10/14/2015cosc237/recursion1 recursion a method of defining a concept which refers to the concept...

04/19/23 cosc237/recursion 1

Recursion

• A method of defining a concept which refers to the concept itself

• A method of solving a problem by reducing it to smaller versions of itself

• A powerful, elegant, efficient way to solve certain classes of programming problems

• A recursive function calls itself


Recursive definitions

Relative: x is a relative of y iff (if and only if)

• x is y’s parent• x is y’s child• x is y’s spouse• x is a relative of a relative of y


Recursive definitions cont’d

Natural numbers• 1 is a natural number• if i is an natural number, then i+1 is

a natural number

• Definition in which a problem is expressed in terms of a smaller version of itself

• Has one or more base cases


Example 1:

void message() //infinite recursive function{ System.out.println("This is a recursive

function.)"; message();}


Example 2:

void message(int times) { if (times > 0) { System.out.println("This is a recursive

function.)"; message(times -1); }}


message (4); //Each call creates an activation record or frame.

1st calltimes: 4

2nd calltimes: 3

3rd calltimes: 2 4th call

times: 15th calltimes: 0


Recursion requires

1. base case2. general case

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General format forMany Recursive Functions

if (some easily-solved condition) // base case

solution statement

else // general case

recursive function call


• Base part– states one or more cases that satisfy

the definition– No base->infinite recursion

• recursive part – states how to apply the definition to

satisfy other cases– with repeated applications, reduces to

base• implicitly: nothing else satisfies the

definition


Example #3:

• Find the sum of all int numbers between 1 and n.

• Sum(5) = 1 + 2 + 3 + 4 + 5 = 15 • Base case: Sum (1) = 1 • General Case:

Sum (2) = 1 + 2 = Sum(1) + 2 = Sum (n-1) + n

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Finding the Sum of the Numbers from 1 to n

int Summation ( int n ){ if ( n == 1) // base case

return 1 ;else // general case

return ( n + Summation ( n - 1 ) ) ;}

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Exercises

• Write a function using recursion to print numbers from n to 0.

• Write a function using recursion to print numbers from 0 to n. (You just need to shift one line in the program above)


Solution:

//n to 0void Print(int n){

System.out.println(n ); if (n <= 0)

return; Print(n-1);

}

//0 to nvoid Print(int n){

if (n > 0)Print(n-1);

System.out.println(n );

}


Example #4:

• Write a recursive function to find n factorial.• n! = n * (n-1) * (n-2)... 3 * 2 * 1• Factorial(5) = 120 = 5 * 4 * 3 * 2 * 1 • Base case:

if (n == 1) return 1;

• General case:n! = n * (n-1) * (n-2)... 3 * 2 * 1

= n * [(n-1) * (n-2)... 3 * 2 * 1] = n * (n-1)! OR:

Factorial(n) = n * Factorial(n-1)


Factorial n! = n x (n-1)!

int factorial(int num){ if (num == 0) return num; else return num * Factorial(num – 1);}

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Recursive Factorial Method (continued)


Example #5:

• Write a recursive function to find xn .

• Base case: 20 = 1 • General Case: 2n = 2 * 2n-1


power function

int Power(int x, int n) // n >0{ if (n == 1) return x; // base case else return x * Power(x,n-1); }


Power, n<0

• Note that 2-3 = 1/23 = 1/8 • In general, xn = 1/ x-n for non-zero

x, and integer n<0.


Power

float Power (float x, int n){

if (n == 0) // base case return 1; else if (n > 0) // first general case return (x * Power(x , n - 1)); else // second general case return (1.0 / Power(x , - n));

}


Use the box method to trace

• Activation record• Represent each call to the function

during the course of the execution by a new box

• Local environment

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Tracing a Recursive Method

• After completing a recursive call:– Control goes back to the calling

environment– Recursive call must execute

completely before control goes back to previous call

– Execution in previous call begins from point immediately following recursive call


void mystery(char param1, char param2){ if (param1 != param2) { param1++; param2--; Mystery(param1, param2); System.out.println(param1); System.out.println(param2); }}• Show all output resulting from the following

invocation:mystery('A','G');• Under what circumstances does mystery result in

infinite recursion?

Exercises


int TermUnknown (int someInt)// This is a classic number theory problem for which// termination for all input is uncertain.{ if (someInt == 1) return 0; if (someInt % 2 == 1) return 1 + TermUnknown(3 * someInt +1); return 100 + TermUnknown(someInt/2);} What value does each of the following invocations

return?a. TermUnknow(1);b. TermUnknown(4);c. TermUnknown(3);

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Largest Value in Arraypublic static int largest(int[] list, int lowerIndex, int upperIndex){ int max; if (lowerIndex == upperIndex) return list[lowerIndex]; else { max = largest(list, lowerIndex + 1, upperIndex); if (list[lowerIndex] >= max) return list[lowerIndex]; else return max; }}

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Largest Value in Array (continued)


Recursive Functions

• Recursive function - invokes itself directly or indirectly– direct recursion - function is invoked

by a statement in its own body– indirect recursion - one function

initiates a sequence of function invocations that eventually invokes the original • A->B->C->A


• When a function is invoked - activated• activation frame - collection of

information maintained by the run-time system(environment)– current contents of local automatic

variables– current contents of all formal parameters– return address– pushed on stack

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Linear Search • array a • two subscripts, low and high• Key• Return -1 if key is not found in the

elements a[low...high]. • Otherwise, return the subscript

where key is found

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// Recursive definitionint LinearSearch ( /* in */ const int a[ ] , /* in */ int low , /* in */ int high, /* in */ int key )

{if ( a [ low ] == key ) // base case

return low ;

else if ( low == high) // second base case

return -1 ;

else // general case

return LinearSearch( a, low + 1, high, key ) ;

}30

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Function BinarySearch( )

BinarySearch that takes sorted array a, and two subscripts, low and high, and a key as arguments. It returns -1 if key is not found in the elements a[low...high], otherwise, it returns the subscript where key is found

BinarySearch can be written using iteration or using recursion

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// Iterative definition

int BinarySearch (int a[ ] , int low , int high,int key ) {

int mid; while ( low <= high ) { // more to examine

mid = (low + high) / 2 ;

if ( a [ mid ] == key ) // found at mid return mid ;

else if ( key < a [ mid ] ) // search in lower half high = mid - 1 ; else // search in upper half

low = mid + 1 ; } return -1 ; // key was not found

}

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// Recursive definition

int BinarySearch (int a[ ], int low, int high, int key) {

int mid = (low + high) / 2 ;

if ( low > high ) // base case -- not foundreturn -1;

if ( a [ mid ] < key ) // search in upper halfreturn BinarySearch( a, mid + 1, high, key ) ;

else if ( key < a [ mid ] ) // search in lower half return BinarySearch ( a, low, mid - 1, key ); else

return mid; }

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"One Small Step/Complete the Solution"

• Each invocation of a recursive function performs one small step towards the entire solution and makes a recursive call to complete the solution.


Recursive Functions

• Non-value-returning– invoked as separate stand-alone

statement

• value-returning– invoked in an expression– often embedded within return

statement

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Recursion or Iteration?• Two ways to solve particular problem:

– Iteration or Recursion• Every loop can be replaced by recursion• not every recursion can be replaced by a loop

• Iterative control structures use looping to repeat a set of statements

• Tradeoffs between two options:– Sometimes recursive solution is easier

– Recursive solution is often slower


Factorial(loop version)

int LoopFactorial (int n){ int i; int fact = 1;

for (i = 1; i <= n; i++) fact = fact * i; return fact;}


Factorial - table lookup

• int factorial[8] = {1,1,2,6,24,120,720,5040};

• faster, can simplify algorithms• more memory space

Java Programming: Program Design Including Data Structures

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Programming Example: Decimal to Binary

public static void decToBin(int num, int base){ if (num > 0) { decToBin(num / base, base); System.out.print(num % base); }}

Java Programming: Program Design Including Data Structures

40

Programming Example: Decimal to Binary


When to Use Recursion

• If the problem is stated recursively• if recursive algorithm is less

complex than recursive algorithm• if recursive algorithm and

nonrecursive algorithm are of similar complexity - nonrecursive is probably more efficient

• consider table-driven code


• References: [1] Programming and Problem Solving with C++, third edition, by Neal Dale, Chip Weems, and Mark Headington, Jones and Bartlett Publishers, 2002. [2] Data Abstraction and Structures Using C++, by Mark R. Headington, and David D. Riley, Jones and Bartlett Publishers, 1997. [3] Starting Out with C++, by Tony Gaddis, Scott Jones Publishers, 2004. Java Programming:

• [4] Program Design Including Data Structures, D.S. Malik

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