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Writing algorithms using the while-statement

Writing algorithms using the while-statement. Previously discussed. Syntax of while-statement:. Previously discussed (cont.). Statements discussed so far:. Assignment statement :. variable = expression ;. Previously discussed (cont.). Conditional statements:.

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Writing algorithms using the while-statement

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  1. Writing algorithms using the while-statement

  2. Previously discussed • Syntax of while-statement:

  3. Previously discussed (cont.) • Statements discussed so far: • Assignment statement: variable = expression ;

  4. Previously discussed (cont.) • Conditional statements: if ( condition ) statement if ( condition ) statement1 else statement2

  5. Previously discussed (cont.) • Loop (while) statements: while ( condition ) { statement1 statement2 ... }

  6. Combining different types of statements • Computer Science Fact: • Every programming language (including Java) has 3 types of statements: • Assignment statements • Conditional statements • Loop statements

  7. Combining different types of statements (cont.) • The good news is: you have learned all of them now ! • However, you still need to learn how to use them effectively

  8. Combining different types of statements (cont.) • Rule of programming languages: • Whenever you see a statement in a syntax construct, you can use any type of statement !!! • Example: while ( condition ) while ( condition ) { { statement ===> if ( condition2 ) } { statement1 } else { statement2 } }

  9. Combining different types of statements (cont.) • Here, we used an if-else (conditional) statement as the statement in the while-body • In fact, the statement1 and statement2 in the then-part and else-part of the if-else (conditional) statement can themselves be a assignment statement, a conditional statement, or a loop statement !!! • Therefore, you can create very complex programs

  10. Combining different types of statements (cont.) • Advice: • The point of computer programming is not writing complex programs... but write out an algorithm in simple steps for a dumb machine (computer). • There are many different ways to write the same computer program, some ways can be very convoluted than others. • You should keep the structure of computer programs simple

  11. Combining different types of statements (cont.) • We will now learn how to use the power of a programming language (Java) by combining (nesting) different statements

  12. Developing computer algorithms • Pre-requisite to developing a computer algorithm: • Before you can write a computer program to solve a problem, youyourself must know what you need to do • Because: • Programming a computer = tell a computer what to do to solve a problem • If you don't know what to do, you cannot tell someone else (or something else like a computer) what to do.... (A blind cannot lead a blind...)

  13. Developing computer algorithms (cont.) • Developing a computer algorithm: • Develop a computer algorithm = Write down the steps that a human must do to solve a problem in such a detail than a dumb machine (computer) can do it !!!

  14. Programming example 1: find all divisors of a number • Problem description: • Write a Java program that reads in an integer n... • and prints all its divisors

  15. Programming example 1: find all divisors of a number (cont.) • A concrete example: • Input: n = 12 • Output: 1, 2, 3, 4, 6, 12

  16. Programming example 1: find all divisors of a number (cont.) • What would you do to solve this problem ? • Suppose: n = 12 • Check if 12 is divisible by 1 • Check if 12 is divisible by 2 • ... • Check if 12 is divisible by 12

  17. Programming example 1: find all divisors of a number (cont.) • Note: • When the remainder of the division is equal to 0, we print out the divisor • We do not need to check numbers > 12 because only number ≤ 12 can be divisors !

  18. Programming example 1: find all divisors of a number (cont.) • Algorithm: The algorithm contains some abstract steps that need to be fleshed out (work out the details)

  19. Programming example 1: find all divisors of a number (cont.) • We have learned the programming technique (trick) to test for divisibility previously: if ( n % a == 0 ) n is divisible by a; else n is not divisible by a;

  20. Programming example 1: find all divisors of a number (cont.) • Apply this programming technique to obtain a refined algorithm: Now the algorithm contain only(concrete) steps that can be easily translated into a Java program !!!

  21. Programming example 1: find all divisors of a number (cont.) • Java program: import java.util.Scanner; public class Divisors01 { public static void main(String[] args) { Scanner in = new Scanner(System.in); int n; int a; System.out.print("Enter a number n: "); n = in.nextInt(); // Read in number

  22. Programming example 1: find all divisors of a number (cont.) a = 1; while ( a <= n ) // Run a = 1, 2, ..., n { if ( n % a == 0 ) { // a is a divisor of n System.out.println(a); // Print a (because it's a divisor) } a++; // Make sure we more to the next number !! // or else: infinite loop !!! } } }

  23. Programming example 1: find all divisors of a number (cont.) • Example Program: (Demo above code) • Prog file: http://mathcs.emory.edu/~cheung/Courses/170/Syllabus/07/Progs/Divisors01.java • How to run the program:             • Right click on link and save in a scratch directory • To compile:   javac Divisors01.java • To run:          java Divisors01

  24. The brute force search method: a commonly used solution method in computer algorithms • The previous algorithm is an example of the brute force search method • The general form of the Brute force search method: for every possible value x do { if ( x is a solution ) print x; }

  25. The brute force search method: a commonly used solution method in computer algorithms (cont.) • Pre-conditions for using the brute force search method: • The number of possible values that needs to be searched must be finite • (I.e.: a finite search space) • There is a method to determine if a value x is a solution

  26. Programming example 2: find all common divisors of 2 numbers • Problem description: • Write a Java program that reads in 2 numbers x and y... • and prints all numbers that are divisors of both x and y

  27. Programming example 2: find all common divisors of 2 numbers (cont.) • A concrete example: • Input: x = 24 and y = 16 • Output: 1, 2, 4, 8

  28. Programming example 2: find all common divisors of 2 numbers (cont.) • What would you do to solve this problem ? • Suppose: x = 24 and y = 16 • The lesser of the values is 16 • Therefore, all divisors are ≤ 16

  29. Programming example 2: find all common divisors of 2 numbers (cont.) • Check if 16 and 24 are divisible by 1 • Check if 16 and 24 are divisible by 2 • ... • Check if 16 and 24 are divisible by 16 • When the remainder of both divisions are equal to 0, we print out the divisor

  30. Programming example 2: find all common divisors of 2 numbers (cont.) • Rough algorithm: input x, y; min = min(x, y); // this is the range of the brute force search for every value a = {1, 2, ...., min} do { if (x and y are divisible by a) { print a; } }

  31. Programming example 2: find all common divisors of 2 numbers (cont.) • Algorithm (structured diagram):

  32. Programming example 2: find all common divisors of 2 numbers (cont.) • Java program: import java.util.Scanner; public class Divisors02 { public static void main(String[] args) { Scanner in = new Scanner(System.in); int x, y, a, min = 0; x = in.nextInt(); // Read in x y = in.nextInt(); // Read in y if ( x < y ) // Find min(x,y) min = x; else min = y;

  33. Programming example 2: find all common divisors of 2 numbers (cont.) a = 1; while ( a <= min ) // Run a = 1, 2, ..., min(x,y) { if ( x % a == 0 && y % a == 0 ) { // a is a divisor of x and y System.out.println(a); // Print a (because it's a common divisor) } a++; // Make sure we move to the next number !! // or else: infinite loop !!! } } }

  34. Programming example 2: find all common divisors of 2 numbers (cont.) • Example Program: (Demo above code) • Prog file: http://mathcs.emory.edu/~cheung/Courses/170/Syllabus/07/Progs/Divisors02.java • How to run the program:             • Right click on link and save in a scratch directory • To compile:   javac Divisors02.java • To run:          java Divisors02

  35. Programming example 3: factor a number (into prime factors) • Problem description: • Write a Java program that reads in an integer number x.... • and prints the prime factorization of the number x

  36. Programming example 3: factor a number (into prime factors) (cont.) • A concrete example: • Input: x = 420 • Output: 2, 2, 3, 5, 7 (because 2 x 2 x 3 x 5 x 7 = 420 and all factors are prime numbers)

  37. Programming example 3: factor a number (into prime factors) (cont.) • What would you do to solve this problem ? • Suppose: x = 420 • Factorization algorithmtaught in High Schools: The number 420 is divisible by 2. Factor out the number 2: 210 ------ 2 / 420

  38. Programming example 3: factor a number (into prime factors) (cont.) The number 210 is divisible by 2. Factor out the number 2: 105 ------ 2 / 210

  39. Programming example 3: factor a number (into prime factors) (cont.) The number 105 is not divisible by 2 ==> try 3 !!! The number 105 is divisible by 3. Factor out the number 3: 35 ------ 3 / 105

  40. Programming example 3: factor a number (into prime factors) (cont.) The number 35 is not divisible by 3 ==> try 4 !!! The number 35 is not divisible by 4 ==> try 5 !!! The number 35 is divisible by 5. Factor out the number 5: 7 ----- 5 / 35

  41. Programming example 3: factor a number (into prime factors) (cont.) The number 7 is not divisible by 5 ==> try 6 !!! The number 7 is not divisible by 6 ==> try 7 !!! The number 7 is divisible by 7. Factor out the number 7: 1 ----- 7 / 7

  42. Programming example 3: factor a number (into prime factors) (cont.) • Rough algorithm: input x; a = 2; <---- Try use a = 2 as factor for x as long as x is not equal to 1 do { if (x is divisible by a) { // a is a factor of x !!! print a; (Because we have found another factor) Remove factor a from the number x (and continue) } else { Try use a+1 as factor } }

  43. Programming example 3: factor a number (into prime factors) (cont.) • Algorithm (structured diagram):

  44. Programming example 3: factor a number (into prime factors) (cont.) • Check for correctness !!! • This algorithm is complicated enough to warrant a correctness check • We do so using a small example: suppose x = 12

  45. Programming example 3: factor a number (into prime factors) (cont.) • Execution of the algorithm: • Iteration 1 through the while-loop: • It has printed the factor 2 and starts a new loop with x = 6

  46. Programming example 3: factor a number (into prime factors) (cont.) • Iteration 2 through the while-loop: (notice that x = 6 in this iteration !) • It has printed anotherfactor 2 and starts a new loop with x = 3

  47. Programming example 3: factor a number (into prime factors) (cont.) • Iteration 3 through the while-loop: (notice that x = 3 in this iteration !) • Because a = 2 is not a factor of x = 3, the else-part is executed. • A new iteration is started with a = 3 (we are trying a new factor)

  48. Programming example 3: factor a number (into prime factors) (cont.) • Iteration 4 through the while-loop: (notice that a = 3 in this iteration !) • It has printed the factor 3 and starts a new loop with x = 1

  49. Programming example 3: factor a number (into prime factors) (cont.) • Iteration 5 through the while-loop: (notice that x = 1 in this iteration !) • The loop-continuation-condition is not true !!! • The while-loopterminates

  50. Programming example 3: factor a number (into prime factors) (cont.) Result: • The program has printed the factors 2, 2, 3 • (Which is correct !!!)

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