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1.3 Conditionals and Loops
Introduction to Programming in Java: An Interdisciplinary Approach · Robert Sedgewick and Kevin Wayne · Copyright © 2008 · February 10, 2009 10:07 PM 2
A Foundation for Programming
objects functions and modules graphics, sound, and image I/O arrays conditionals and loops Math text I/O primitive data types assignment statements equivalent to a calculator any program you might want to write 3
A Foundation for Programming
objects functions and modules graphics, sound, and image I/O arrays any program you might want to write to infinity conditionals and loops and beyond! Math text I/O primitive data types assignment statements 4
Control Flow
Control flow.
! Sequence of statements that are actually executed in a program.
! Conditionals and loops: enable us to choreograph control flow.
statement 2 statement 1 statement 4 statement 3 boolean 2^ true false statement 2 boolean 1 statement 3 false statement 1 true straight-line control flow control flow with conditionals and loops
Conditionals
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If Statement
The if statement. A common branching structure.
! Evaluate a boolean expression.
! If true, execute some statements.
! If false, execute other statements.
if ( boolean expression ) { statement T ; } else { statement F ; } can be any sequence of statements statement T true false boolean expression statement F 7
If Statement
The if statement. A common branching structure.
! Evaluate a boolean expression.
! If true, execute some statements.
! If false, execute other statements.
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If Statement
Ex. Take different action depending on value of variable.
public class Flip { public static void main(String[] args) { if (Math.random() < 0.5) System.out.println("Heads"); elseMath.random() < 0.5) System.out.println("Tails"); } } % java Flip Heads % java Flip Heads % java Flip Tails % java Flip Heads
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Powers of Two
public class PowersOfTwo { public static void main(String[] args) { // last power of two to print int N = Integer.parseInt(args[ 0 ]); int i = 0 ; // loop control counter int v = 1 ; // current power of two while (i <= N) { System.out.println(i + " " + v); i = i + 1 ; v = 2 * v; } } } % java PowersOfTwo 4 0 1 1 2 2 4 3 8 % java PowersOfTwo 6 0 1 1 2 2 4 3 8 4 16 5 32 6 64 print i and ith power of two 14
While Loop Challenge
Q. Anything wrong with the following code for printing powers of 2?
int i = 0 ; int v = 1 ; while (i <= N) System.out.println(i + " " + v); i = i + 1 ; v = 2 * v; 16
A Wonderful Square Root
Copyright 2004, Sidney Harris, http://www.sciencecartoonsplus.com % java Sqrt 60481729
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While Loops: Square Root
Q. How might we implement Math.sqrt()?
A. To compute the square root of c:
! Initialize t 0 = c.
! Repeat until ti = c / ti, up to desired precision:
set ti+1 to be the average of ti and c / ti.
t 0 = 2. t 1 = 12 ( t 0 + (^) t^20 ) = 1. t 2 = 12 ( t 1 + (^2) t 1 ) = 1. t 3 = 12 ( t 2 + (^) t^22 ) = 1. t 4 = 12 ( t 3 + (^) t^23 ) = 1. t 5 = 12 ( t 4 + (^) t^24 ) = 1. computing the square root of 2
18 public class Sqrt { public static void main(String[] args) { double epsilon = 1e-15; double c = Double.parseDouble(args[ 0 ]); double t = c; while (Math.abs(t - c/t) > t*epsilon) { t = (c/t + t) / 2.0; } System.out.println(t); } } (^) % java Sqrt 2.
relative error tolerance 15 decimal digits of accuracy in 5 iterations
While Loops: Square Root
Q. How might we implement Math.sqrt()?
A. To compute the square root of c:
! Initialize t 0 = c.
! Repeat until ti = c / ti, up to desired precision:
set ti+1 to be the average of ti and c / ti.
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Square root method explained.
! Goal: find root of any function f(x).
! Start with estimate t 0.
! Draw line tangent to curve at x= ti.
! Set ti+1 to be x-coordinate where line hits x-axis.
! Repeat until desired precision.
Caveat. f(x) must be smooth; t 0 must be good estimate.
Newton-Raphson Method
f(x) = x^2 - c to compute "c 20
The For Loop
Copyright 2004, FoxTrot by Bill Amend www.ucomics.com/foxtrot/2003/10/ 21
For Loops
The for loop. Another common repetition structure.
! Execute initialization statement.
! Evaluate a boolean expression.
! If true, execute some statements.
! And then the increment statement.
! Repeat.
for ( init ; boolean expression ; increment ) { statement 1 ; statement 2 ; } statement 1 true false boolean expression statement 2 init increment body loop continuation condition
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Nesting
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Nesting Conditionals and Loops
Conditionals enable you to do one of 2n sequences of operations with n lines.
More sophisticated programs.
! Nest conditionals within conditionals.
! Nest loops within loops.
! Nest conditionals within loops within loops.
if (a0 > 0 ) System.out.print( 0 ); if (a1 > 0 ) System.out.print( 1 ); if (a2 > 0 ) System.out.print( 2 ); if (a3 > 0 ) System.out.print( 3 ); if (a4 > 0 ) System.out.print( 4 ); if (a5 > 0 ) System.out.print( 5 ); if (a6 > 0 ) System.out.print( 6 ); if (a7 > 0 ) System.out.print( 7 ); if (a8 > 0 ) System.out.print( 8 ); if (a9 > 0 ) System.out.print( 9 ); Loops enable you to do an operation n times using only 2 lines of code. double sum = 0.0; for (int i = 1 ; i <= 1024 ; i++) sum = sum + 1.0 / i; 210 = 1024 possible results, depending on input computes 1/1 + 1/2 + ... + 1/ 28
Nested If Statements
Ex. Pay a certain tax rate depending on income level.
double rate; if (income < 47450 ) rate = 0.22; else if (income < 114650 ) rate = 0.25; else if (income < 174700 ) rate = 0.28; else if (income < 311950 ) rate = 0.33; else if (income < 311950 ) rate = 0.35; graduated income tax calculation 0 - 47,450 22% Income Rate 47,450 – 114,650 25% 114,650 – 174,700 28% 174,700 – 311,950 33% 311,950 - 35% 5 mutually exclusive alternatives 29
Nested If Statements
is shorthand for
Be careful when nesting if-else statements. (See Q+A on p. 75.)
if (income < 47450 ) rate = 0.22; else if (income < 114650 ) rate = 0.25; else if (income < 174700 ) rate = 0.28; else if (income < 311950 ) rate = 0.33; else if (income < 311950 ) rate = 0.35; if (income < 47450) rate = 0.22; else { if (income < 114650) rate = 0.25; else { if (income < 174700) rate = 0.28; else { if (income < 311950) rate = 0.33; else if (income < 311950) rate = 0.35; } } }
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Nested If Statement Challenge
Q. Anything wrong with the following for income tax calculation?
double rate = 0.35; if (income < 47450 ) rate = 0.22; if (income < 114650 ) rate = 0.25; if (income < 174700 ) rate = 0.28; if (income < 311950 ) rate = 0.33; wrong graduated income tax calculation 0 - 47,450 22% Income Rate 47,450 – 114,650 25% 114,650 – 174,700 28% 174,700 – 311,950 33% 311,950 - 35% 31
Monte Carlo Simulation
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Gambler's Ruin
Gambler's ruin. Gambler starts with $stake and places $1 fair bets
until going broke or reaching $goal.
! What are the chances of winning?
! How many bets will it take?
One approach. Monte Carlo simulation.
! Flip digital coins and see what happens.
! Repeat and compute statistics.
33 public class Gambler { public static void main(String[] args) { int stake = Integer.parseInt(args[ 0 ]); int goal = Integer.parseInt(args[ 1 ]); int T = Integer.parseInt(args[ 2 ]); int wins = 0 ; System.out.println(wins + " wins of " + T); } } // repeat experiment N times for (int t = 0 ; t < T; t++) { } // do one gambler's ruin experiment int cash = stake; while (cash > 0 && cash < goal) { } if (cash == goal) wins++; // flip coin and update if (Math.random() < 0.5) cash++; else cash--;
Gambler's Ruin
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95% of Program Development
Debugging. Cyclic process of editing, compiling, and fixing errors.
! Always a logical explanation.
! What would the machine do?
! Explain it to the teddy bear.
You will make many mistakes as you write programs. It's normal.
“ If I had eight hours to chop down a tree, I would spend six hours sharpening an axe. ” — Abraham Lincoln “As soon as we started programming, we found out to our surprise that it wasn't as easy to get programs right as we had thought. I can remember the exact instant when I realized that a large part of my life from then on was going to be spent in finding mistakes in my own programs. ” — Maurice Wilkes 39
Debugging Example
Factor. Given an integer N > 1, compute its prime factorization.
Application. Break RSA cryptosystem (factor 200-digit numbers).
3,757,208 = 2^3 # 7 # 132 # 397
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Debugging Example
Factor. Given an integer N > 1, compute its prime factorization.
Brute-force algorithm. For each putative factor i = 2, 3, 4, …,
check if N is a multiple of i, and if so, divide it out.
3757208/ 41
Debugging: 95% of Program Development
Programming. A process of finding and fixing mistakes.
! Compiler error messages help locate syntax errors.
! Run program to find semantic and performance errors.
public class Factors { public static void main(String[] args) { long N = Long.parseLong(args[ 0 ]) for (i = 0 ; i < N; i++) { while (N % i == 0 ) System.out.print(i + " ") N = N / i } } } this program has many bugs! as long as i is a factor, divide it out check if i is a factor
42 public class Factors { public static void main(String[] args) { long N = Long.parseLong(args[ 0 ]) for (i = 0 ; i < N; i++) { while (N % i == 0 ) System.out.print(i + " ") N = N / i } } } % javac Factors.java Factors.java:6: ';' expected for (i = 2; i < N; i++) ^ 1 error
Debugging: Syntax Errors
Syntax error. Illegal Java program.
! Compiler error messages help locate problem.
! Goal: no errors and a file named Factors.class.
the first error 43 public class Factors { public static void main(String[] args) { long N = Long.parseLong(args[ 0 ]); for (int i = 0 ; i < N; i++) { while (N % i == 0 ) System.out.print(i + " "); N = N / i; } } }
Debugging: Syntax Errors
Syntax error. Illegal Java program.
! Compiler error messages help locate problem.
! Goal: no errors and a file named Factors.class.
syntax (compile-time) errors need to declare variable i need terminating semicolons 44 public class Factors { public static void main(String[] args) { long N = Long.parseLong(args[ 0 ]); for (int i = 0 ; i < N; i++) { while (N % i == 0 ) System.out.print(i + " "); N = N / i; } } }
Debugging: Semantic Errors
Semantic error. Legal but wrong Java program.
! Run program to identify problem.
! Add print statements if needed to produce trace.
% javac Factors.java % java Factors Exception in thread "main" java.lang.ArrayIndexOutOfBoundsException: 0 at Factors.main(Factors.java:5) oops, no argument 45 public class Factors { public static void main(String[] args) { long N = Long.parseLong(args[ 0 ]); for (int i = 0 ; i < N; i++) { while (N % i == 0 ) System.out.print(i + " "); N = N / i; } } }
Debugging: Semantic Errors
Semantic error. Legal but wrong Java program.
! Run program to identify problem.
! Add print statements if needed to produce trace.
% javac Factors.java % java Factors 98 Exception in thread "main" java.lang.ArithmeticExeption: / by zero at Factors.main(Factors.java:8) need to start at 2 because 0 and 1 cannot be factors
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Performance error. Correct program, but too slow.
public class Factors { public static void main(String[] args) { long N = Long.parseLong(args[ 0 ]); for (int i = 2 ; i < N; i++) { while (N % i == 0 ) { System.out.print(i + " "); N = N / i; } } if (N > 1) System.out.println(N); else System.out.println(); } }
Debugging: Performance Error
% java Factors 11111111 11 73 11 137 % java Factors 11111111111 21649 51329 % java Factors 11111111111111 11 239 4649 909091 % java Factors 11111111111111111 2071723 very long wait (with a surprise ending) 51 public class Factors { public static void main(String[] args) { long N = Long.parseLong(args[ 0 ]); for (int i = 2 ; i <= N/i; i++) { while (N % i == 0 ) { System.out.print(i + " "); N = N / i; } } if (N > 1) System.out.println(N); else System.out.println(); } }
Performance error. Correct program, but too slow.
Solution. Improve or change underlying algorithm.
Debugging: Performance Error
% java Factors 11111111 11 73 11 137 % java Factors 11111111111 21649 51329 % java Factors 11111111111111 11 239 4649 909091 % java Factors 11111111111111111 2071723 5363222357 fixes performance error: if N has a factor, it has one less than or equal to its square root 52
Q. How large an integer can I factor?
Note. Can't break RSA this way (experts are still trying).
% java Factors 3757208 2 2 2 7 13 13 397 % java Factors 9201111169755555703 9201111169755555703
Program Development: Analysis
† estimated largest factor (^3) instant digits (i^ <=^ N) 6 0.15 seconds 9 77 seconds 12 21 hours † instant (i*i <= N) instant instant 0.16 seconds 15 2.4 years † 18 2.4 millennia † 2.7 seconds 92 seconds after a few minutes of computing…. 53
Debugging
Programming. A process of finding and fixing mistakes.
1. Create the program.
2. Compile it.
Compiler says: That’s not a legal program.
Back to step 1 to fix syntax errors.
3. Execute it.
Result is bizarrely (or subtly) wrong.
Back to step 1 to fix semantic errors.
4. Enjoy the satisfaction of a working program!
5. Too slow? Back to step 1 to try a different algorithm.
