Chapter 14

Recursion

Chapter Objectives

• Learn about recursive definitions

• Explore the base case and the general case of a recursive definition

• Learn about recursive algorithms

Chapter Objectives

• Learn about recursive methods

• Become aware of direct and indirect recursion

• Explore how to use recursive methods to implement recursive algorithms

Recursive Definitions

• Recursion– Process of solving a problem by reducing it to

smaller versions of itself

• Recursive definition– Definition in which a problem is expressed in

terms of a smaller version of itself– Has one or more base cases

Recursive Definitions• Recursive algorithm

– Algorithm that finds the solution to a given problem by reducing the problem to smaller versions of itself

– Has one or more base cases– Implemented using recursive methods

• Recursive method– Method that calls itself

• Base case– Case in recursive definition in which the solution is

obtained directly – Stops the recursion

Recursive Definitions

• General solution– Breaks problem into smaller versions of itself

• General case– Case in recursive definition in which a smaller

version of itself is called– Must eventually be reduced to a base case

Tracing a Recursive Method

Recursive method– Has unlimited copies of itself– Every recursive call has its own

• code• set of parameters• set of local variables

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

Recursive Definitions

• Directly recursive: a method that calls itself• Indirectly recursive: a method that calls another

method and eventually results in the original method call

• Tail recursive method: recursive method in which the last statement executed is the recursive call

• Infinite recursion: the case where every recursive call results in another recursive call

Designing Recursive Methods

• Understand problem requirements

• Determine limiting conditions

• Identify base cases

Designing Recursive Methods

• Provide direct solution to each base case

• Identify general case(s)

• Provide solutions to general cases in terms of smaller versions of general cases

Recursive Factorial Method

public static int fact(int num){ if(num = = 0) return 1; else return num * fact(num – 1);}

Recursive Factorial Trace

Recursive Implementation: Largest Value in Array

Execution of largest(list, 0, 3)

Recursive Fibonacci

public static int rFibNum(int a, int b, int n){ if(n = = 1) return a; else if(n = = 2) return b; else return rFibNum(a, b, n -1) + rFibNum(a, b, n - 2);}

Execution of rFibonacci(2,3,5)

Towers of Hanoi Problem with Three Disks

Towers of Hanoi: Three Disk Solution

Towers of Hanoi: Three Disk Solution

Towers of Hanoi: Recursive Algorithm

Recursion or Iteration?

• Two ways to solve particular problem– Iteration– Recursion

• 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

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); }}

Execution of decToBin(13,2)

Sierpinski Gaskets of Various Orders

Programming Example:Sierpinski Gasket

• Input: non-negative integer indicating level of Sierpinski gasket

• Output: triangle shape displaying a Sierpinski gasket of the given order

• Solution includes– Recursive method drawSierpinski– Method to find midpoint of two points

Recursive Algorithm to Draw Sierpinski Gasket

Programming Example: Sierpinski Gasket Input

Programming Example: Sierpinski Gasket Input

Chapter Summary

• Recursive Definitions

• Recursive Algorithms

• Recursive Methods

• Base cases

• General cases

Chapter Summary

• Tracing recursive methods

• Designing recursive methods

• Varieties of recursive methods

• Recursion vs. Iteration

• Various recursive functions explored