max on above constructed max heap.
A. Y X T V M U A N I J
B. Y X T V U M A J I N
C. U N T J I M A
D. U N T I J M A

21. Quiz What’s the time complexity of making a heap?
What’s the time complexity of heap sort?
A. Θ(n)
B. Θ(nlogn)
C. Θ(n2)
D. none of above

22. Applications Event-driven simulation. [customers in a line]
Data compression. [Huffman codes]
Graph searching.[Dijkstra's algorithm, Prim's algorithm]
Statistics. [maintain largest M values in a sequence]
Operating systems.[load balancing, interrupt handling]
Discrete optimization. [bin packing, scheduling]
......

23. Questions: Priority queue
Data structure = data representation + manipulation
Explain what is Priority queue.
Why not use array or sorted array to implement Priority queue?
9/21/2018
Xiaojuan

24. Questions: Heap
Does this array represent a max-heap?
Give the result array after following operations. (1) insert 15 (2) max (3) delete-max
9/21/2018
Xiaojuan

25. Questions: makeheap and heapsort
1. Give the array after heap construction on the following array of 10 keys:
X U M V Y T A J I N
2. Is heapsort stable? In place?
9/21/2018
Xiaojuan

26. Questions: time complexity
Give the time complexity of following operations.
Insert
Max
Delete-max
Makeheap
Heapsort
9/21/2018
Xiaojuan

27. Sort summary inplace? stable? worst average best remarks selection x
N exchanges
insertion
N 2 / 4 N
use for small N or partially ordered
shell ?
tight code, subquadratic
quick
2 N ln N
N lg N
N log N probabilistic guarantee fastest in practice
3-way quick
improves quicksort in presence of duplicate keys
merge
N log N guarantee, stable
heap
2 N lg N
N log N guarantee, in-place
???
holy sorting grail

28. Complexity of Sort a < b yes no compares
height of tree = worst-case number of compares
b < c
yes no
a < c
yes no
a b c
a < c
yes no
b < c
yes no
b a c
a c b
c a b
b c a
c b a
(at least) one leaf for each possible ordering

29. Complexity of Sort
a < b
yes no
compares
height of tree = worst-case number of compares
b < c
yes no
a < c
yes no
b < c
yes no
a b c
a < c
yes no
b a c
Proposition. Any compare-based sorting algorithm must use at least log (n!) ~ nlogn compares in the worst-case.
a c b
c a b
b c a
c b a
(at least) one leaf for each possible ordering

30. Sort summary inplace? stable? worst average best remarks selection x
N exchanges
insertion
N 2 / 4 N
use for small N or partially ordered
shell ?
tight code, subquadratic
quick
2 N ln N
N lg N
N log N probabilistic guarantee fastest in practice
3-way quick
improves quicksort in presence of duplicate keys
merge
N log N guarantee, stable
heap
2 N lg N
N log N guarantee, in-place
radix kN
for numbers with k digits,
or for string
???
holy sorting grail

31. Which sort algorithm?

32. Which sort algorithm?