Lecture 2 Sorting

Sorting Problem e.g., Insertion Sort, Merge Sort

Efficiency Running time from receiving the input to producing the output. Running time Insertion Sort Merge Sort

Quiz Sample Is array a data structure?

Is array a data structure? No! A data structure is a standard part in construction of algorithms. What data structures do you know on array?

Is array a data structure? No! A data structure is a standard part in construction of algorithms. What data structures do you know on array? Stack, queue, list, …, heap.

Heapsort Heap, a data structure Max-Heapify procedure Building a heap

A Data Structure Heap A heap is an array object that can be viewed as a nearly complete binary tree. 1 6 2 3 5 3 6 5 3 2 4 1 4 5 6 2 4 1

Max-Heap

Min-Heap

Max-Heapify Max-Heapify(A,i) is a subroutine. When it is called, two subtrees rooted at Left(i) and Right(i) are max-heaps, but A[i] may not satisfy the max-heap property. Max-Heapify(A,i) makes the subtree rooted at A[i] become a max-heap by letting A[i] “float down”.

4 14 7 14 7 4 2 8 1 2 8 1 14 8 7 2 1 4

Running time

Building a Heap e.g., 4, 1, 3, 2, 16, 9, 10, 14, 8, 7.

4 1 3 10 2 16 9 8 7 14 Proof.

4 1 3 10 2 16 9 8 7 7 14

4 1 3 9 10 2 16 8 7 14

4 1 3 9 10 2 16 8 7 14

Building a Max-Heap e.g., 4, 1, 3, 2, 16, 9, 10, 14, 8, 7.

4 1 3 9 10 2 16 8 7 14

4 1 3 9 10 2 16 8 7 14

4 1 3 9 10 16 14 2 8 7

4 1 3 9 10 14 16 2 8 7

4 1 10 3 14 16 9 2 8 7

4 1 10 3 14 16 9 2 8 7

4 16 10 3 14 1 9 2 8 7

4 16 10 3 14 7 9 2 8 1

16 4 10 3 14 7 9 2 8 1

16 14 10 3 4 7 9 2 8 1

16 14 10 3 7 9 8 2 4 1

Analysis

16 14 10 3 7 9 8 2 4 1

Running time

Heapsort

Input: 4, 1, 3, 2, 16, 9, 10, 14, 8, 7. Build a max-heap 16, 14, 10, 8, 7, 9, 3, 2, 4, 1.

1 14 10 3 7 9 8 2 4 16

1 14 10 3 7 9 8 2 4 16 1, 14, 10, 8, 7, 9, 3, 2, 4, 16.

14 8 10 3 7 9 4 2 1 16

14 8 10 3 7 9 4 2 1 16 14, 8, 10, 4, 7, 9, 3, 2, 1, 16.

1 8 10 3 7 9 4 2 14 16

1 8 10 3 7 9 4 2 14 16 1, 8, 10, 4, 7, 9, 3, 2, 14, 16.

10 8 9 3 7 1 4 2 14 16

10 8 9 3 7 1 4 2 14 16 10, 8, 9, 4, 7, 1, 3, 2, 14, 16.

2 8 9 1 3 7 4 14 16 10

2 8 9 3 7 1 4 14 16 10 2, 8, 9, 4, 7, 1, 3, 10, 14, 16.

9 8 3 1 2 7 4 14 16 10

9 8 3 2 7 1 4 14 16 10 9, 8, 3, 4, 7, 1, 2, 10, 14, 16.

2 8 3 1 9 7 4 14 16 10

8 7 3 9 2 1 4 14 16 10

1 7 3 9 2 8 4 14 16 10

7 4 3 9 2 8 1 14 16 10

2 4 3 9 7 8 1 14 16 10

4 2 3 9 7 8 1 14 16 10

1 2 3 9 4 7 8 14 16 10

3 2 1 9 4 7 8 14 16 10

1 2 3 9 4 7 8 14 16 10

2 1 3 9 4 7 8 14 16 10

1 2 3 9 4 7 8 14 16 10

Running Time O(n) O(lg n)

Quicksort Worst-case running time Expected running time The best practical choice (why?)

Divide and Conquer Divide the problem into subproblems. Conquer the subproblems by solving them recursively. Combine the solutions to subproblems into the solution for original problem.

Idea of Quicksort

Example

Quicksort

How to find such a partition? Such a partition may not exist. e.g., 5, 4, 3, 2, 1. Hence, we may need to make such a partition. Take a A[i]. Classify other A[j] by comparing it with A[i].

Expected Partition

Expected Running Time

Randomized Quicksort

Randomized Quicksort

What we learnt in this lecture? What is heap, max-heap and min-heap? Heapsort and Quicksort. What is expected running time?