3. Java - Language Constructs I

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1 Names and Identifiers A program (that is, a class) needs a name public class SudokuSolver { Java - Language Constructs I Names and Identifiers, Variables, Assignments, Constants, Datatypes, Operations, Evaluation of Expressions, Type Conversions Convention for class names: use CamelCase Words are combined into one word, each starting with a capital letter Allowed names for entities in a program: Names begin with a letter or _ or $ Then, sequence of letters, numbers or _ or $ Names - what is allowed Keywords The following words are already used by the language and cannot be used as names: _myname AN$WE4_1S_42 $bling$ TheCure strictfp 49ers?! side-swipe Ph.D s abstract continue for new switch assert default goto package synchronized boolean do if private this break double implements protected throw byte else import public throws case enum instanceof return transient catch extends int short try char final interface static void class finally long strictfp volatile const float native super while 64 65

2 Variables and Constants Constants Variables are buckets for a value Have a data type and a name The data type determines what kind of values are allowed in the variable int x 23 int y 42 float f 0.0f char c a Declaration in Java: int x = 23, y = 42; float f; char c = a ; Keyword final The value of the variable can be set exactly once final int maxsize = 100; Hint: Always use final, unless the value actually needs to change over time. Initialization Standard Types Types and Memory Usage Data Type Definition Value Range Initial Value byte 8-bit integer 128,..., short 16-bit integer ,..., int 32-bit integer 2 31,..., long 64-bit integer 2 63,..., L float 32-bit floating point ±1.4E 45,..., ±3.4E f double 64-bit floating point ±4.9E 324,..., ±1.7E d Reminder: Memory cells contain 1 Byte = 8 bit boolean byte short, char boolean logical value true, false false char unicode-16 character \u0000,..., a, b,..., \uffff \u0000 String string null int, float long, double 68 69

3 Literals: Integer Numbers Type int (or short, byte) 12 : value 12-3 : value 3 Type long 25_872_224L : value Literals: Floating Point Numbers are different from integers by providing decimal comma 1.0 : type double, value f : type float, value 1.27 and / or exponent. 1e3 : type double, value 1000 integer part 1.23e-7f fractional part exponent Hint: Underscores between digits are allowed! e-7 : type double, value e-7f : type float, value Literals: Characters and Strings Character: In ASCII Table Individual characters: a : Type char, value 97 Strings: "Hello There!" : Type String "a" : Type String Mind: Characters and Strings are two different things! 72 73

4 Value Assignment Copies a value into variable x In pseudo code: x value In Java: x = value value = is the assignment operator and not a comparison! x (copy) value Therefore, int y = 42 is both a declaration + an assignment. Value Assignment s int a = 3; double b; b = 3.141; int c = a = 0; String name = "Inf"; A nested assignment: The expression a = 0 stores the value 0 into variable a. and then returns the value Arithmetic Binary Operators Arithmetic Binary Operators Infix notation: x op y with the following operators op: + / % modulo Precedence: Multiplication, division, and modulo first, then addition and subtraction Associativity: Evaluation from left to right Division x / y: Integer division if x and y are integer. Division x / y: Floating-point division if x or y is a floating-poing number! s Integer division and modulo 5 / 3 evaluates to 1 5 / 3 evaluates to 1 5 % 3 evaluates to 2 5 % 3 evaluates to

5 Arithmetic Assignment Arithmetic Unary Operators x = x + y x += y Analogous for,, /, % s: x = 3; // x = x 3 name += "x" // name = name + "x" num = 2; // num = num 2 Prefix notation: + x or x Precedence: Unary operators bind stronger than binary operators s Assuming x is 3 2 x evaluates to 6 x +1 evaluates to Increment/Decrement Operators Increment operators ++x and x++ have the same effect: x x + 1.But different return values: Prefix operator ++x returns the new value: a = ++x; x = x + 1; a = x; Postfix operator x++ returns the old value: a = x++; temp = x; x = x + 1; a= temp; Precedence: Increment and decrement operators bind stronger than unary operators Analogous for x and x. Increment/Decrement Operators s Assuming x is initially set to 2 y = ++x 3 evaluates to: x is 3 and y is 9 y = x++ 3 evaluates to: x is 3 and y is

6 Expressions Expressions represent computations are either primary or composed from other expressions, using operators are statically typed s primary: 4.1d or x or "Hi" composed: x + y or f 2.1f The type of f is float Analogy: Construction kit Celsius to Fahrenheit Celsius to Fahrenheit - Analysis public class Main { } public static void main(string[] args) { Out.print("Celsius: "); int celsius = In.readInt(); float fahrenheit = 9 * celsius / ; Out.println("Fahrenheit: " + fahrenheit); } : 15 Celsius are 59 Fahrenheit 9 * celsius / Arithmetic expression, contains three literals, one variable, three operator symbols Where are the brackets in this expression? 84 85

7 Rule 1: Precedence Rule 2: Associativity Multiplicative operators (*, /, %) have a higher precedence ("bind stronger") than additive operators (+, -). Arithmetic operators (*, /, %, +, -) are left-associative: in case of the same precedence, the evaluation happens from left to right. 9 * celsius / means (9 * celsius / 5) * celsius / means ((9 * celsius) / 5) Rule 3: Arity Bracketing Unary operators +, - before binary operators +, -. 9 * celsius / means 9 * celsius / (+5) + 32 Any expression can be bracketed unambiguously using the associativities precedences arities (number of operands) of the involved operators

8 Expression Trees Bracketing leads to an expression tree (((9 * celsius) / 5) + 32) 9 celsius 5 32 * / + Evaluation Order From leafs to the root in the expression tree 9 * celsius / celsius 5 32 * / Expression Trees Notation Usual notation: root on top 9 * celsius / Type System Java festures a static type system: All types must be declared If possible, the compiler checks the typing otherwise it s checked at run-time / 32 Advantages of a static type system * 9 celsius 5 Fail-fast Bugs in the program are often found already by the compiler Understandable code 92 93

9 Type errors int pi_ish; float pi = 3.14f; pi_ish = pi; Explicit Type Conversion int pi_ish; float pi = 3.14f; pi_ish = (int) pi; Compiler error:./root/main.java:12: error: incompatible types: possible lossy conversion from float to int pi_ish = pi; ^ Explicit type conversion using casts (type) Statically type-correct, compiler is happy Run-time behavior: depends on the situation Here: loss of precision: Can crash a program at run-time 95 Type Conversion - Visually for Integer Numbers Mixed Expressions, Conversion explicit cast byte short int long implicit conversion Floating point numbers are more general than integers. In mixed expressions integers are converted to floating point numbers. 9 * celsius / Potential loss of information when casting explicitly, because less memory available to represent the number 96 97

10 Type Conversions for Binary Operations Numeric operands in a binary operation are being converted according to the following rules: If both operands have the same type, no conversion will happen If one operand is double, the other operand is converted to double as well If one operand is float, the other operand is converted to float as well If one operand is long, the other operand is converted to long as well Otherwise: Both operands are being converted to int 4. Efficiency of algorithms Efficiency of Algorithms, Random Access Machine Model, Function Growth, Asymptotics [Cormen et al, Kap. 2.2,3, Ottman/Widmayer, Kap. 1.1] Efficiency of Algorithms Technology Model Random Access Machine (RAM) Goals Quantify the runtime behavior of an algorithm independent of the machine. Compare efficiency of algorithms. Understand dependece on the input size. Execution model: instructions are executed one after the other (on one processor core). Memory model: constant access time. Fundamental operations: computations (+,,,...) comparisons, assignment / copy, flow control (jumps) Unit cost model: fundamental operations provide a cost of 1. Data types: fundamental types like size-limited integer or floating point number

11 Size of the Input Data Asymptotic behavior An exact running time can normally not be predicted even for small input data. Typical: number of input objects (of fundamental type). Sometimes: number bits for a reasonable / cost-effective representation of the data. We consider the asymptotic behavior of the algorithm. And ignore all constant factors. An operation with cost 20 is no worse than one with cost 1 Linear growth with gradient 5 is as good as linear growth with gradient Superficially 4.2 Function growth O, Θ, Ω [Cormen et al, Kap. 3; Ottman/Widmayer, Kap. 1.1] Use the asymptotic notation to specify the execution time of algorithms. We write Θ(n 2 ) and mean that the algorithm behaves for large n like n 2 : when the problem size is doubled, the execution time multiplies by four

12 More precise: asymptotic upper bound Graphic provided: a function g : N R. Definition: O(g) = {f : N R c > 0, n 0 N : 0 f(n) c g(n) n n 0 } g(n) = n 2 f O(g) Notation: O(g(n)) := O(g( )) = O(g). h O(g) 106 n s Property O(g) = {f : N R c > 0, n 0 N : 0 f(n) c g(n) n n 0 } f(n) f O(?) 3n + 4 O(n) c = 4, n 0 = 4 2n O(n) c = 2, n 0 = 0 n n O(n 2 ) c = 2, n 0 = 100 n + n O(n) c = 2, n 0 = 1 f 1 O(g), f 2 O(g) f 1 + f 2 O(g)

13 Converse: asymptotic lower bound Given: a function g : N R. Definition: Ω(g) = {f : N R c > 0, n 0 N : 0 c g(n) f(n) n n 0 } h Ω(g) f Ω(g) g(n) = n n Asymptotic tight bound Given: function g : N R. Definition: g(n) = n 2 f Θ(n 2 ) Θ(g) := Ω(g) O(g). Simple, closed form: exercise. h(n) = 0.5 n

14 Notions of Growth Small n O(1) bounded array access O(log log n) double logarithmic interpolated binary sorted sort O(log n) logarithmic binary sorted search O( n) like the square root naive prime number test O(n) linear unsorted naive search O(n log n) superlinear / loglinear good sorting algorithms O(n 2 ) quadratic simple sort algorithms O(n c ) polynomial matrix multiply O(2 n ) exponential Travelling Salesman Dynamic Programming O(n!) factorial Travelling Salesman naively n 4 2 n n 2 n ln n Larger n Large n n n n n 2 log n 0.2 n 4 n log n

15 Logarithms 1, n 2 n 3/2 n log n n log n Time Consumption Assumption 1 Operation = 1µs. problem size log 2 n 1µs 7µs 13µs 20µs 30µs n 1µs 100µs 1/100s 1s 17 minutes n log 2 n 1µs 700µs 13/100µs 20s 8.5 hours n 2 1µs 1/100s 1.7 minutes 11.5 days 317 centuries 2 n 1µs centuries A good strategy?... Then I simply buy a new machine If today I can solve a problem of size n, then with a 10 or 100 times faster machine I can solve... Komplexität (speed 10) (speed 100) log 2 n n n 10 n n 100 n n 10 n n 100 n n 2 n 3.16 n n 10 n 2 n n n n n s n O(n 2 ) correct, but too imprecise: n O(n) and even n Θ(n). 3n 2 O(2n 2 ) correct but uncommon: Omit constants: 3n 2 O(n 2 ). 2n 2 O(n) is wrong: 2n2 cn = 2 c n n! O(n) O(n 2 ) is correct Θ(n) Θ(n 2 ) is wrong n Ω(n 2 ) Θ(n 2 )

16 Useful Tool Theorem Let f, g : N R + be two functions, then it holds that f(n) 1 lim n g(n) f(n) 2 lim n g(n) 3 f(n) g(n) = 0 f O(g), O(f) O(g). = C > 0 (C constant) f Θ(g). n g O(f), O(g) O(f). About the Notation Common notation f = O(g) should be read as f O(g). Clearly it holds that f 1 = O(g), f 2 = O(g) f 1 = f 2! Beispiel n = O(n 2 ), n 2 = O(n 2 ) but naturally n n Algorithms, Programs and Execution Time Program: concrete implementation of an algorithm. Execution time of the program: measurable value on a concrete machine. Can be bounded from above and below. Beispiel 3GHz computer. Maximal number of operations per cycle (e.g. 8). lower bound. A single operations does never take longer than a day upper bound. From an asymptotic point of view the bounds coincide. 124 Complexity Complexity of a problem P : minimal (asymptotic) costs over all algorithms A that solve P. Complexity of the single-digit multiplication of two numbers with n digits is Ω(n) and O(n log 3 2 ) (Karatsuba Ofman). : Problem Complexity O(n) O(n) O(n 2 ) Algorithm Costs 2 3n 4 O(n) Θ(n 2 ) Program Execution time Θ(n) O(n) Θ(n 2 ) 2 Number funamental operations 125

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