Lecture 2 Sorting Sorting Problem e g Insertion

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

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

Quiz Sample • Is array a data structure?

Is array a data structure? • No! • A data structure is a data organization associated with a set of operations (standard algorithms) for efficiently using the data. • 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.

A Data Structure Heap • A heap is a nearly complete binary tree which can be easily implemented on an 1 array. 6 2 3 5 3 4 2 5 4 6 1 6 5 3 2 4 1

Nearly complete binary tree • Every level except bottom is complete. • On the bottom, nodes are placed as left as possible. 1 1 6 6 2 3 5 4 2 6 1 No! 4 2 3 5 3 4 2 5 4 Yes! 6 1

Algorithms associated with Heap

Two Special Heaps • Max-Heap • Min-Heap

Max-Heap

Min-Heap

Algorithms associated with Max-Heap • Three algorithms associated with heap. • In addition, it is associated with two more algorithms: Max-Heapify and Build-Max. 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”.

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

Running time

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

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

4 1 2 14 3 9 16 8 7 7 10

4 1 2 14 3 16 8 7 9 10

4 1 2 14 3 16 8 7 9 10

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

4 1 2 14 3 16 8 7 9 10

4 1 2 14 3 16 8 7 9 10

4 1 16 14 2 3 8 7 9 10

4 1 14 2 3 16 8 7 9 10

4 1 14 2 10 16 8 7 9 3

4 1 14 2 10 16 8 7 9 3

4 16 14 2 10 1 8 7 9 3

4 16 14 2 10 7 8 1 9 3

16 4 14 2 10 7 8 1 9 3

16 14 4 2 10 7 8 1 9 3

16 14 7 8 2 10 4 1 9 3

Analysis

16 14 7 8 2 10 4 1 9 3

Running time

Heapsort

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

1 14 7 8 2 10 4 16 9 3

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

14 8 7 4 2 10 1 16 9 3

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

1 8 7 4 2 10 14 16 9 3

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

10 8 7 4 2 9 14 16 1 3

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

2 8 7 4 10 9 14 16 1 3

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

9 8 7 4 10 3 14 16 1 2

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

2 8 7 4 10 3 14 16 1 9

8 7 2 4 10 3 14 16 1 9

1 7 2 4 10 3 14 16 8 9

7 4 2 1 10 3 14 16 8 9

2 4 7 1 10 3 14 16 8 9

4 2 7 1 10 3 14 16 8 9

1 2 4 10 3 7 14 16 8 9

3 2 4 10 1 7 14 16 8 9

1 2 4 10 3 7 14 16 8 9

2 1 4 10 3 7 14 16 8 9

1 2 4 10 3 7 14 16 8 9

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?