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CSC240 :: Lecture Note :: Week 05
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Assignment(s): #stats1 (due 10/20/2017) and #RecursiveReverse (due 10/20/2017)

Code: alloc.cpp | factorial.cpp | power.cpp | printld.cpp


Recursion is any solution technique in which large problems are solved by reducing them to smaller problems of the same form.

According to multiple sources, the text Thinking Recursively by Eric Roberts (John Wiley 1986) is an excellent resource on recursion.

Internally, recursive functions are supported using a stack.

Almost every recursive function has the same basic structure:

   if (test for simple case)
      return (simple solution computed w/o using recursion)
      return (recursive solution involving a call to the same function)

A recursive function is a function that calls itself in its function body. The call uses different arguments that ultimately read values that allow the function to return a result.

A recursive algorithm must have a well-defined stopping state, or termination condition.

Tail recursion is when the only recursive call appears in the function and that recursive call is the last operation performed at that procedural level. Typically, tail recursive functions can be easily rewritten as non-recursive functions. The traverse linked-list example provides an example of tail recursion.

Some final comments:

Interesting Recursion Related Tidbit

Headline: "Microsoft's Java JIT compiler bungles tail recursion."

A programmer from England has discovered a flaw in Microsoft's Java environment running in Internet Explorer under Windows. On 10/20/1998 Andrew Kennedy posted a note to Java programming newsgroups showing that jview, Microsoft's command-line interface to its just-in-time compiler, calculates the factorial of 5 (i.e., 5 * 4 * 3 * 2 * 1 * 0!) to be 16. Kennedy writes:
> Microsoft seem to have confused addition with multiplication. > Such an easy mistake to make.
Followup postings pin down the likely source of the bug as incorrectly handled tail recursion in Microsoft's JIT compiler. Unfortunately there seems to be no way to turn off JIT. Kennedy speculates that this error may be responsible for the failure of a large number of Java applets to run in the IE 4 environment.

The Factorial Example

Computing the factorial of a number is popular example of using recursion.

   Mathematically:  n! = { 1  if n = 0; otherwise,  n * (n-1)! }

To compute n! you must:

  1. remember what  n  is
  2. go compute  (n-1)! 
  3. once  (n-1)!  is computed, multiply by  n  to get the final answer
   To compute (n-1)!, you 1st compute (n-2)!.  Computing (n-2)!
   requires finding (n-3)! and so on.  Eventually we end up at 0!
   and 0! is defined to be the value 1.
   In C:
      printf("%i\n", factorial(4));
      int factorial(int n) {
         if (n == 0) return 1;
         return n * factorial(n -1);

   4 is pushed onto the stack
      3 is pushed onto the stack
         2 is pushed onto the stack
            1 is pushed onto the stack
               0 is pushed onto the stack
               returns 1
            1 * 1 returns 1
         2 * 1 returns 2
      3 * 2 returns 6
   4 * 6 returns 24
See factorial.cpp and Factorial.java.

Recall: When a function is called, EXPRs are evaluated and are passed by value. An implicit argument that is passed to a function is a return address. Function return values, if any, are placed in a secure location from which the calling program may retrieve it.

The Power Example

Raising a number to a power can also be solved recursively.
   Power function:   n**k = { 1  if k is 0; otherwise, n * (n**(k-1)) }
Computing  xn , where  x  is a real number and  n  is a non-negative integer, is called the power function and its calculation is usually accomplished with repeated multiplication:
   xn = x * x * x * ... *x * x

      where  x  multiplied with itself  n  times

   20 = 1
   21 = 2 * 1 = 2
   22 = 2 * 2 = 4
   23 = 2 * 2 * 2 = 8
   24 = 2 * 2 * 2 * 2 = 16
The power function can be implemented as follows:

   power(x, n) = xn = { 1, n = 0 | x * xn-1, n > 0 }

   double power(double x, int n) {
      if (n == 0) return 1;
      return x * power(x, n-1);

See power.cpp and Power.java.

Coverting an Integer to a String

   printd(int n) {
      if (n / 10)
         printd(n / 10);
      putchar(n % 10 + '0');
   150 / 10 -- 15 pushed onto the stack
      15 / 10 -- 1 pushed onto the stack
         1 / 10  
         1 is printed
       5 is printed
   0 is printed
See printld.cpp and PrintLD.java.

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An array is a collection of individual data values with two characteristics: it is ordered and homogeneous.

The following are some terms commonly used when talking about arrays:

An array is homogeneous because each element must be of the same type. For example, an array of int's, an array of float's, an array of char's.

An array is ordered -- it has a 1st element, a 2nd element, a 3rd element, and so on. Array elements are stored in contiguous memory locations.

Just like any other variable, an array must be defined before it is used.

   Syntax:  element-data-type array-name [ length ];

      int intArray[10]; 
         name:  intArray
         type:  int
         length:  10
         size:  10 * sizeof(int)

      char someString[32];
         name:  someString
         type: char
         length: 32
         size:  32 * sizeof(char)

Each element of the array is identified by a numeric value called its index (index numbers always start at 0).

When defining an array, its length must be specified at compile-time. In addition, the specification of the length must be a constant integral EXPR.

   int i = 10;
   /* double salaries[i];   // illegal (not a constant) */
   /* char name[15.5];      // illegal (not an integral type) */
   const int MAX_ELEMENTS = 10;
   short scores[MAX_ELEMENTS];  /* okay in C++, not C */
   #define LENGTH 5
   float someArrayOfFloats[LENGTH];

Typically, array lengths are defined to be manifest constants.

Array Initialization

Arrays can be initialized at the time they are defined.

   int evenNumbers[5] = { 2, 4, 6, 8, 10 };

   note:  it is a syntax error if the # of initializers is
          greater than the array length (i.e. # of elements)

   The elements of the initializer list must be constant
   EXPRs.  If the number of initializers is less than the
   array length, then remaining elements are set to zero.

If an array is initialized when defined, then the length is not needed -- the compiler will set the length depending on the number of initializers. If the length of the array needs to be greater than the number of initializers specified, then the length must be specified.

   float radioStations[] = { 103.1, 93.3, 100.7 };

   radioStations  has a length of 3; to figure out the length
   of the array using code:

      sizeof(radioStations) / sizeof(radioStations[0])

   sizeof(radiostations)  evaluates to the size of the array
   and  sizeof(radioStations[0])  evaluates to the size of a
   single element of the array (recall, an array size is equal 
   to the length of the array times the size of the array type)

   the following EXPR also works to determine the length of
   an array    sizeof(radioStations) / sizeof(float)    but could 
   result in the defect if the type of the array is changed and
   the EXPR is not

There is not a convenient mechanism for initializing all elements of an array to a single value (exception: it is easy to set all elements of an array to zero -- int a[10] = { 0 };).

For efficiency, locally declared arrays that are initialized at definition may be declared to be static.

   static short tvStations[99] = { 3, 10, 12, 61 };

   tvStations  is an array of length 99; elements 0, 1, 2, 3
   have non-zero values, whereas elements 4 through 98 equal 0;
   the array is initialized once -- at program load time

Once an array has been defined, its length (i.e. number of elements) cannot be altered.

Elements of the array are accessed using the unary array operator [].

   const int LEN = 10;
   int a[LEN];

   a[3] = 150;  /* set element #4 to 150 */
   a[1] = 200;  /* assign the value 200 to element #2 */

   if (a[1] == a[3]) /* compare the value of element #2 with element #4*/

Array indicies must be integral values, but they are not restricted to being constants.

   int i, j, k;

   a[0] = 100;
   a[i] = 150;
   a[i * j - k] = 210;
   a[a[a[i]]] = 250;
   a[rand() % LEN] = 178;

The language does not protect against indexing beyond the ends of an array. Typically, if this happens, then a run-time error is encountered.

   a[-1] = 100;
   a[sizeof(a)/sizeof(a[0]) + 2] = 200;

Array 'a' cannot be copied to array 'b' using simple assignment.

   #define LEN 5 
   int a[LEN] = { 100, 200, 210, 120, 240 }, b[LEN];

   /* b = a;   //syntax error -- array name not a lvalue */
   for (int i = 0; i < LEN; i++)
      b[i] = a[i];
Arrays and Functions

Arrays are passed to functions "by reference." (Think about the overhead if arrays were passed by value.)

A function that receives an array as a parameter, obtains a constant pointer to the first element of the array.

Unless specifically qualified to be const, the function can modify the content of the array.

When a function receives an array as a parameter, it does not know, nor can it figure out, the length or size of the array. In many cases, the caller passes the array length (size_t) to the function.

   void func1(int[]);         //do *not* specify the array length
   void func2(const int[]);   //func2() not allowed to modify the array
   void func3(int [], int);
   #define A_LEN 10
   int a[A_LEN];
   func1(a);  //using array name w/o [] operator "decays" into a
              //constant pointer to the 1st element of the array
   func3(a, A_LEN);  //pass the array length to the function
   void func1(int i[]) {  //again, array length not specified
      int len = sizeof(i) / sizeof(i[0]);
         // does not work -- sizeof(i) evaluates to the size
         // allocated for pointer variables, which is typically
         // the sizeof(int)...
   void func2(const int i[]) {
      //! i[0] = 100;    //use of  const  implies the content of the
                         //array cannot be changed...
   void func3(int i[], int len) {
      for (int i = 0; i < len; i++)
         //loop through each element of the array

Pointer notation can be used when prototyping and defining functions that receive arrays as parameters.

   void func1(int*);  //this syntax indicates that the function receives
                      //a pointer to an array
   void func2(const int*);  //array content cannot be modified

A Tutorial on Pointers and Arrays in C

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A pointer is a variable whose value is an address of another variable.

Defining and Initializing Pointer Variables
When defining a variable, prefixing the variable's name with an asterik * causes the variable to be a pointer.

   int* iptr;   /*define variable iptr that will point to an int variable*/
   float * fptr; /*define variable fptr that will point to a float variable*/
   char *cptr;  /*define variable cptr that will point to a character*/
   int i, j;
   float f;
   char c;

   iptr = &i;  /*assign the address-of variable 'i' to iptr*/
   fptr = &f;  /*assign the address-of variable 'f' to fptr*/
   cptr = &c;  /*assign the address-of variable 'c' to cptr*/
   int* iptr2 = &j; /*define and initialize an int pointer*/
   iptr = iptr2;  /*now iptr points to the variable 'j'*/

   note:  Placement of the  *  when defining a pointer variable is
          a matter of style -- I like to place the  *  next to the
          data type, but others prefer to place it next to the 
          variable name.  Placing it next to the data type does
          require caution when multiple variables are defined on
          the same declaration statement.  Example:

              int* ip1, ip2;  
                   // ip1  is a pointer to an  int  , but  ip2  is a
                      regular  int

              int* ip1, *ip2;  
                   // ip1  and  ip2  are both pointers to an  int

The address-of operator only applies to objects in memory: variables and array elements.

Locally defined non-static pointers, unless explicitly initialized, are garbage and using them without initialization can cause a program to execute incorrectly (or abort).

Global and statically defined pointers are initialized to the NULL pointer.

Accessing Data Using Pointers

The unary operator * is the indirection or dereferencing operator; when applied to a pointer, it accesses the object the pointer points to.

   int i = 200;
   int* iptr = &i;

   cout << *iptr;    /*prints 200 -- the value of 'i' which is the object
                       iptr points to*/
Pointers and Function Arguments

Since C passes arguments to functions by value, there is no way for the called function to alter a variable in the calling function. A way to obtain the desired effect is for the calling program to pass pointers to the values to be changed.

   void swap(int*, int*);

   int i, j;

   swap(&i, &j);

   void swap(int* a, int* b) {
      int tmp = *a;
      *a = *b;
      *b = tmp;

Using pointers is similar to using reference variables; however, reference variables are part of C++ and they are not part of C.

Advantage of using reference variables over pointers.

Introduction to Pointers and Arrays

There is a strong relationship between pointers and arrays.

When an array name is used by itself, it evaluates to a constant pointer to the first element of the array.

Everywhere you use arrayName[index] you can use *(ptr + index) (assuming ptr points to some part of the array).

   int scores[10];

   scores[0] = 2;      // or  *scores = 2
   scores[1] = 3;      // or  *(scores + 1) = 3
   *(scores + 2) = 5;  // or  scores[2] = 5

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