Chapter 14: Recursion

J ava P rogramming: From Problem Analysis to Program Design, Second Edition

Chapter Objectives

      Learn about recursive definitions.

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

Learn about recursive algorithms.

Learn about recursive methods.

Become aware of direct and indirect recursion.

Explore how to use recursive methods to implement recursive algorithms.

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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.

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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.

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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.

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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 Java Programming: From Problem Analysis to Program Design, Second Edition 6

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.

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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.

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Designing Recursive Methods

 Understand problem requirements.

 Determine limiting conditions.

 Identify base cases.

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Designing Recursive Methods

 Provide direct solution to each base case.

 Identify general cases.  Provide solutions to general cases in terms of smaller versions of general cases. Java Programming: From Problem Analysis to Program Design, Second Edition 10

Recursive Factorial Method

public static int { if (num = = 0) fact( int 1; else return return num) num * fact(num – 1); } Java Programming: From Problem Analysis to Program Design, Second Edition 11

Recursive Factorial Method Java Programming: From Problem Analysis to Program Design, Second Edition 12

} { Largest Value in Array public static int largest( int [] list, int int lowerIndex, upperIndex) int if max; (lowerIndex == upperIndex) return else list[lowerIndex]; { max = largest(list, lowerIndex + 1, upperIndex); if (list[lowerIndex] >= max) return list[lowerIndex]; else return max; } Java Programming: From Problem Analysis to Program Design, Second Edition 13

Largest Value in Array Java Programming: From Problem Analysis to Program Design, Second Edition 14

Recursive Fibonacci

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Recursive Fibonacci

public static int rFibNum( int int { a, n) int if(n = = 1) return a; else if (n = = 2) return else return b; rFibNum(a, b, n -1) + rFibNum(a, b, n - 2); b, } Java Programming: From Problem Analysis to Program Design, Second Edition 16

Recursive Fibonacci Java Programming: From Problem Analysis to Program Design, Second Edition 17

Towers of Hanoi: Three Disk Problem Java Programming: From Problem Analysis to Program Design, Second Edition 18

Towers of Hanoi: Three Disk Solution Java Programming: From Problem Analysis to Program Design, Second Edition 19

Towers of Hanoi: Three Disk Solution Java Programming: From Problem Analysis to Program Design, Second Edition 20

Towers of Hanoi: Recursive Algorithm public static void moveDisks( int int { if { (count > 0) count, int needle1, needle3, int needle2) moveDisks(count - 1, needle1, needle2, needle3); System.out.println( " Move disk " + count + " from needle + needle1 + " " to needle " + needle3 + " . " ); moveDisks(count - 1, needle2, needle3, needle1); } } Java Programming: From Problem Analysis to Program Design, Second Edition 21

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.

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Programming Example: Decimal to Binary

public static void decToBin( int int { if { (num > 0) num, base) decToBin(num / base, base); System.out.print(num % base); } } Java Programming: From Problem Analysis to Program Design, Second Edition 23

Programming Example: Decimal to Binary Java Programming: From Problem Analysis to Program Design, Second Edition 24

Programming Example: Sierpinski Gasket Java Programming: From Problem Analysis to Program Design, Second Edition 25

Programming Example: Sierpinski Gasket

 Input: Non-negative integer that indicates level of Sierpinski gasket.

 Output: Triangle shape that displays a Sierpinski gasket of the given order.

 Solution includes:  Recursive method drawSierpinski.

 Method to find midpoint of two points.

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Programming Example: Sierpinski Gasket private void drawSierpinski(Graphics g, int lev, Point p1, Point p2, Point p3) { Point midP1P2; Point midP2P3; Point midP3P1; if (lev > 0) { g.drawLine(p1.x, p1.y, p2.x, p2.y); g.drawLine(p2.x, p2.y, p3.x, p3.y); g.drawLine(p3.x, p3.y, p1.x, p1.y); midP1P2 = midPoint(p1, p2); midP2P3 = midPoint(p2, p3); midP3P1 = midPoint(p3, p1); drawSierpinski(g, lev - 1, p1, midP1P2, midP3P1); drawSierpinski(g, lev - 1, p2, midP2P3, midP1P2); drawSierpinski(g, lev - 1, p3, midP3P1, midP2P3); } } Java Programming: From Problem Analysis to Program Design, Second Edition 27

Programming Example: Sierpinski Gasket Java Programming: From Problem Analysis to Program Design, Second Edition 28

Chapter Summary

 Recursive definitions  Recursive algorithms  Recursive methods  Base cases  General cases Java Programming: From Problem Analysis to Program Design, Second Edition 29

Chapter Summary

 Tracing recursive methods  Designing recursive methods  Varieties of recursive methods  Recursion vs. iteration  Various recursive functions Java Programming: From Problem Analysis to Program Design, Second Edition 30