ESc101 : Fundamental of Computing
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1 ESc101 : Fundamental of Computing I Semester Lecture 36 Announcement : Extra sessions for lab test Sorting algorithms based on recursion Quick Sort (did in lst class) Merge Sort Introduction to Time Complexity 1
2 Extra session for lab On Sunday- 9th November B1 and B2 : Nikhi Jain CS :30 PM B3 and B4 : Kshitiz Garg CS :30 PM. B5 and B6 : Abhinav/Nitin CS101 3:45-5:15PM B7 and B8 : Vikas Mards CS102 3:45-5:15PM B9 and B10: Paras Tikamani CS101 5:15-6:45PM 2
3 Optional extra class Saturday- 8th November at 5:15 PM in CS101 3
4 Tower of Hanoi problem Hints: You can see that for 1 or 2 discs, there is a straight forward solution. Now for the problem with n + 1 discs, assume you have a black box method which can transfer n discs from one tower to another. How can you solve the given problem (that is, n + 1 discs) using that method. Now try to fill in the details of the recursive method. The final solution (recursive method) is very small... 4
5 Quick Sort 5
6 Select an element x. Quick Sort : sorting a set S of numbers Partition the set S of numbers into two parts: S <x : the subset consisting of numbers less than x. S >x : the subset consisting of numbers greater than x. Recursively sort the two sets separately S <x and S >x, and concatenate them. 6
7 Quick Sort : when set is represented as array public static void Qsort(int[] A, int left, int right) if(left<right) int mid = partition(a, left, right); Qsort(A,left,mid-1); Qsort(A,mid+1,right); You can observe that the size of problem corresponding to recursive calls decreases always. Hence the program will eventually terminate. 7
8 Merge Sort 8
9 Merging two sorted arrays Problem : Given two sorted arrays A and B, produce another sorted array C A B. Trivial Solution : Copy the elements of A and B into C, then sort C Missing Point : A and B are already sorted 9
10 Merging two sorted arrays Problem : Given two sorted arrays A and B, produce another sorted array C A B. Trivial Solution : Copy the elements of A and B into C, then sort C Missing Point : A and B are already sorted 10
11 Merging two sorted arrays Problem : Given two sorted arrays A and B, produce another sorted array C A B. Trivial Solution : Copy the elements of A and B into C, then sort C Missing Point : A and B are already sorted 11
12 Merging two sorted arrays : a better solution start scanning A and B from left, compare two elements of A and B, copy the smaller one into C and continue... Let us consider an example... 12
13 Merging two sorted arrays : a better solution start scanning A and B from left, compare two elements of A and B, copy the smaller one into C and continue... Let us consider an example... 13
14 Example : Merging two sorted arrays A B C 14
15 Example : Merging two sorted arrays A B C 3 15
16 Example : Merging two sorted arrays A B C
17 Example : Merging two sorted arrays A B C
18 Example : Merging two sorted arrays A B C
19 Example : Merging two sorted arrays A B C
20 Merge Sort Key Idea : Merging two sorted arrays is easier than sorting their union. 20
21 Merge Sort on an array A Sort the first half of array A recursively Sort the second half of the array recursively merge the two halves. 21
22 Merge Sort on an array A public static void mergesort(int[] A, int left, int right) if(?? )?? ;?? ;?? ;?? ; Convince yourself that each recursive call makes some progress, that is,approaches the base case. This is very important rule that each recursive method must obey. 22
23 Merge Sort on an array A public static void mergesort(int[] A, int left, int right) if(left<right)?? ;?? ;?? ;?? ; Convince yourself that each recursive call makes some progress, that is,approaches the base case. This is very important rule that each recursive method must obey. 23
24 Merge Sort on an array A public static void mergesort(int[] A, int left, int right) if(left!=right) int mid = (left+right)/2;?? ;?? ;?? ; Convince yourself that each recursive call makes some progress, that is,approaches the base case. This is very important rule that each recursive method must obey. 24
25 Merge Sort on an array A public static void mergesort(int[] A, int left, int right) if(left!=right) int mid = (left+right)/2; mergesort(a, left, mid);?? ;?? ; Convince yourself that each recursive call makes some progress, that is,approaches the base case. This is very important rule that each recursive method must obey. 25
26 Merge Sort on an array A public static void mergesort(int[] A, int left, int right) if(left!=right) int mid = (left+right)/2; mergesort(a, left, mid); mergesort(a, mid+1, right);?? ; Convince yourself that each recursive call makes some progress, that is,approaches the base case. This is very important rule that each recursive method must obey. 26
27 Merge Sort on an array A public static void mergesort(int[] A, int left, int right) if(left!=right) int mid = (left+right)/2; mergesort(a, left, mid); mergesort(a, mid+1, right); merge(a,left,mid,right); Convince yourself that each recursive call makes some progress, that is,approaches the base case. This is very important rule that each recursive method must obey. 27
28 Merge Sort on an array A public static void mergesort(int[] A, int left, int right) if(left!=right) int mid = (left+right)/2; mergesort(a, left, mid); mergesort(a, mid+1, right); merge(a,left,mid,right); Convince yourself that each recursive call makes progress, that is, it approaches the base case. This is very important rule that each recursive method must obey. 28
29 Recursion Tree When a method makes two or more recursive calls to itself, it is better to view the execution as a tree. 29
30 Recursion Tree for Fibonacci number public static int fib(int n) if(n==0) return 0; else if(n==1) return 1; else return fib(n-1)+fib(n-2); Example : n=4 30
31 Recursion Tree for Fibonacci number for n = 4 2 fib(4) 1 1 fib(3) 1 1 fib(2) 0 1 fib(2) 0 fib(1) fib(1) fib(0) fib(1) fib(0) 31
32 Recursion Tree for Merge sort public static void mergesort(int[] A, int left, int right) if(left!=right) int mid = (left+right)/2; mergesort(a, left, mid); mergesort(a, mid+1, right); merge(a,left,mid,right); Example : A =99, 7, 5, 1, 67, 11, 4, 2 32
33 for A =99, 7, 5, 1, 67, 11, 4, 2 Recursion Tree for Merge sort Note : In the next few slides, for sake of compactness we shall use MSort() to denote mergesort(). We show the status of A as we move up the recursion tree level by level. 33
34 Base case : Level 0 (no processing required) MSort(A,0,7) Level 3 MSort(A,0,3) MSort(A,4,7) Level 2 MSort(A,0,1) MSort(A,2,3) MSort(A,4,5) MSort(A,6,7) Level 1 MSort(A,0,0) MSort(A,1,1) MSort(A,2,2) MSort(A,3,3) MSort(A,4,4) MSort(A,5,5) MSort(A,6,6) MSort(A,7,7) Level 0 A
35 Passing results to Level 1 MSort(A,0,7) Level 3 MSort(A,0,3) MSort(A,4,7) Level 2 MSort(A,0,1) MSort(A,2,3) MSort(A,4,5) MSort(A,6,7) Level 1 MSort(A,0,0) MSort(A,1,1) MSort(A,2,2) MSort(A,3,3) MSort(A,4,4) MSort(A,5,5) MSort(A,6,6) MSort(A,7,7) Level 0 A
36 Merging at Level 1 MSort(A,0,7) Level 3 MSort(A,0,3) MSort(A,4,7) Level 2 MSort(A,0,1) MSort(A,2,3) MSort(A,4,5) MSort(A,6,7) Level 1 MSort(A,0,0) MSort(A,1,1) MSort(A,2,2) MSort(A,3,3) MSort(A,4,4) MSort(A,5,5) MSort(A,6,6) MSort(A,7,7) Level 0 A
37 Passing results to Level 2 MSort(A,0,7) Level 3 MSort(A,0,3) MSort(A,4,7) Level 2 MSort(A,0,1) MSort(A,2,3) MSort(A,4,5) MSort(A,6,7) Level 1 Level 0 MSort(A,0,0) MSort(A,1,1) MSort(A,2,2) MSort(A,3,3) MSort(A,4,4) MSort(A,5,5) MSort(A,6,6) MSort(A,7,7) A
38 Merging at Level 2 MSort(A,0,7) Level 3 MSort(A,0,3) MSort(A,4,7) Level 2 MSort(A,0,1) MSort(A,2,3) MSort(A,4,5) MSort(A,6,7) Level 1 Level 0 MSort(A,0,0) MSort(A,1,1) MSort(A,2,2) MSort(A,3,3) MSort(A,4,4) MSort(A,5,5) MSort(A,6,6) MSort(A,7,7) A
39 Passing result to Level 3 MSort(A,0,7) Level 3 MSort(A,0,3) MSort(A,4,7) Level 2 MSort(A,0,1) MSort(A,2,3) MSort(A,4,5) MSort(A,6,7) Level 1 MSort(A,0,0) MSort(A,1,1) MSort(A,2,2) MSort(A,3,3) MSort(A,4,4) MSort(A,5,5) MSort(A,6,6) MSort(A,7,7) Level 0 A
40 Merging at Level 3 MSort(A,0,7) Level 3 MSort(A,0,3) MSort(A,4,7) Level 2 MSort(A,0,1) MSort(A,2,3) MSort(A,4,5) MSort(A,6,7) Level 1 MSort(A,0,0) MSort(A,1,1) MSort(A,2,2) MSort(A,3,3) MSort(A,4,4) MSort(A,5,5) MSort(A,6,6) MSort(A,7,7) Level 0 A
41 Recursion Tree for Quick sort Do it as homework 41
42 Comparing Three sorting algorithms for(n=1000,n<20000;n=n+1000) Generate an array A of size n; Fill it with random integers; Create copies B and C of array A; Execute Quick sort on A and measure time. Execute Merge sort on A and measure time. Execute SelectionSort on B and measure time. 42
43 Measuring time taken a method M long start = System.currentTimeMillis(); M(); long stop = System.currentTimeMillis(); System.out.println(stop-start); Note : System.currentTimeMillis() returns a long which corresponds to current time in milliseconds. 43
44 Comparing Three sorting algorithms Experimental Observations Quick sort is more efficient than merge sort Merge sort is more efficient than Selection Sort Please study the program : Three sorting algos.java 44
45 What is the reason for different running times? Given that all of them has same input and output all of them are executed on the same machine We need to analyze the number of steps/instruction taken by each sorting algorithm 45
46 How many steps/instructions are executed by the following loop? for(int i=1; i<=n; i=i+1) sum = sum + i; Steps = 1+3n+1 46
47 How many steps/instructions are executed by the following loop? for(int i=1; i<=n; i=i+1) sum = sum + i; Steps = 1 + 3n
48 How many steps/instructions are executed by the following loop? for(int n=1;n<=m;n=n+1) for(int i=1; i<=n; i=i+1) sum = sum + i; Steps =1 + m + n=m n=1 (1 + 3n + 1) + m
49 How many steps/instructions are executed by the following loop? for(int n=1;n<=m;n=n+1) for(int i=1; i<=n; i=i+1) sum = sum + i; Steps = 1 + m + n=m n=1 (1 + 3n + 1) + m
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