Basic C Programming: Examples. Bin Li Assistant Professor Dept. of Electrical, Computer and Biomedical Engineering University of Rhode Island

Transcription

1 Basic C Programming: Examples Bin Li Assistant Professor Dept. of Electrical, Computer and Biomedical Engineering University of Rhode Island

2 Selection Sort Given n numbers to sort: Repeat the following n-1 times: Mark the first unsorted number Find the smallest unsorted number Swap the marked and smallest numbers

3 Selection Sort Given n numbers to sort: Repeat the following n-1 times: Mark the first unsorted number Find the smallest unsorted number Swap the marked and smallest numbers

4 Selection Sort Given n numbers to sort: Repeat the following n-1 times: Mark the first unsorted number Find the smallest unsorted number Swap the marked and smallest numbers How efficient is selection sort? In general, given n numbers to sort, it performs n 2 comparisons Why might selection sort be a good choice? Simple to write code Intuitive

5 Selection Sort Given n numbers to sort: Repeat the following n-1 times: Mark the first unsorted number Find the smallest unsorted number Swap the marked and smallest numbers Try one!

6 Bubble Sort Given n numbers to sort: Repeat the following n-1 times: For each pair of adjacent numbers: If the number on the left is greater than the number on the right, swap them.

7 Bubble Sort Given n numbers to sort: Repeat the following n-1 times: For each pair of adjacent numbers: If the number on the left is greater than the number on the right, swap them.

8 Bubble Sort Given n numbers to sort: Repeat the following n-1 times: For each pair of adjacent numbers: If the number on the left is greater than the number on the right, swap them How efficient is bubble sort? In general, given n numbers to sort, it performs n 2 comparisons The same as selection sort Is there a simple way to improve on the basic bubble sort? Yes! Stop after going through without making any swaps This will only help some of the time

9 Bubble Sort Given n numbers to sort: Repeat the following n-1 times: For each pair of adjacent numbers: If the number on the left is greater than the number on the right, swap them Try one!

10 Bubble Sort #include <stdio.h> #define MAXSIZE 10 void main() { int array[maxsize]; int i, j, num, temp; printf("enter the value of num \n"); scanf("%d", &num); printf("enter the elements one by one \n"); for (i = 0; i < num; i++) { scanf("%d", &array[i]); } printf("input array is \n"); for (i = 0; i < num; i++) { printf("%d\n", array[i]); } /* Bubble sorting begins */ for (i = 0; i < num; i++) { for (j = 0; j < (num - i - 1); j++) { if (array[j] > array[j + 1]) { temp = array[j]; array[j] = array[j + 1]; array[j + 1] = temp; } } } printf("sorted array is...\n"); for (i = 0; i < num; i++) { printf("%d\n", array[i]); } } Given n numbers to sort: Repeat the following n-1 times: For each pair of adjacent numbers: If the number on the left is greater than the number on the right, swap them.

11 Merge Sort Given n numbers to sort: Divide: Divide the n-element sequence to be sorted into two subsequences of n/2 elements each Conquer: Sort the two subsequences recursively using merge sort. Combine: Merge the two sorted subsequences to produce the sorted answer.

12 Merge-Sort (A, p, r) INPUT: a sequence of n numbers stored in array A OUTPUT: an ordered sequence of n numbers MergeSort (A, p, r) // sort A[p..r] by divide & conquer 1 if p < r 2 then q (p+r)/2 3 MergeSort (A, p, q) 4 MergeSort (A, q+1, r) 5 Merge (A, p, q, r) // merges A[p..q] with A[q+1..r] Initial Call: MergeSort(A, 1, n)

13 Procedure Merge Merge(A, p, q, r) 1 n 1 q p n 2 r q 3 for i 1 to n 1 4 do L[i] A[p + i 1] 5 for j 1 to n 2 6 do R[j] A[q + j] 7 L[n 1 +1] 8 R[n 2 +1] 9 i 1 10 j 1 11 for k p to r 12 do if L[i] R[j] 13 then A[k] L[i] 14 i i else A[k] R[j] 16 j j + 1 Input: Array containing sorted subarrays A[p..q] and A[q+1..r]. Output: Merged sorted subarray in A[p..r].

14 Execution Example Partition

15 Execution Example (Cont ) Recursive call, partition

16 Execution Example (Cont.) Recursive call, partition

17 Execution Example (Cont ) Recursive call, base case

18 Execution Example (Cont ) Recursive call, base case

19 Execution Example (Cont ) Merge

20 Execution Example (Cont ) Recursive call,, base case, merge

21 Execution Example (Cont ) Merge

22 Execution Example (Cont ) Recursive call,, merge, merge

23 Execution Example (Cont ) Merge

24 Analysis of Merge-Sort The height h of the merge-sort tree is O(log n) at each recursive call we divide in half the sequence, The overall amount or work done at the nodes of depth i is O(n) we partition and merge 2 i sequences of size n/2 i we make 2 i+1 recursive calls Thus, the total running time of merge-sort is O(n log n) depth #seqs size 0 1 n 1 2 n/2 i 2 i n/2 i 24

25 Summary of Sorting Algorithms Algorithm Time Notes selection-sort O(n 2 ) bubble-sort O(n 2 ) merge-sort O(n log n) slow in-place for small data sets (< 1K) slow in-place for small data sets (< 1K) fast sequential data access for huge data sets (> 1M) 25

26 Tower of Hanoi There are three towers 64 gold disks, with decreasing sizes, placed on the first tower You need to move all of the disks from the first tower to the last tower Larger disks can not be placed on top of smaller disks The third tower can be used to temporarily hold disks

27 Tower of Hanoi (Cont ) The disks must be moved within one week. Assume one disk can be moved in 1 second. Is this possible? To create an algorithm to solve this problem, it is convenient to generalize the problem to the N-disk problem, where in our case N = 64.

28 Recursive Solution

29 Recursive Solution

30 Recursive Solution

31 Recursive Solution

32 Tower of Hanoi

33 Tower of Hanoi

34 Tower of Hanoi

35 Tower of Hanoi

36 Tower of Hanoi

37 Tower of Hanoi

38 Tower of Hanoi

39 Tower of Hanoi

40 Recursive Algorithm void Hanoi(int n, char source, char destination, char spare) { if (n == 1) /* base case */ printf( Move top disk from pole %c, source); printf( to pole %c\n, destination); else { /* recursion */ Hanoi(n-1,source,spare,destination); Hanoi(1, source, destination, spare); Hanoi(n-1,spare, destination, source); } }

41 Some Puzzles Rubik s Cube Sudoku

42 Monte-Carlo Methods 1953, Nicolaus Metropolis Monte Carlo method refers to any method that makes use of random numbers Simulation of natural phenomena Simulation of experimental apparatus Numerical analysis 42

43 Estimating π using Monte Carlo The probability of a random point lying inside the unit circle: Pr xx 2 + yy 2 < 1 = AA cccccccccccc AA ssssssssssss = ππ 4 (x,y) If pick a random point N times and M of those times the point lies inside the unit circle: MM NN ππ 4 as N goes to infinity N M 43

44 The Strong Law of Large Numbers Let XX 1, XX 2, be a sequence of independent identically distributed random variables with mean μμ. Then, the sequence of sample means MM nn = (XX XX nn )/nn converges to μμ, with probability 1, in the sense that Pr XX XX nn lim nn nn = μμ = 1

45 Estimating π using Monte Carlo (Cont ) double x, y, pi; const long m_nmaxsamples = 1e8; long count=0; for (long k=0; k<m_nmaxsamples; k++) { x=2.0*rand() 1.0; // Map to the range [-1,1] y=2.0*rand() 1.0; if (x*x+y*y<=1.0) count++; } pi=4.0 * (double)count / (double)m_nmaxsamples; 45

46 The Central Limit Theorem Theorem (The Central Limit Theorem) The CDF of ZZ nn = XX 1 + +XX nn nnμμ converges to standard normal CDF σσ nn Φ zz = 1 zz 2ππ ee xx2 /2 dddd in the sense that lim P ZZ nn zz = Φ(zz) nn

47 Queueing Model qq kk : Queue length (number of jobs in the queue) at the beginning of time slot kk aa kk : Variable which takes on a value 1 if there was an arrival in time slot k and 0 otherwise, i.e., it is an indicator variable indicating if an arrival occurred ss kk : Indicator variable indicating if there was a service in time slot kk or not.

48 Queueing Dynamics where qq kk + 1 = max{qq kk + aa kk ss kk, 0} 1 wwwwwww pppppppppppppppppppppp λλ aa kk = 0 wwwwwww pppppppppppppppppppppp 1 λλ ss kk = 1 wwwwwww pppppppppppppppppppppp μμ 0 wwwwwww pppppppppppppppppppppp 1 μμ

49 Queue Simulations T = 1e6; //simulation time lambda = 0.8; // arrival rate mu = 0.9; //service rate for (t = 0; t < T; t++) { A(t) = (rand() < lambda); S(t) = (rand() < mu); Q(t+1) = max(q(t)+a(t)-s(t), 0); } Get the average queue length

50 Opening and Closing Files Files can be opened by calling fopen: FILE *fopen(const char *filename, const char *mode); See the next slide to see different modes. If the file cannot be opened, fopen returns NULL. All subsequent file-processing functions after the file is opened must refer to the file with the appropriate file pointer. Files can be closed by calling fclose: int fclose(file *stream); If function fclose is not called explicitly, the operating system normally will close the file when program execution terminates.

51 Example 1 #define _CRT_SECURE_NO_DEPRECATE #include <stdio.h> int main() { FILE *fp; fp = fopen("test.txt", "w"); fprintf(fp, "Testing...\n"); fclose (fp); } 51

52 Example 2 #define _CRT_SECURE_NO_DEPRECATE #include <stdio.h> #define NumofPoints 10 int main() { FILE *fp; // fp = output.txt file pointer double ArrialRate[NumofPoints] = { 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95}; double meanqueue[numofpoints] = { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}; int i; fp = fopen("output.txt", "w"); fprintf(fp, "ArrivalRate MeanQueue\n"); for (i = 0; i < NumofPoints; i++) { meanqueue[i] = 1 / (1 - ArrialRate[i]); fprintf(fp, "%f %f\n", ArrialRate[i], meanqueue[i]); } fclose(fp); }