Chap14

Java Programming: From Problem Analysis to Program Design, 3e 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

Java Programming: From Problem Analysis to Program Design, 3e

Chapter Objectives (continued)

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 (continued) Java Programming: From Problem Analysis to Program Design, 3e

Recursive Definitions (continued)

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 (continued)

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

Logically, you can think of a recursive method having unlimited copies of itself

Every recursive call has its own

Code

Set of parameters

Set of local variables

Tracing a Recursive Method (continued)

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 (continued)

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 Java Programming: From Problem Analysis to Program Design, 3e public static int fact( int num) { if (num = = 0) return 1; else return num * fact(num – 1); }

Recursive Factorial Method (continued) Java Programming: From Problem Analysis to Program Design, 3e

Largest Value in Array Java Programming: From Problem Analysis to Program Design, 3e

Largest Value in Array (continued) Java Programming: From Problem Analysis to Program Design, 3e

if the size of the list is 1

- the largest element in the list is the only element in the list

else

- to find the largest element in list[a]...list[b]

a. find the largest element in list[a + 1]...list[b]

and call it max

b. compare list[a] and max

if ( list[a] >= max )

the largest element in list[a]...list[b] is

list[a]

else

the largest element in list[a]...list[b] is max

Largest Value in Array (continued) Java Programming: From Problem Analysis to Program Design, 3e public static int largest( int [] list, int lowerIndex, int upperIndex) { int max; if (lowerIndex == upperIndex) return list[lowerIndex]; else { max = largest(list, lowerIndex + 1, upperIndex); if (list[lowerIndex] >= max) return list[lowerIndex]; else return max; } }

Execution of largest (list, 0, 3) Java Programming: From Problem Analysis to Program Design, 3e

Recursive Fibonacci Java Programming: From Problem Analysis to Program Design, 3e

Recursive Fibonacci (continued) Java Programming: From Problem Analysis to Program Design, 3e 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); }

Recursive Fibonacci (continued) Java Programming: From Problem Analysis to Program Design, 3e

Towers of Hanoi Problem with Three Disks Java Programming: From Problem Analysis to Program Design, 3e

Towers of Hanoi: Three Disk Solution Java Programming: From Problem Analysis to Program Design, 3e

Towers of Hanoi: Three Disk Solution (continued) Java Programming: From Problem Analysis to Program Design, 3e

Towers of Hanoi: Recursive Algorithm Java Programming: From Problem Analysis to Program Design, 3e public static void moveDisks( int count, int needle1, int needle3, int needle2) { if (count > 0) { moveDisks(count - 1, needle1, needle2, needle3); System.out.println( " Move disk " + count + " from needle " + needle1 + " to needle " + needle3 + " . " ); moveDisks(count - 1, needle2, needle3, needle1); } }

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 Java Programming: From Problem Analysis to Program Design, 3e 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) Java Programming: From Problem Analysis to Program Design, 3e

Sierpinski Gaskets of Various Orders Java Programming: From Problem Analysis to Program Design, 3e

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

Programming Example: Sierpinski Gasket (continued) Java Programming: From Problem Analysis to Program Design, 3e 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); } }

Programming Example: Sierpinski Gasket Input Java Programming: From Problem Analysis to Program Design, 3e

Chapter Summary