Algorithms and Programming ChinSung Lin Eleanor Roosevelt High

Algorithms and Programming Chin-Sung Lin Eleanor Roosevelt High School

Algorithms and Programming • Types of Algorithms – Brute-Force (or Exhaustive Search) – Divide and Conquer – The Greedy Method – Recursion • Algorithms of Skyscrapers Project – Review of Pseudo Code – Decimal-to-Binary Conversion Algorithm • Programming: A Crash Course of C • Sorting Algorithms and Implementations • Searching Algorithms and Implementations

Brute-Force Algorithms

Brute-Force (or Exhaustive Search) • Brute-force is a straightforward approach to solving a problem, usually directly based on the problem’s statement and definitions of the concepts involved. • Try out all possible values or combinations to solve a problem, or try every possible solution to see which is best.

Brute-Force (or Exhaustive Search) Example • Example: Try out all possible combinations to open a lock. • Example: Try out all possible password values to log into an account.

Brute-Force (or Exhaustive Search) Example • Example: Compute an for a given number a and a nonnegative integer n. By the definition of exponentiation, an = a a a. n times • Example: Build 250 -block skyscraper by adding one block each week.

Brute-Force (or Exhaustive Search) Though inefficient in general, the brute-force algorithm should not be overlooked as an important algorithm design strategy for the following reasons: • It is applicable to a very wide variety problems, and seems to be the only general approach for all problems. • The expense of designing a more efficient algorithm may be unjustifiable if only a few instances of a problem need to be solved and a brute-force algorithm can solve those instances with acceptable speed. • It is useful for solving small-size instances of a problem. • It can serve as an important theoretical or education purpose, e. g. , as a benchmark to judge more efficient alternatives algorithms.

Divide and Conquer Algorithms

Divide and Conquer • Repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually recursively) until the instances are small enough to solve easily. The solutions to the subproblems are then combined to give a solution to the original problem.

Divide and Conquer The most well known algorithm design strategy: 1. Divide the problem into two or more smaller subproblems. 2. Conquer the subproblems by solving them recursively. 3. Combine the solutions to the subproblems into the solutions for the original problem.

Divide and Conquer Example: Large-integer multiplication (The Grade-School Algorithm) x a 1 a 2 … a n b 1 b 2 … b n (d 10) d 11 d 12 … d 1 n (d 20) d 21 d 22 … d 2 n ………………… (dn 0) dn 1 dn 2 … dnn This process requires n 2 single digit multiplications. For 1231 2012, this process requires 16 multiplications.

Divide and Conquer Example Large-integer multiplication (Divide and Conquer Algorithm) 1231 2012 = (12*102 + 31) * (20*102 + 12) = (12*20)*104 + c 1*102 + 31*12 where c 1 = (12+31)*(20+12) – 12*20 – 31*12 = 764 12*20 = (1*10 + 2) * (2*10 + 0) = (1*2)*102 + c 2*10 + 2*0 where c 2 = (1+2)*(2+0) – 1*2 – 2*0 = 4 31*12 = (3*10 + 1) * (1*10 + 2) = (3*1)*102 + c 3*10 + 1*2 where c 3 = (3+1)*(1+2) – 3*1 – 1*2 = 7 (12+31)*(20+12) = 43 * 32 = (4*10 + 3) * (3*10 + 2) = (4*3)*102 + c 4*10 + 3*2 where c 4 = (4+3)*(3+2) – 4*3 – 3*2 = 17 This process requires 9 digit multiplications as opposed to 16.

Greedy Algorithms

Greedy Algorithms • An optimization problem is one in which you want to find, not just a solution, but the best solution. • It is not exhaustive, and does not give an accurate answer to many problems. But when it works, it is the fastest method. • A greedy algorithm works in phases. At each phase: – You take the best you can get right now, without regard for future consequences – You hope that by choosing a local optimum at each step, you will end up at a global optimum 14

Greedy Algorithm Example • Example: Pay out the exact amount of money using US monetary system. • Example: The knapsack problem – Given a set of items, each with a mass and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

Greedy Algorithm Example • Example: Dijkstra's algorithm (single-source shortest path problem) Works on both directed and undirected graphs. However, all edges must have nonnegative weights. Input: Weighted graph and source vertex (v) such that all edge weights are nonnegative Output: Lengths of shortest paths (or the shortest paths themselves) from a given source vertex (v) to all other vertices

Greedy Algorithm Example

Greedy Algorithm Example

Greedy Algorithm Example

Greedy Algorithm Example

Greedy Algorithm Example

Greedy Algorithm Example

Greedy Algorithm Example

Greedy Algorithm Example

Greedy Algorithm Example

Greedy Algorithm Example

Recursive Algorithms Recursive Algorithms

Recursion A recursive function is one which calls itself. To prevent infinite recursion, you need an if-else statement (of some sort) where one branch makes a recursive call, and the other branch does not.

Recursion Dream (problem) { If problems solved return solution else dream(problem) }

Recursion Example • Example: Calculate the factorial of N: N! = N x (N-1) x (N-2) x ……. x 3 x 2 x 1 Traditional algorithm using Loop: Factorial () 1. Let index = 1 2. Let product = 1 3. For index = 1 to N product = product x index 4. end-For 5. return product

Recursion Example • Example: Calculate the factorial of N: N! = N x (N-1) x (N-2) x ……. x 3 x 2 x 1 Recursive algorithm: Factorial(long n) { if (n==1) } return(n * return(n); else factorial(n-1));

Recursion Example A traditional algorithm for finding the Fibonacci Numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ……) Fibonacci () 1. Input N 2. Let result 0 = 0 3. Let result 1 = 1 4. If N = 1 or N = 2 5. result = N - 1 6. else 7. for loop counter = 2 to N – 1 8. result = result 1 + result 0 9. result 0 = result 1 10. result 1 = result 11. end-for loop 12. end-if 13. output result

Recursion Example A recursive algorithm for finding the Fibonacci Numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ……) Fibonacci (N) 1. If N = 1 or N = 2 2. return (N – 1) 3. else 4. return Fibonacci (N – 1) + Fibonacci (N – 2) 5. end-if

Algorithms of Skyscrapers Project

Algorithms of Skyscrapers Project Rules: • Using prefabricated-modular blocks of 100 meters long, 100 meters wide, and 5 meters tall. • The blocks interlock on top and bottom (like Legos), and they cannot be stacked sideways. • Using special lifters, putting stacks of blocks at the same height on ground or on top of another set of equal-height stacks takes one week regardless of how tall the stacks are or how many stacks are lifted. • The prefabrication time of the blocks doesn’t count since they are already in stock. • No resource/budget limitations (i. e. , you can have as many stacks as possible at the same time),

Algorithms of Skyscrapers Project Problem Part A: • If a client wants to build a 100 -meter long, 100 -meter wide, and 1250 -meter high tower as quickly as possible, what is the shortest amount of time that it will take to build the tower? • Show your algorithm in both flowchart and pseudocode forms. Problem Part B: • Develop a general algorithm for skyscrapers of 100 -meter long, 100 -meter wide, and N-meter high (where N is a multiple of 5) in pseudocode format.

Review of Pseudocode

Convention of Pseudocode Write pseudocode for: • calculating total number of blocks needed for N-meter high skyscraper. • calculating the remainder of number of stacks divided by 2. • creating a while loop to work as long as the number of stacks is not equal to 1. • creating a counter to increment the week count in a while loop.

Convention of Pseudocode Write pseudocode for: • calculating total number of blocks needed for N-meter high skyscraper. 1. number. Block = N/5

Convention of Pseudocode Write pseudocode for: • calculating the remainder of number of stacks divided by 2. 1. Remainder = number. Stack % 2

Convention of Pseudocode Write pseudocode for: • creating a while loop to work as long as the number of stacks is not equal to 1. While loop number. Stack ≠ 1 2. …… 3. end-loop

Convention of Pseudocode Write pseudocode for: • creating a counter to increment the week count in a while loop. 1. week. Counter = 0 2. While loop number. Stack ≠ 1 3. week. Counter = week. Counter + 1 4. end-loop

Group Presentation of Skyscraper Algorithms

Decimal-to-Binary Conversion Algorithm

Decimal-to-Binary Conversion 240 0 120 0 60 0 30 0 2 15 1 2 7 1 2 3 1 2 2 1 Decimal: 240 Binary: 11110000 No. of Division 7 No. of One’s: 4 No. of Weeks: 7 + 4 = 11

Programming: A Crash Course of C

C Example: Sum of 1+ 2+…+N 1. // Calculate the Sum of 1 + 2 + 3 + …. . + N 2. #include <stdio. h> 3. int main () 4. { 5. int n, i; 6. int sum = 0; 7. printf("Enter N: "); 8. scanf("%d", &n); 9. for (i = 1; i <= n; i++) 10. sum = sum + i; 11. printf("Sum = %dn", sum); 12. return 0; 13. }

C Example: Average of Numbers 1. // Calculate the average of three numbers 2. // average = (a + b + c)/3 3. #include <stdio. h> 4. int main() 5. { 6. float a, b, c; 7. float average; 8. printf ("Averaging three numbers n"); 9. printf ("Enter three numbers a, b, and c: "); 10. scanf ("%f %f %f", &a, &b, &c); 11. average = (a + b + c)/3. 0; 12. printf("Average = %. 2 fn", average); 13. return 0; 14. }

C Example: Linear Equation 1. // Solve Linear Equation 2. // ax + b = c, x = (c - b)/a 3. #include <stdio. h> 4. int main() 5. { 6. float a, b, c; 7. float x; 8. printf("Solve linear equation: ax + b = cn"); 9. printf("Enter coefficients: a, b, c: "); 10. scanf ("%f %f %f", &a, &b, &c); 11. x = (c - b)/a; 12. printf ("x = %. 2 fn", x); 13. return 0; 14. }

C Example: Skyscraper Algorithm 1. // Skyscraper Algorithm 2. #include <stdio. h> 3. int main() 4. { 5. int no. Week, building. Height, no. Stack, remainder; 6. no. Week = 1; 7. printf ("Hello! Please enter the building height: "); 8. scanf ("%d", &building. Height); 9. no. Stack = building. Height/5; 10. while (no. Stack != 1) 11. { 12. remainder = no. Stack%2; 13. no. Stack = no. Stack/2; 14. no. Week = no. Week + remainder + 1; 15. } 16. printf("Number of Week to Build the Skyscraper is: %dn", no. Week); 17. return 0; 18. }

C Example: Square Root 1. #include <stdio. h> 2. #include <math. h> 3. int main() 4. { 5. float a; 6. float x; 7. printf("Please enter the number: "); 8. scanf("%f", &a); 9. if (a < 0) 10. printf("No solution. n"); 11. else 12. { 13. x = sqrt(a); 14. printf("sqrt(%. 2 f) = %. 5 fn", a, x); 15. } 16. return 0; 17. }

C Example: Integer Array 1. // Integer Array 2. #include <stdio. h> 3. #define MAX 5 4. int main() 5. { 6. int array[MAX]; 7. float sum = 0. ; 8. int i; 9. printf("Enter 5 integers: "); 10. for (i =0; i < MAX; i++) 11. { 12. scanf("%d", &array[i]); 13. sum = sum + array[i]; 14. } 15. for (i =0; i < MAX; i++) 16. printf("array[%d] = %dn", i, array[i]); 17. printf("Average = %. 2 fn", sum/MAX); 18. return 0; 19. }

C Example: Integer Array Output: Enter 5 integers: 1 2 3 4 5 Enter 5 integers: 1 2 3 4 6 array[0] = 1 array[1] = 2 array[2] = 3 array[3] = 4 array[4] = 5 array[4] = 6 Average = 3. 00 Average = 3. 20

C Example: Character Strings 1. // Strings are Array of Characters. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. #include <stdio. h> #include <string. h> #define MAX_STRING_LENGTH 80 int main() { char str[MAX_STRING_LENGTH]; unsigned long length; int i; str[0] = 'H'; str[1] = 'e'; str[2] = 'l'; str[3] = 'l'; str[4] = 'o'; str[5] = '!'; str[6] = ''; length = strlen(str); for (i = 0; i < length; i++) printf("str[%d] = %cn", i, str[i]); printf("nstring =t%sn", str); printf("length =t%lun", length); return 0; }

C Example: Character Strings 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. // Strings are Array of Characters. #include <stdio. h> #include <string. h> #define MAX_STRING_LENGTH 80 int main() { char str[MAX_STRING_LENGTH]; unsigned long length; int i; str[0] = 'H'; str[1] = 'e'; str[2] = 'l'; str[3] = 'l'; str[4] = 'o'; str[5] = '!'; str[6] = ''; length = strlen(str); for (i = 0; i < length; i++) printf("str[%d] = %cn", i, str[i]); printf("nstring =t%sn", str); printf("length =t%lun", length); return 0; } Output: str[0] = H str[1] = e str[2] = l str[3] = l str[4] = o str[5] = ! string = Hello! length = 6

C Example: Strings Operations 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. // Strings IO from Command Line // String Comparison, Concatenation & Copy #include <stdio. h> #include <string. h> #define MAX_STRING_LENGTH 80 int main() { char str 1[MAX_STRING_LENGTH]; char str 2[MAX_STRING_LENGTH]; char str 3[MAX_STRING_LENGTH]; printf("n. String 1: "); scanf("%s", str 1); printf("n. String 2: "); scanf("%s", str 2); printf("n. String 1 =t%sn", str 1); printf("n. String 2 =t%sn", str 2); 17. if(strcmp(str 1, str 2) > 0) 18. { 19. printf("n. String 1 > String 2n"); 20. strcat(str 1, str 2); 21. printf("n. String 1=t%sn", str 1); 22. printf("n. String 2=t%sn", str 2); 23. } 24. else if (strcmp(str 1, str 2) < 0) 25. { 26. printf("n. String 1 < String 2n"); 27. strcat(str 2, str 1); 28. printf("n. String 1=t%sn", str 1); 29. printf("n. String 2=t%sn", str 2); 30. } 31. else 32. { 33. printf("n. String 1 = String 2n"); 34. strcpy(str 3, str 1); 35. printf("n. String 3: %sn", str 3); 36. } 37. return 0; 38. }

ASCII Code

C Example: Strings Operations Output: String 1: Ally String 1: Alexia String 1: ELRO String 2: Henry String 2: Zoe String 2: ELRO String 1 = Ally String 1 = Alexia String 1 = ELRO String 2 = Henry String 2 = Zoe String 2 = ELRO String 1 < String 2 String 1 = String 2 String 1= Ally String 1= Alexia String 3: ELRO String 2= Henry. Ally String 2= Zoe. Alexi

C Example: File R/W, Strings & Array 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. // Read / Write Strings and Integers from / to Files #include <stdio. h> #define FILESIZE 10 #define STRINGLENGTH 15 int main() { FILE* file. Input = NULL; FILE* file. Output = NULL; char str[FILESIZE][STRINGLENGTH]; int number[FILESIZE]; int i; file. Input = fopen("input. txt", "r"); file. Output = fopen("output. txt", "w"); for (i = 0; i < FILESIZE; i++) fscanf (file. Input, "%s %d", str[i], &number[i]); for (i = 0; i < FILESIZE; i++) fprintf(file. Output, "string[%d] = %13 s number[%d] = %3 dnn", i, str[i], i, number[i]*2) ; for (i = 0; i < FILESIZE; i++) fprintf(file. Output, "%s ", str[i]); fprintf(file. Output, "nn"); fclose(file. Input); fclose(file. Output); }

C Example: File R/W, Strings & Array In Xcode, after copying and pasting the program, follow the following steps: 1. Drag and drop the input files into the project folder 2. Set Working Directory through: Product > Scheme > Edit Scheme > Working Directory 3. Build and then run the current scheme 4. Display the output file through: Control Click Project folder > Add Files to “project folder”……

C Example: File R/W, Strings & Array FILE: in. txt FILE: output. txt The 10 purpose 20 of 30 this 40 project 50 is 60 to 70 have 80 fun 90 : ) 100 string[0] = The number[0] = 20 string[1] = purpose number[1] = 40 string[2] = of number[2] = 60 string[3] = this number[3] = 80 string[4] = project number[4] = 100 string[5] = is number[5] = 120 string[6] = to number[6] = 140 string[7] = have number[7] = 160 string[8] = fun number[8] = 180 string[9] = : ) number[9] = 200 The purpose of this project is to have fun : )

C Example: File R/W, Strings & Array FILE: input. txt FILE: output. txt ELRO 1 Computational 2 Service 3 Company 4 is 5 based 6 in 7 New 8 York 9 City. 10 string[0] = ELRO number[0] = 2 string[1] = Computational number[1] = 4 string[2] = Service number[2] = 6 string[3] = Company number[3] = 8 string[4] = is number[4] = 10 string[5] = based number[5] = 12 string[6] = in number[6] = 14 string[7] = New number[7] = 16 string[8] = York number[8] = 18 string[9] = City. number[9] = 20 ELRO Computational Service Company is based in New York City.

Sorting Algorithms and Implementations

Sorting Algorithms • Selection Sort I • Bubble Sort • Selection Sort 2 • Insertion Sort

Sort Main Functions 1. // This is the main function which can call different sort functions 2. #include <stdio. h> 3. int main(int argc, const char * argv[]) 4. { 5. int i, number; 6. int a[20]; 7. printf("Number of integers: "); 8. scanf("%d", &number); 9. printf("Enter %d integers: ", number); 10. for (i = 0; i< number; i++) 11. scanf("%d", &a[i]); 12. sort(a, number); 13. for (i = 0; i< number; i++) 14. printf("%d ", a[i]); 15. printf("n"); 16. return 0; 17. }

Selection Sort I • Selection Sort I compares the element in the first position with the elements in the rest of the array in order. • Swap them if the first one is greater than any other element in the array, and result in the smallest element in the first position. • Find the second smallest element through the same procedure. • Continue until the array is sorted

Selection Sort I Example 3 5 1 2 4 1 5 3 2 4 1 3 5 2 4 1 2 5 3 4 1 2 3 5 4 1 2 3 4 5

Selection Sort I Code 1. // Selection Sort: Compare the elements with the elements in the rest of the 2. // array in order. Swap them until the whole array is sorted. 3. 4. void Selection. Sort (int a[], int array_size) 5. { 6. int i, j, temp; 7. for (i = 0; i < array_size-1; i++) 8. for (j = i+1; j < array_size; j++) 9. if (a[i] > a[j]) 10. { 11. temp = a[i]; 12. a[i] = a[j]; 13. a[j] = temp; 14. } 15. } 3 5 1 2 4 1 5 3 2 4 1 3 5 2 4 1 2 5 3 4 1 2 3 5 4 1 2 3 4 5

Selection Sort II • It basically determines the minimum (or maximum) of the list and swaps it with the element at the index where its supposed to be. • Find the smallest element in the array. • Exchange it with the element in the first position. • Find the second smallest element and exchange it with the element in the second position. • Continue until the array is sorted.

Selection Sort II Example 3 5 1 2 4 1 5 3 2 4 1 2 3 5 4 1 2 3 4 5

Selection Sort II Code 1. // Selection Sort: It basically determines the minimum (or maximum) of the list and 2. // swaps it with the element at the index where its supposed to be. 3. 4. void Selection. Sort (int a[], int array_size) 5. { 6. int i; 7. int j, min, temp; 8. for (i = 0; i < array_size - 1; i++) 9. { 10. min = i; 11. for (j = i+1; j < array_size; j++) 12. { 3 5 1 2 4 1 5 3 2 4 1 2 3 5 4 13. if (a[j] < a[min]) 1 2 3 4 5 14. min = j; 1 2 3 4 5 15. } 16. temp = a[i]; 17. a[i] = a[min]; 18. a[min] = temp; 19. 20. } }

Bubble Sort • Bubble Sort works by comparing each element of the list with the element next to it and swapping them if required. • When we compare pairs of adjacent elements and none are out of order, the list is sorted. • If any are out of order, we must have to swap them to get an ordered list. • Bubble sort will make passes though the list swapping any adjacent elements that are out of order.

Bubble Sort 3 5 1 2 4 3 1 5 2 4 3 1 2 5 4 3 1 2 4 5 1 3 2 4 5 1 2 3 4 5 1 2 3 4 5

Bubble Sort Code 1. // Bubble Sort: comparing each element of the list with the element 2. // next to it and swapping them if required. 3. 4. void Bubble. Sort (int a[], int array_size) 5. { 6. int i, j, temp; 7. for (i = 0; i < (array_size - 1); ++i) 8. for (j = 0; j < array_size - 1 - i; ++j ) 9. if (a[j] > a[j+1]) 10. { 11. temp = a[j+1]; 12. a[j+1] = a[j]; 13. a[j] = temp; 14. 15. } } 3 5 1 2 4 3 1 5 2 4 3 1 2 5 4 3 1 2 4 5 1 3 2 4 5 1 2 3 4 5 1 2 3 4 5

Insertion Sort • Insertion Sort starts from the beginning, traverses through the list and as you find elements misplaced by precedence you remove them and insert them back into the right position. Eventually what you have is a sorted list of elements. • Adding a new element to a sorted list will keep the list sorted if the element is inserted in the correct place. • A single element list is sorted. • Inserting a second element in the proper place keeps the list sorted. • This is repeated until all the elements have been inserted into the sorted part of the list.

Insertion Sort 3 5 1 2 4 1 3 5 2 4 1 2 3 5 4 1 2 3 4 5

Insertion Sort Code 1. // Insertion Sort: You start from the beginning, traverse through the list 2. // and as you find elements misplaced by precedence you remove them 3. // and insert them back into the right position. 4. 5. void insertion. Sort (int a[], int array_size) 6. { 7. int i, j, key; 8. for (i = 1; i < array_size; i++) 3 5 1 2 4 9. { 3 5 1 2 4 10. key = a[i]; 1 3 5 2 4 11. for (j = i; j > 0 && a[j-1] > key; j--) 1 2 3 5 4 1 2 3 4 5 12. a[j] = a[j-1]; 13. 14. 15. } a[j] = key; }

Sort Main Functions - Strings 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. // This is the main function which can call different sort functions #include <stdio. h> #include <string. h> #define STRINGLENGTH 15 int main(int argc, const char * argv[]) { int i, number; char str[20][STRINGLENGTH]; printf("Number of strings: "); scanf("%d", &number); printf("Enter %d strings: ", number); for (i = 0; i< number; i++) scanf("%s", str[i]); Sort. Str(str, number); for (i = 0; i< number; i++) printf("%s ", str[i]); printf("n"); return 0; }

Selection Sort I Code - Strings 1. // Selection Sort: Compare the elements with the elements in the rest of the 2. // array in order. Swap them until the whole array is sorted. 3. 4. void Selection. Sort (char str[][STRINGLENGTH], int array_size) 5. { 6. int i, j; 7. char temp[STRINGLENGTH]; 8. for (i = 0; i < array_size-1; i++) 9. for (j = i+1; j < array_size; j++) 10. if (strcmp(str[i], str[j]) > 0) 11. { 12. strcpy(temp, str[i]); 13. strcpy(str[i], str[j]); 14. strcpy(str[j], temp); 15. 16. } }

Searching Algorithms and Implementations

Searching Algorithms • Linear Search • Binary Search

Search Functions 1. // This is the main function which can call different search functions 2. #include <stdio. h> 3. int main() 4. { 5. int i, number, key; 6. int a[20]; 7. printf("Number of integers: "); 8. scanf("%d", &number); 9. printf("Enter %d integers: ", number); 10. for (i = 0; i< number; i++) 11. scanf("%d", &a[i]); 12. printf("The array of integers: n"); 13. for (i = 0; i< number; i++) 14. printf("%d ", a[i]); 15. printf("n"); 16. printf("Enter your search key: "); 17. scanf("%d", &key); 18. search (key, a, number); 19. return 0; 20. }

Linear Search • Linear Search is the simplest searching algorithm. • It uses a loop to sequentially step through an array, starting with the first element. • It compares each element with the value being searched for and stops when that value is found or the end of the array is reached.

Linear Search Code 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. // Linear Search int Linear. Search (int key, int a[], int array_size) { int i, found = 0; for(i = 0; i < array_size; i++) { if (a[i] == key) { found = 1; printf ("%2 d] %2 dnn", i, a[i]); } } if (found == 1) return 1; else { printf ("Can not find %dn", key); return -1; } }

Binary Search • Binary search is much more efficient than the linear search. • It requires the list to be in order. • The algorithm starts searching with the middle element. • If the item is less than the middle element, it starts over searching the first half of the list. • If the item is greater than the middle element, the search starts over starting with the middle element in the second half of the list. • It then continues halving the list until the item is found.

Binary Search

Binary Search Code 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. // Binary Search: int binsearch(int x, int v[], int n) { int low, high, mid; low = 0; high = n - 1; while(low<=high) { mid = (low + high) / 2; if(x < v[mid]) { high = mid + 1; } else if(x > v[mid]) { low = mid + 1; } else { return mid; } return -1; } }

Algorithms and Programming • Types of Algorithms – Brute-Force (or Exhaustive Search) – Divide and Conquer – The Greedy Method – Recursion • Algorithms of Skyscrapers Project – Review of Pseudo Code – Decimal-to-Binary Conversion Algorithm • Programming: A Crash Course of C • Sorting Algorithms and Implementations • Searching Algorithms and Implementations

Q&A