Advanced Data Structure Recursion

Presented by:-A.RAVINDRA KUMAR RAO (B.TECH (C.E))

Madanapalle Institute Of Technology & Science

CONTENTS RECURSION RECURSIVE ALGORITHMAS

RECURSIVE TYPES

RECURSIVE FUNCTIONS

RECURSIVE DATATYPES

ORDER OF EXECUTION

EXAMPLE

RECURSIONWhat is recursion?

It’s when a function calls itself.

how does this happen?When we talk about recursion, we are really talking about creating a loop.

Let’s start by looking at a basic loop.

for(int i=0; i<10;i++) { cout<<“the number is:”<<i<< end1; }Output:

the number is :0the number is:1

the number is:2 the number is:3

RECURSIVE ALGORITHMImplementation and use of Recursive Algorithms: space for all local, automatic (not static) variables space for all formal parameters the return address any other system information about the function or the function that called it.

The function must have a selection construct which caters for the base case The recursive call must deal with a simpler/smaller version of the data Recursion uses selection construct and achieves repetition through repeated function

calls. Recursion has a base case. Any problem that can be solved recursively can be solved iteratively.

RECURSIVE ALGORITHMA recursive algorithm is an algorithm which calls itself with "smaller or simpler" input values, and which obtains the result for the current input by applying simple operations to the returned value for the smaller input.

Example 1: Algorithm for finding the k-th even natural number Algorithm 1:   Even(positive integer k)

Input: k , a positive integer

Output: k-th even natural number (the first even being 0)

if k = 1, then return 0;

else

return Even(k-1) + 2 .

RECURSIVE TYPES

Single recursion:Recursion that only contains a single self-

reference is known as single recursion.Example: Factorial function.

Multiple recursion: Recursion that contains multiple self-references is known as

multiple recursion.Ex: Fibonacci sequence

Direct recursion:In which a function calls itself.

Ex: ƒ calls ƒ i.e. direct recursion

Indirect recursion:Indirect recursion occurs when a function is called not by

itself but by another function that it called (either directly or indirectly).Ex: ƒ calls g, which calls ƒ

RECURSIVE TYPES

Anonymous recursion:Recursion is usually done by explicitly calling a function by name.

However, recursion can also be done via implicitly calling a function based on the current context, which is particularly useful for anonymous function and is known as anonymous recursion.

Structural Recursion:Functions that consume structured data, typically decompose their

arguments into their immediate structural components and then process those components. If one of the immediate components belongs to the same class of data as the input, the function is recursive.EX: Factorial.

Generative Recursion:Many well-known recursive algorithms generate an entirely new

piece of data from the given data and recur on it.Ex: GCD, Binary Search

RECURSIVE FUNCTIONRECURSION FUNCTION:

Recursion is often seen as an efficient method of programming since it requires the least amount of code to perform the necessary functions. However, recursion must be incorporated carefully, since it can lead to an infinite loop if no condition is met that will terminate the function.

MATHMATICAL FUNCTION:

In mathematics we often define a function in terms of itself.For example: The factorial function f(n)=n!, for n is an integer, is defined as follows; ƒ(n)={1 n≤1 { n ƒ(n-1) n>1

TAIL RECURSIVE: In a tail-recursive function, none of the recursive call do additional work after the recursive call is complete (additional work includes printing, etc), except to return the value of the recursive call. return rec_func( x, y ) ; // no work after recursive call, just return the value of call

MUTUALLY RECURSIVE: For example, function f() can call function g() and function g() can call function f(). This is still considered recursion because a function can eventually call itself. In this case, f() indirectly calls itself.

RECURSIVE FUNCTIONS

FACTORIAL FUNCTION: int Fact(int n) { if (n == 0) { return 1; } else { return n * Fact(n - 1); } }

RECURSIVE FUNCTION

RECURSIVE DATA TYPES

Inductively defined data:An inductively defined recursive data definition is one that

specifies how to construct instances of the data. Ex: linked list.Co-inductively defined data type:

A co-inductive data definition is one that specifies the operations that may be performed on a piece of data.

A co-inductive definition of infinite streams of stringsEx: A stream of strings is an object s such that head(s) is a string, and tail(s) is a stream of strings.

ORDER OF EXECUTION

In the simple case of a function calling itself only once, instructions placed before the recursive call are executed once per recursion before any of the instructions placed after the recursive call. The latter are executed repeatedly after the maximum recursion has been reached.Function 1void recursiveFunction(int num) { printf("%d\n", num); if (num < 4) recursiveFunction(num + 1); }

ORDER OF EXECUTIONFunction 2 with swapped lines:

void recursiveFunction(int num) { if (num < 4) recursiveFunction(num + 1); printf("%d\n", num);}

SYSTEM STACK OF RECURSION

OBJECT HEAP UNUSED MEMORY

CALL STACK

STATIC VARS

BYTECODES

OS / JVM

Method n Activation on Record . . . . Method 3 Activation on Record Method 2 Activation on Record Method 1 Activation on Record

LOCAL VARS

PARAMETERS

RETURN ADDRESS (PC Values)PREVIOUS BASE POINTER

RETURN VALUE

BASE POINTER

PROGRAM COUNTER

REAL TIME EXAMPLE

THANK YOU FOR ALL!