Quicksort
Quicksort as a pivot-selection and partition-sorting algorithm.
Kevin Lin, with thanks to many others.
1

Valid Partitions
Given the array [4, 5, 1, 3, 7, 2, 6], which arrays are valid partitions on the pivot 4?
[4, 5, 1, 3, 7, 2, 6]
[1, 2, 3, 4, 5, 6, 7]
[1, 7, 5, 3, 4, 2, 6]
[1, 3, 4, 2, 5, 7, 6]
[1, 3, 4, 2, 5, 6, 7]
[2, 1, 3, 4, 5, 7, 6]
[3, 2, 1, 4, 7, 5, 7]
2
Q
Given the array [4, 5, 1, 3, 7, 2, 6], which arrays are valid partitions on the pivot 4?

Q1: [4, 5, 1, 3, 7, 2, 6]


Q2: [1, 2, 3, 4, 5, 6, 7]


Q3: [1, 7, 5, 3, 4, 2, 6]


Q4: [1, 3, 4, 2, 5, 7, 6]


Q5: [1, 3, 4, 2, 5, 6, 7]


Q6: [2, 1, 3, 4, 5, 7, 6]


Q7: [3, 2, 1, 4, 7, 5, 7]

Valid Partitions
Given the array [4, 5, 1, 3, 7, 2, 6], which arrays are valid partitions on the pivot 4?
[4, 5, 1, 3, 7, 2, 6]
[1, 2, 3, 4, 5, 6, 7]
[1, 7, 5, 3, 4, 2, 6]
[1, 3, 4, 2, 5, 7, 6]
[1, 3, 4, 2, 5, 6, 7]
[2, 1, 3, 4, 5, 7, 6]
[3, 2, 1, 4, 7, 5, 7]
3
A

Hoare Partitioning
Given the array [4, 5, 1, 3, 7, 2, 6] and pivot 6, give the array after the first swap. Then, give the array after the entire partition operation is complete.




Is Hoare partitioning stable?
4
Q
Q1: Given the array [4, 5, 1, 3, 7, 2, 6] and pivot 6, give the array after the first swap.







Q2: Then, give the array after the entire partition operation is complete.







Q3: Is Hoare partitioning stable?

Hoare Partitioning
Given the array [4, 5, 1, 3, 7, 2, 6] and pivot 6, give the array after the first swap. Then, give the array after the entire partition operation is complete.




Is Hoare partitioning stable?
5
A

Selection Algorithms
Selection. Given an array of N items, find item of rank K.
For finding the median, choose K = N / 2.

Characterize the difficulty of this problem.
Why is the time complexity of selection in Ω(N)?
Describe an O(N log N)	runtime algorithm for selection for any K.
Describe an O(N)		runtime algorithm for selection for K = 0, 1, 2.
6
Algorithms (Robert Sedgewick, Kevin Wayne/Princeton)
Q
Q1: Why is the time complexity of selection in Ω(N)?







Q2: Describe an O(N log N) runtime algorithm for selection for any K.









Q3: Describe an O(N) runtime algorithm for selection for K = 0, 1, 2.

Selection Algorithms
Why is the time complexity of selection in Ω(N)?

Describe an O(N log N) runtime algorithm for selection for any K.

Describe an O(N) runtime algorithm for selection for K = 0, 1, 2.

7
A

Partition Sort
Consider a generalization of quicksort: PartitionSort(input, lo, hi, subsort).
If hi - lo ≤ 1, return.
Partition around input[lo], the leftmost item of the current subproblem.
Call subsort on the left and right subproblems.
Give the worst-case runtime for each call below.
PartitionSort(input, 0, N, InsertionSort)
PartitionSort(input, 0, N, MergeSort)
PartitionSort(input, 0, N, PartitionSort)

8
Q
Give the worst-case runtime for each call below.

Q1: PartitionSort(input, 0, N, InsertionSort)







Q2: PartitionSort(input, 0, N, MergeSort)







Q3: PartitionSort(input, 0, N, PartitionSort)

Partition Sort
PartitionSort(input, 0, N, InsertionSort)

PartitionSort(input, 0, N, MergeSort)

PartitionSort(input, 0, N, PartitionSort)

9
A

10
Sort
Best-Case
Worst-Case
Space
Stable
Notes
Selection sort
Θ(N2)
Θ(N2)
Θ(1)
No
Heapsort
Θ(N)
Θ(N log N)
Θ(1)
No
Slow in practice.
Merge sort
Θ(N log N)
Θ(N log N)
Θ(N)
Yes
Fastest stable sort.
Insertion sort
Θ(N)
Θ(N2)
Θ(1)
Yes
Best for small or almost sorted inputs.
Naive quicksort
Θ(N log N)
Θ(N2)
Θ(N)
Yes
2x or more slower than merge sort.
Java quicksort
Θ(N)
Θ(N2)
O(N)
No
Fastest comparison sort.