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Recursion

Tuesday, September 26, 2006

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Recursion

A recursive function is one that makes use ofitself

Every recursive function involves base case(s), simple instance of the problem that

can be solved directly crucial for termination recursive case(s).

divide the problem into one or more simpler orsmaller parts call the function (recursively) on each part combine the solutions of the parts into a

solution for the problem

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Designing a Recursive Algorithm

Consider the following questions:

1. How could you define the problem in terms of asmaller problem of the same type?

2. How would each recursive call affect the size of theproblem? (size should decrease)

3. What instance(s) of the problem can serve as thebase case?

4. As the problem size decreases, will you reach thebase case?

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Recursion: Advantages

For some problems, recursive solution may bemore natural and simpler to write than using

non-recursive solutions Example: The factorial function:

n! = 1 (if n = 0)

n! = n * (n-1)! (otherwise)

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Recursion : Disadvantages

Calling a recursive function consumes moretime and memory than non-recursive solutionof the same problem

It is easy to write inefficient recursivealgorithms

==> High performance applications hardlyever use recursion

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Watch Out

When using recursion, we must be careful not tocreate an infinite chain of function calls:

int factorial(int numb){return numb * factorial(numb-1);

} A recursive function must

Contain at least one non-recursive branch Eventually lead to a non-recursive branch

Oops!Oops!

No termination conditionNo termination condition

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Fibonacci Sequence Example I

Fibonacci Sequence

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Fibonacci Sequence Example II

int fib(int number)

{

if (number == 0)return 0;

if (number == 1)

return 1;return (fib(number-1) + fib(number-2));

}

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Fibonacci Sequence Example III

Reference: [6]

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Linear Recursion

A Linear Recursion is the simplest form ofrecursion. Has one recursive call

Is used when an algorithm has a simplerepetitive structure

Includes some basic step (usually first or lastelement)

The remaining set has the same structure as theoriginal one

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Tail Recursion

A Tail Recursion is a special case of linearrecursion

With the last operation being a recursive call Tail Recursions can very often be easily

transformed into an iteration

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Binary Recursion

A Binary Recursion is a recursion that callsitself twice

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Quick Test

Can you think of an example for

Linear Recursion

Tail Recursion Binary Recursion

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Exercice 1

Write a recursive function that prints thedigits of a non-negative integer number

vertically

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Solution 1

void recursive_write_vertical(int number){

if (number < 10) // stop condition// Write the only digit, and stop recursion

System.out.println(number);else // recursive calls{// Print all digits except the last digit

recursive_write_vertical(number/10);

// Print the last digit

System.out.println(number % 10);}

}

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Exercise 2

Instead of printing only non-negativeintegers, print any integer number vertically(i.e., including negative numbers)

Hint:

You can have more than one recursivecalls or stopping conditions in yourrecursive function

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Solution 2

void super_write_vertical(int number)// If number is negative, then a negative sign appears on the top.

{ if (number < 0) {System.out.println(-); // print a negative sign

super_write_vertical(abs(number));// abs computes the absolute value

} else if (number < 10) {System.out.println(number); // Write the one digit

} else {super_write_vertical(number/10);

// Write all but the last digit

System.out.println(number % 10);

// Write the last digit}

}

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Exercise 3

Write a recursive function that counts thenumber of zero digits in an integer

Example:zeros(10200) returns 3

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Solution 3

int zeros(int numb){

if(numb==0)

return 1;

else if(numb < 10 && numb > -10)return 0;

else // > 1 digits: recursionreturn zeros(numb/10) + zeros(numb%10);

}

zeros(10200)zeros(1020) + zeros(0)

zeros(102) + zeros(0) + zeros(0)

zeros(10) + zeros(2) + zeros(0) + zeros(0)

zeros(1) + zeros(0) + zeros(2) + zeros(0) + zeros(0)

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Recurrence Relation

A recurrence relation is an equation defininga function f(n) recursively in terms of smaller

values of n Can help to determine the Big-O running time of

recursive function

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Example: Recurrence Relation

The running time of Merge-Sort is: (assumingthat n is a power of 2)

T(n) = O(1) if n = 1 T(n) = 2 T(n/2) + O(n) if n > 1

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Solving Recurrence Relation - I

To simplify the algebra, we write

n instead of O(n)

1 instead of O(1) So we have

T(1) = 1

T(n) = 2 T(n/2) + n

as the recurrence relation

Reference: [10]

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Solving Recurrence Relation - II

T(n) = 2 T(n/2) + n

= 2 [2 T(n/4) + n/2] + n

= 4 T(n/4) + 2n

= 4 [2 T(n/8) + n/4] + 2n= 8 T(n/8) + 3n

...

= 2k

T(n/2k

) + k nWe know that T(1) = 1 and this is how we could terminate the

derivation

n/2k = 1 n = 2k k = log2n

Reference: [10]

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Solving Recurrence Relation - III

...

= 2k T(n/2k) + k n // k = log2n

= 2log2

n T(1) + (log2

n) n

= 2log2n * 1 + n log2n

= n + n log2n

= O(n log n)

Reference: [10]

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Exercise 4

Solve the following recurrence relation:

T(N) = T(N/2) + 1

T(1) = 1 You may assume that N is a power of 2.

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Solution 4

Recurrence Relation: T(N) = T(N/2) + 1; T(1) = 1

T(N) = T(N/2) + 1

= T(N/4) + 1 + 1

= T(N/8) + 1 + 1 + 1

= T(N/2k) + 1 + 1 ++ 1

= T(1) + 1 + 1 + + 1 = 1 + log2N

log2N times

k times

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References

[1] Wikipedia Encyclopedia http://en.wikipedia.org/[2] Dictionary of Algorithms and Data Structures . National Institute of Standards and

Technology. http://www.nist.gov/dads/[3] M.T. Goodrich, R. Tamassia. Data Structures and Algorithms in Java, 4th edition. John

Wiley & Sons, 2006. ISBN 0-471-73884-0. http://java4.datastructures.net/[4] M.T. Goodrich, R. Tamassia, D. Mount. Data Structures and Algorithms in C++. John Wiley

& Sons, 2004. ISBN 0-471-20208-8. http://cpp.datastructures.net/[5] SparkNotes: Examples of Recursion. http://www.sparknotes.com/cs/recursion/examples/[6] Basic Math and Recursion. Brooks/Cole Publishing Company , 2000.

http://www.cs.ust.hk/~huamin/COMP171/recursion.ppt[7] Walter Savitch, Problem Solving with C++ - The Object of Programming, Chapter 13

Recursion, 4th Edition, Pearson Education Inc., 2003

[8] Andrew Horner, Comp104 notes on Recursion, CS Dept, HKUST.[9] Tao Jiang, Recurrence relations and analysis of recursive algorithms, Lecture notes for UCRCS 141: Intermediate Data Structures and Algorithms.http://www.cs.ucr.edu/~jiang/cs141/recur.pdf

[10] Owen L. Astrachan, Big-Oh for Recursive Functions: Recurrence Relations, DukeUniversity, 2003. http://www.cs.duke.edu/~ola/ap/recurrence.html

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Thank you !See you next time