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: if ( condition ) statement if ( condition ) statement1 else statement2

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

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

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

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 }

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

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

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

Developing computer algorithms Pre-requisite to developing a computer algorithm: Before you can write a computer program to solve a problem, you yourself 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...)

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 !!!

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

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

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

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 !

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)

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;

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 !!!

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

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 !!! }

Programming example 1: find all divisors of a number (cont.) Example Program: (Demo above code) –Prog file: 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

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; }

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

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

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

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

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

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; }

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

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;

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 !!! }

Programming example 2: find all common divisors of 2 numbers (cont.) Example Program: (Demo above code) –Prog file: 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

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

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)

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

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

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: / 105

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: / 35

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: / 7

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 }

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

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

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

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 another factor 2 and starts a new loop with x = 3

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)

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

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-loop terminates

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

Programming example 3: factor a number (into prime factors) (cont.) Java program: import java.util.Scanner; public class Factor01 { public static void main(String[] args) { Scanner in = new Scanner(System.in); int x; int a; System.out.print("Enter a number x: "); x = in.nextInt(); // Read in number

Programming example 3: factor a number (into prime factors) (cont.) a = 2; while ( x > 1 ) { if ( x % a == 0 ) { // a is a factor of x System.out.println(a); // Print a (because it's a divisor) x = x / a; // Remove factor a from x } else { a++; // Use next number as factor }

Programming example 3: factor a number (into prime factors) (cont.) Example Program: (Demo above code) –Prog file: Factor01.java How to run the program: Right click on link and save in a scratch directory To compile: javac Factor01.java To run: java Factor01