CS212: DATASTRUCTURES

Lecture 4: Recursion

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Lecture Contents

The Concept of Recursion Why recursion? Factorial – A case study Content of a Recursive Method Recursion vs. iteration

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The Concept of Recursion

repetition can be achieved by writing loops, such as for loops and while loops.

Another way to achieve repetition is through recursion, which occurs when a function calls itself.

A recursive function is a function that calls itself , either directly or indirectly ( through another function).

For example :void myFunction( int counter)

{if(counter == 0)     return;else       {             myFunction(--counter);       return;       }

}

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Why recursion?

Sometimes, the best way to solve a problem is by solving a smaller version of the exact same problem first

Recursion is a technique that solves a problem by solving a smaller problem of the same type

Allows very simple programs for very complex problems

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Factorial – A case study

Factorial definition: the factorial of a number is product of the integral values from 1 to the number.

factorial 3

3*2*1=

6=

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Factorial – A case study

Iteration algorithm definition (non-recursive)

factorial 4

product [4..1]=

product [4,3,2,1]=

4*3*2*1=

24=

1 if n = 0 Factorial(n) =

n*(n-1)*(n-2)*…*3*2*1 if n > 0

Factorial – A case study

Recursion algorithm definition

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factorial 3

3 * factorial 2=

3 * (2 * factorial 1)=

3 * (2 * (1 * factorial 0))

=

3 * (2 * (1 * 1))=

3 * (2 * 1)=

=

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3 * 2=

1 if n = 0 Factorial(n) =

n * ( Factorial(n-1) ) if n > 0

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Content of a Recursive Method

Base case(s). One or more stopping conditions that can be directly

evaluated for certain arguments there should be at least one base case.

Recursive calls. One or more recursive steps, in which a current value of

the method can be computed by repeated calling of the method with arguments (general case).

The recursive procedure call must use a different argument that the original one: otherwise the procedure would always get into an infinite loop…

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Content of a Recursive Method

void myFunction( int counter){if( counter == 0)     …........else {      ………       myFunction(--counter);

  }}

Use an if-else statement to distinguish between a Base case

and a recursive step

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Iterative factorial algorithm

public static int recursiveFactoria(int n) {

i = 1

factN = 1

for(int i=1; i<= n; i++)

factN = factN * i

i = i + 1

return factN

}

public static int recursiveFactoria(int n) {

if (n == 0)return 1; // base case

else return n * recursiveFactorial(n-1): // recursive case

}

Recursive factorial Algorithm

recursive implementation of the factorial function is somewhat simpler than the iterative version in this case there is no compelling reason for preferring recursion over iteration. For some problems, however, a recursive implementation can be significantly simpler and easier to understand than an iterative implementation.

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Tracing Recursive Method

• Each recursive call is made with a new arguments Previous calls are suspended

• Each entry of the trace corresponds to a recursive call.

• Each new recursive function call is indicated by an arrow to the newly called function.

• When the function returns, an arrow showing this return is drawn and the return value may be indicated with this arrow.

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Recursion vs. iteration

Iteration can be used in place of recursion An iterative algorithm uses a looping

construct A recursive algorithm uses a branching

structure(if-statement) Recursive solutions are often less

efficient, in terms of both time and space, than iterative solutions

Recursion can simplify the solution of a problem, often resulting in shorter, more easily understood source code

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Recursion

1 Linear Recursion

2 Binary Recursion

3 Multiple Recursion

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1 Linear Recursion

The simplest form of recursion is linear recursion, where a method is defined so that it makes at most one recursive call each time it is invoked.

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Summing the Elements of an Array Recursively

Algorithm LinearSum(A , n )input A integer array A and an

integer n>=1, such that A has at least n elements

output The sum of the first n

integers in A

if (n = 1) then return A[0]else return LinearSum(A, n - 1) +

A[n - 1]

4 3 6 2 5A =

n = 5

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Summation (Sigma) NotationAlgorithm sum ( n <integer>)Input

n is an integer number

o

is returned if (n == 1) return 1;else return n + sum(n-1);End IF

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Summation (Sigma) Notation con..

Sum (3)return 3+Sum(2)return 3+3 Sum(2)

return 2+ sum(1) return 2+1

Sum(1) return 1

return 1

return 3

return 6

For n =3

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Printing “Hello World” n Times Using Recursion

Algorithm printHelloWorld ( n <integer>)Input

n is an integer numberOutput

“Hello World” printed n times

If ( n >0 ) {

Print “Hello World”

printHelloWorld( n – 1 );

}

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Printing “Hello World” n Times Using Recursion

printHelloWorld(3)Print hello worldprintHelloWorld(2)

hello world

printHelloWorld(2)Print hello worldprintHelloWorld(1)

hello world

printHelloWorld(1)Print hello world printHelloWorld(0)

hello world

printHelloWorld(0)

return

return

return

return

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2 Binary Recursion

When an algorithm makes two recursive calls, we say that it uses binary recursion. These calls can, for example, be used to solve two similar halves of some problem,

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Computing Fibonacci Numbers via Binary Recursion

Let us consider the problem of computing the kth Fibonacci number.

{0, 1, 1, 2, 3, 5, 8, 13, 21, 34,….} Fibonacci numbers are recursively defined

as follows: F0 = 1 F1 = 1 Fi = Fi−1 + Fi−2.

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Computing Fibonacci Numbers con..

Algorithm BinaryFib(k):Input:

Nonnegative integer k

Output: The kth Fibonacci number f

if (k< I) then return kelsereturn BinaryFib(k - 1) + BinaryFib(k -2)

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Calculate BinaryFib(3)

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Exercise

what is the output of fun(16)?

Algorithm fun(int X)

if ( x <=4 ) return 1

elsereturn fun(x/4) + fun (x/2)

Result = 3

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3 Multiple Recursion

Generalizing from binary recursion, we use multiple recursion when a method may make multiple recursive calls, with that number potentially being more than two.

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