Lecture Java methods: Object-oriented programming and data structures (2nd AP edition): Chapter 23 - Maria Litvin, Gary Litvin

Chapter 23 - Recursion revisited. After you have mastered the material in this chapter, you will be able to: Take a fresh look at recursion, learn when to use recursion and when to stay away from it, learn to prove the correctness of recursive methods, get ready for the Game of Hex lab.

and Data Structures

Maria Litvin ● Gary Litvin

2nd AP edition with GridWorld

Objectives:

• Take a fresh look at recursion

• Learn when to use recursion and when to stay away from it

• Learn to prove the correctness of recursive methods

• Get ready for the Game of Hex lab

Recursion Basics

• A recursive method has a base case (or

several base cases) and a recursive case

for a “smaller” task

• Recursive calls must eventually converge to a base case, when recursion stops

A

Append the first char

DCBA

Take substring

Base case(nothing to do)

Recursive case

Caution: NOTdouble y = pow(x, n / 2) *pow(x, n / 2);

Example 4

public boolean degreeOfSeparation (

Set<Person> people, Person p1, Person p2, int n)

p1

p2p

When to Use Recursion

• Recursion is especially useful for handling nested structures and branching processes

ABCD DCA BCDA B

CAD B ACD B DAC B ADC B

ABCD BDA CDBA C

DAB C ADB C BAD C ABD C

ABCD

BCA D CBA D CAB D ACB D BAC D ABC D

Use Recursion

• When it significantly simplifies code

without significant perfomance penalty

Do Not Use Recursion

• When a method allocates large local arrays

• When a method unpredictably changes fields

• When iterative solutions is just as simple

Just As Easy With Iterations

public String reverse (String s)

Not Easy Without Recursion

public void traverse (TreeNode root)

Need your own stack to do this without recursion

Very Inefficient Recursive Code

public long fibonacci (int n)

{ next = f1 + f2;

f1 = f2;

f2 = next;

n ;

} return f2;

Recursion and Math Induction

• Recursive methods are hard to trace in a

conventional way.

• A recursive method can be proven correct using math induction.

• Other properties of a recursive method

(running time, required space, etc.) can be obtained by using math induction.

Math Induction Basics

• You have a sequence of statements

P1, P2, P3, Pn-1 , Pn ,

• Suppose P1 is true (“base case”).

• Suppose you can prove that for any n > 1, if

P1, Pn-1 are all true then Pn must be true, too.

• Then you can conclude (“by math induction”)

that Pn is true for any n 1

It is often possible to prove that if P n-1

is true then P n is also true

Math Induction Example:

Prove that for any n 1

Math Induction and Recursion

public String reverse (String s)

Let us verify that this method works, that is, reverse(s)

indeed returns the reverse of s We will use math

induction “over the length of s.”

To be continued 

2 Suppose (induction hypothesis)

reverse works for any string of length

n-1 Then it works for s.substring(1).

So we reverse that substring and then

append the first char of s at the end.

We get the reverse of s.

By math induction, reverse works for

a string of any length, q.e.d

public String reverse (String s) {

if (s.length < 2) return s;

return reverse (s.substring(1)) + s.charAt(0);

}

The Tower of Hanoi

• Objective: move the tower from one peg to another, moving one disk at a time and

never placing a larger disk on top of a

smaller one; you can use the spare peg.

• This puzzle was invented by François

Edouard Anatole Lucas and published in 1883

The Tower of Hanoi: Recursive Solution

For n disks:

disk to the desired peg.

• If n > 1

to a spare peg (recursive

step)

the desired peg

the spare to the desired

peg (recursive step)

The Game of Hex

• Two players take turns placing a stone of their color

• Objective: connect your pair of the opposite sides of the board with stones of your color

The Game of Hex (cont’d)

• Computer representation of the board

in a 2-D array

The Game of Hex (cont’d)

• To detect a win, determine whether any

“blob” (connected group of stones)

touches the opposite sides

The Game of Hex (cont’d)

• This kind of task falls into the category of

“area fill” tasks

• What kinds of applications especially

benefit from recursion?

• Give an example of a task that can be

programmed recursively and also, as

easily, with iterations.

Review (cont’d):

• What is the number of moves necessary

to solve the Tower of Hanoi puzzle with

1, 2, 3, , n disks? How would you prove

your hypothesis?

• Can you come up with an idea for a

recursive algorithm for “area fill”?