This section under major construction.

Our study of algorithms is based upon implementing them as programs written in the Java
programming language. We do so for several reasons:

  • Our programs are concise, elegant, and complete descriptions of algorithms.
  • You can run the programs to study properties of the algorithms.
  • You can put the algorithms immediately to good use in applications.

Primitive data types and expressions.

A data type is a set of values and a set of operations on those values.
The following four primitive data types are the basis of the Java language:

  • Integers, with arithmetic operations (int)
  • Real numbers, again with arithmetic operations (double)
  • Booleans, the set of values { true, false }
    with logical operations (boolean)
  • Characters, the alphanumeric characters and symbols that you type (char)

A Java program manipulates variables that are named with
identifiers. Each variable is associated with a data
type and stores one of the permissible data-type values.
We use expressions to apply the operations associated with each type.

basic building blocks for Java programs

The following table summarizes the set of values and most common operations on those
values for Java’s int, double, boolean, and char
data types.

primitive data types in Java

  • Expressions. Typical expressions are infix.
    When an expression contains more than one operator, the
    precedence order specifies the order in which they are
    The operators * and / (and %)
    have higher precedence than (are applied before)
    the + and - operators; among logical operators, !
    is the highest precedence, followed by && and and then ||.
    Generally, operators of the same precedence are
    left associative (applied left to right).
    You can use parentheses to override these rules.
  • Type conversion.
    Numbers are automatically promoted to a more inclusive type if no
    information is lost.For example, in the expression 1 + 2.5,
    the 1 is promoted to the double value 1.0 and
    the expression evaluates to the double value 3.5.
    A cast is a directive to convert a value of one type into a value of another
    type. For example (int) 3.7 is 3.
    Casting a double to an int truncates toward zero.

  • Comparisons.
    The following mixed-type operators compare two values of the same
    type and produce a boolean value:

    • equal (==)
    • not equal (!=)
    • less than (<)
    • less than or equal (<=)
    • greater than (>)
    • greater than or equal (>=)
  • Other primitive types.
    Java’s int has a 32-bit representation; Java’s double type
    has a 64-bit representation.
    Java has five additional primitive data types:

    • 64-bit integers, with arithmetic operations (long)
    • 16-bit integers, with arithmetic operations (short)
    • 16-bit characters, with arithmetic operations (char)
    • 8-bit integers, with arithmetic operations (byte)
    • 32-bit single-precision real numbers, with arithmetic operations (float)


A Java program is composed of statements,
which define the computation by creating and manipulating variables,
assigning data-type values to them,
and controlling the flow of execution of such operations.

  • Declarations
    create variables of a specified type and name them with identifiers.
    Java is a strongly typed language because the Java compiler
    checks for consistency.
    The scope of a variable is the part of the program where it is
  • Assignments
    associate a data-type value (defined by an expression) with a variable.
  • Initializing declarations
    combine a declaration with an assignment to initialize a variable at the same
    time it is declared.
  • Implicit assignments.
    The following shortcuts are available when our purpose is to modify a variable’s value
    relative to the current value:

    • Increment/decrement operators: the code i++ is shorthand for i = i + 1.
      The code ++i is the same except that the expression value is taken after
      the increment/decrement, not before.
    • Other compound operators: the code i /= 2 is shorthand for
      i = i/2.
  • Conditionals
    provide for a simple change in the flow of execution—execute the
    statements in one of two blocks, depending on a specified condition.
  • Loops
    provide for a more profound change in the flow of execution—execute
    the statements in a block as long as a given condition is true.
    We refer to the statements in the block in a loop as the body of the loop.
  • Break and continue.
    Java supports two additional statements for use within while loops:

    • The break statement, which immediately exits the loop
    • The continue statement, which immediately begins the next iteration of the
  • For notation.
    Many loops follow this scheme: initialize an index variable to some value and then use a
    while loop to test a loop continuation condition involving the index variable,
    where the last statement in the while loop increments the index variable.
    You can express such loops compactly with Java’s for notation.
  • Single-statement blocks.
    If a block of statements in a conditional or a loop has only a single statement, the curly
    braces may be omitted.

The following table illustrates different kinds of Java statements.

Java statements


An array stores a sequence of values that are all of the same type.
If we have N values, we can use the notation a[i] to refer
to the ith value for any value of i from 0 to

  • Creating and initializing an array.
    Making an array in a Java program involves three distinct steps:

    • Declare the array name and type.
    • Create the array.
    • Initialize the array values.
  • Default array initialization.
    For economy in code, we often take advantage of Java’s
    default array initialization convention
    and combine all three steps into a single statement.
    The default initial value is zero for numeric types and false
    for type boolean.
  • Initializing declaration.
    We can specify the initialization values at compile time,
    by listing literal values between curly braces, separated by commas.

    declaring, creating, and initializing an array

  • Using an array.
    Once we create an array, its size is fixed.
    A program can refer to the length of an array a[] with the code a.length.
    Java does automatic bounds checking—if you access an array with
    an illegal index your program will terminate with an
  • Aliasing.
    An array name refers to the whole array—if we assign one array name to another,
    then both refer to the same array, as illustrated in the following code fragment.

    int[] a = new int[N];
    a[i] = 1234;
    int[] b = a;
    b[i] = 5678;   // a[i] is now 5678.

    This situation is known as aliasing and can lead to subtle bugs.

  • Two-dimensional arrays.
    A two-dimensional array in Java is an array of one-dimensional arrays.
    A two-dimensional array may be ragged
    (its arrays may all be of differing lengths), but we most often work with
    (for appropriate parameters M and N) M-by-N two-dimensional arrays.
    To refer to the entry in row i and column j
    of a two-dimensional array a[][], we use the notation

Static methods.

Static methods are called functions
in many programming languages, since they can behave like
mathematical functions.
Each static method is a sequence of statements that are executed,
one after the other, when the static method is called.

  • Defining a static method.
    A method encapsulates a computation that is defined as a
    sequence of statements.
    A method takes arguments
    (values of given data types) and computes a return value
    of some data type or causes a side effect.
    Each static method is composed of a signature
    and a body.

    anatomy of a static method

  • Invoking a static method.
    A call on a static method is its name followed by
    expressions that specify argument values in parentheses, separated by commas.
    When a method is called, its argument variables are initialized
    with the values of the corresponding expressions in the call.
    A return statement terminates a static method, returning control to the caller.
    If the static method is to compute a value,
    that value must be specified in a return statement.
  • Properties of methods.
    Java methods have the following features:

    • Arguments are passed by value.
      When calling a function, the argument value is fully evaluated and the
      resulting value is copied into argument variable.
      This is known as pass by value. Array (and other object) references
      are also passed by value: the method cannot change the reference, but it
      can change the entries in the array (or value of the object).
    • Method names can be overloaded.
      Methods within a class can have the same name, provided they have different
      signatures. This features is known as overloading.
    • A method has a single return value but may have multiple return statements.
      A Java method can provide only one return value.
      Control goes back to the calling program as soon as the
      first return statement is reached.
    • A method can have side effects.
      A method may use the keyword void as its return type,
      to indicate that it has no return value and produces side effects
      (consume input, produce output, change entries in an array,
      or otherwise change the state of the system).
  • Recursion.
    A recursive method is a method that calls itself either directly or indirectly.
    There are three important rules of thumb in developing recursive programs:

    • The recursion has a base case.
    • Recursive calls must address subproblems that are smaller
      in some sense, so that recursive calls converge to the base case.
    • Recursive calls should not address subproblems that overlap.
  • Basic programming model.
    A library of static methods is a set of static methods that
    are defined in a Java class.
    A basic model for Java programming is to develop a program that
    addresses a specific computational task by creating a library of static methods,
    one of which is named main().
  • Modular programming.
    Libraries of static methods enable modular programming,
    where static methods in one library can call static methods defined in
    other libraries.
    This approach has many important advantages.

    • Work with modules of reasonable size
    • Share and reuse code without having to reimplement it
    • Substitute improved implementations
    • Develop appropriate abstract models for addressing programming problems
    • Localize debugging
  • Unit testing.
    A best practice in Java programming is to include a main()
    in every library of static methods that tests the methods in the library.
  • External libraries.
    We use static methods from three different kinds of libraries,
    each requiring (slightly) differing procedures for code reuse.

    • Standard system libraries in java.lang,
      including java.lang.Math, java.lang.Integer, and
      java.lang.Double. This libraries are always available in Java.
    • Imported system libraries such as java.util.Arrays.
      An import statement at the beginning of the program is
      needed to use such libraries.
    • Libraries in this book.
      Follow these instructions for adding
      algs4.jar to your Java classpath.

    To invoke a method from another library, we prepend the
    library name to the method name for each call:
    Math.sqrt(), Arrays.sort(),
    BinarySearch.rank(), and StdIn.readInt().


  • Java libraries.
  • Our standard libraries.
  • Your own libraries.


Java's string data type

  • Concatenation.
  • Conversion.
  • Automatic conversion.
  • Command-line arguments.

Input and output.

  • Commands and arguments.

    anatomy of a command

  • Standard output.
  • Formatted output.

    printf formatting conventions

  • Standard input.
  • Redirection and piping.

    redirection and piping from the command line

  • Input and output from a file.
  • Standard drawing.

Binary search.

Below is a complete Java program
that illustrates many of the
basic features of our programming model. It implement a classic
algorithm known as binary search and tests it for an
application known as whitelist filtering.

anatomy of binary search

The static method rank() takes an integer key and a sorted
array of int values as arguments and returns the index of the key
if it is present in the array, -1 otherwise.
It accomplishes this task by maintaining variables lo
and hi such that the key is in a[lo..hi]
if it is in the array, then entering into a loop that tests the
middle entry in the interval (at index mid).
If the key is equal to a[mid], the return value is mid;
otherwise the method cuts the interval size about in half,
looking at the left half if the key is less than a[mid]
and at the right half if the key is greater than a[mid].
The process terminates when the key is found or the interval is empty.

binary search in an ordered array

Input and output libraries.

Here is a list of the input and output libraries that we use throughout
the textbook and beyond.

1.5 read numbers and text from standard input
1.5 write numbers and text to standard output
1.5 draw geometric shapes in a window
1.5 create, play, and manipulate sound
2.2 generate random numbers
2.2 compute statistics
2.2 read and write 1D and 2D arrays
3.1 read numbers and text from files and URLs
3.1 write numbers and text to files
3.1 draw geometric shapes
3.1 support keyboard and mouse events in Draw
3.1 process digital images
3.2 measure running time read bits from standard input write bits to standard output read bits from files and URLs write bits to files

We briefly describe the input and output libraries and include a sample client.

Standard input and standard output. and are libraries
for reading in numbers and text from standard input and printing out
numbers and text to standard output.
Our versions have a simpler interface than the corresponding Java ones
(and provide a few technical improvements). generates random numbers
in a given range. reads in
a sequence of real numbers from standard input and prints their
average on standard output.

% java Average
10.0 5.0 6.0 3.0 7.0 32.0
3.14 6.67 17.71
Average is 10.05777777777778 and are object-oriented
versions that support multiple input and output streams,
including reading from a file or URL and writing to a file.

Standard drawing. is an easy-to-use library
for drawing geometric shapes, such as points, lines, and circles.
draws a right triangle and a circumscribing circle. is an object-oriented
versions that support drawing in multiple windows.

Standard audio. is an easy-to-use library
for synthesizing sound. reads in a frequency
and duration from the command line, and it sonifies a sine wave of the
given frequency for the given duration.

Image processing. is an easy-to-use library
for image processing. takes
the name of a picture file and two integers (width w and height h)
as command-line arguments and scales the image to w-by-h.

% java Scale mandrill.jpg 298 298


% java Scale mandrill.jpg 200 200


% java Scale mandrill.jpg 200 400


Q + A

How important is it to use a good shuffling algorithm?


Here’s an
amusing anecdote
of what happens when you don’t do it
correctly (and you’re business is online poker!).
If you’re running an online casino, here’s the recommended
approach for shuffling a deck of cards: (i) get a cryptographically
secure pseudo-random number generator, (ii) assign a random 64-bit
number to each card, (iii) sort the cards according to their

Creative Problems

  1. Binomial distribution.
    Estimate the number of recursive calls that would be used by
    the method call binomial1(100, 50, .25)
    Develop a better
    implementation that is based on saving computed values in an array.

Web Exercises

  1. Sattolo’s algorithm.
    Write a program
    that generates a unifomly distributed
    cycle of length N using Sattolo’s algorithm.
  2. Wget.
    Write a program that reads in
    data from the URL specified on the command line and saves it
    in a file with the same name.

    % java Wget
    % more codes.csv
    United States,USA,00
  3. Right triangle.
    Write a client
    that draws a right triangle and a circumscribing circle.

      % java RightTriangle
    right triangle and circumscribing circle

  4. Bouncing ball.
    Write a program
    that illustrates the motion of a bouncing ball.

    Your browser can not display this movie.
    Be sure that Javascript is enabled and
    that you have Flash 9.0.124 or better.

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