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Recursion

- Chapter 8

Chapter Contents

- What Is Recursion?
- Tracing a Recursive Method
- Recursive Methods That Return a Value
- Recursively Processing an Array
- Recursively Processing a Linked Chain
- The Time Efficiency of Recursive Methods
- Time Efficiency of countDown
- A Simple Solution to a Difficult Problem
- A Poor Solution to a Simple Problem

What Is Recursion?

- It is a problem-solving process involves

repetition - Breaks a problem into identical but smaller

problems - Eventually you reach a smallest problem
- Answer is obvious or trivial
- Using the solution to smallest problem enables

you to solve the previous smaller problems - Eventually the original big problem is solved
- An alternative to iteration
- An iterative solution involves loops

What Is Recursion?

New Year Eve counting down from 10.

What Is Recursion?

- A method that calls itself is a recursive method
- Base case a known case in a recursive

definition. The smallest problem - Eventually, one of the smaller problems must be

the base case

/ Task Counts down from a given positive

integer. _at_param integer an integer gt 0

/ public static void countDown(int

integer) System.out.println(integer) if

(integer gt 1) countDown(integer - 1) // end

countDown

When Designing Recursive Solution

- Four questions to ask before construct recursive

solutions. If you follow these guidelines, you

can be assured that it will work. - How can you define the problem in terms of a

smaller problem of the same type? - How does each recursive call diminish the size of

the problem? - What instance of the problem can serve as the

base case? - As the problem size diminishes, will you reach

this base case?

When Designing Recursive Solution

- For the method countDown, we have the following

answers. - countDown displays the given integer as the part

of the solution that it contribute directly. Then

call countDown with smaller size. - The smaller problem is counting down from integer

-1. - The if statement asks if the process has reached

the base case. Here, the base case occurs when

integer is 1.

Recursive Solution Guidelines

- Method definition must provide parameter
- Leads to different cases
- Typically includes an if or a switch statement
- One or more of these cases should provide a non

recursive solution( infinite recursion if dont) - The base or stopping case
- One or more cases includes recursive invocation
- Takes a step towards the base case

Tracing a Recursive Method

- Given

public static void countDown(int

integer) System.out.println(integer) if

(integer gt 1) countDown(integer - 1) // end

countDown

The effect of method call countDown(3)

Tracing a Recursive Method

Tracing the recursive call countDown(3)

Compare Iterative and Recursive Programs

- //Recursive version
- public static void countDown( int integer)
- if ( integer gt 1)
- System.out.println(integer)
- countDown(integer -1)
- //Iterative version
- public static void countDown( int integer)
- while ( integer gt 1)
- System.out.println(integer)
- integer--

Question?

- Could you write an recursive method that skips n

lines of output, where n is a positive integer.

Use System.out.println() to skip one line. - Describe a recursive algorithm that draws a given

number of concentric circles. The innermost or

outmost circle should have a given diameter. The

diameter of each of the other circles should be

three-fourths the diameter of the circle just

outside it.

Answer to skipLines

- pubic static void skipLines ( int givenNumber)
- if (givenNumber gt 1)
- System.out.println()
- skipLines(givenNumber 1)
- drawConcentricCircle( givenNumber, givenDiameter)
- // what should be put inside??.

Tracing a Recursive Method

- Each call to a method generate an activation

record that captures the state of the methods

execution and that is placed into a ADT stack. - The activation-record stack remembers the history

of incompleted method calls. A snapshot of a

methods state. - The topmost activation record holds the data

values for the currently executing method. - When topmost method finishes, its activation

record is popped - In this way, Java can suspend the execution of a

recursive method and re-invoke it when it appears

on the top.

Tracing a Recursive Method

The stack of activation records during the

execution of a call to countDown(3) continued ?

Tracing a Recursive Method

Note the recursive method will use more memory

than an iterative method due to the stack of

activation records

ctd. The stack of activation records during the

execution of a call to countDown(3)

Tracing a Recursive Method

- Too many recursive calls can cause the error

message stack overflow. Stack of activation

records has become full. Method has used too much

memory. - Infinite recursion or large-size problems are the

likely cause of this error.

Recursive Methods That Return a Value

- Task Compute the sum1 2 3 n for an

integer n gt 0

public static int sumOf(int n) int sum if (n

1) sum 1 // base case else sum

sumOf(n - 1) n // recursive call return

sum // end sumOf

Recursive Methods That Return a Value

The stack of activation records during the

execution of a call to sumOf(3)

Recursively Processing an Array

- When processing array recursively, divide it into

two pieces - Last element one piece, rest of array another
- First element one piece, rest of array another
- Divide array into two halves
- A recursive method part of an implementation of

an ADT is often private - Its necessary parameters make it unsuitable as an

ADT operation

Recursively Processing a Linked Chain

- To write a method that processes a chain of

linked nodes recursively - Use a reference to the chain's first node as the

method's parameter - Then process the first node
- Followed by the rest of the chain

public void display() displayChain(firstNode)

System.out.println() // end display private

void displayChain(Node nodeOne) if (nodeOne !

null) System.out.print(nodeOne.data "

") displayChain(nodeOne.next) // end

displayChain

Recursively Divide the Array in Half

- public static void displayArray( int array, int

first, int last) - if (first last)
- System.out.print(arrayfirst)
- else
- int mid (first last) /2
- displayArray(array, first, mid)
- displayArray(array, mid1, last)

A Simple Solution to a Difficult Problem

The initial configuration of the Towers of Hanoi

for three disks

A Simple Solution to a Difficult Problem

- Rules for the Towers of Hanoi game
- Move one disk at a time. Each disk you move must

be a topmost disk. - No disk may rest on top of a disk smaller than

itself. - You can store disks on the second pole

temporarily, as long as you observe the previous

two rules.

A Simple Solution to a Difficult Problem

The sequence of moves for solving the Towers of

Hanoi problem with three disks. Continued ?

A Simple Solution to a Difficult Problem

(ctd) The sequence of moves for solving the

Towers of Hanoi problem with three disks

A Simple Solution to a Difficult Problem

The smaller problems in a recursive solution for

four disks

A Simple Solution to a Difficult Problem

- Algorithm for solution with 1 disk as the base

case

Algorithm solveTowers(numberOfDisks, startPole,

tempPole, endPole) if (numberOfDisks

1) Move disk from startPole to

endPole else solveTowers(numberOfDisks-1,

startPole, endPole, tempPole) Move disk from

startPole to endPole solveTowers(numberOfDisks-1

, tempPole, startPole, endPole)

Recursion Efficiency

- How many moves occur for n disks?
- m(1) 1 for ngt1, two recursive calls to solve

problems that have n-1 disks. - m(n) m(n-1) m(n-1) 1 2m(n-1) 1
- Lets evaluate the recurrence for m(n) for a few

values of n - m(1) 1 m(2) 3 m(3) 7m(4) 15 m(5) 31

m(6) 63. - m(n) 2n -1

Mathematical Induction

- Prove this conjecture m(n) 2n -1 by using

mathematical induction - We know that m(1) 1, which equals to 21-11, so

the conjecture is true for n 1. - Now assume that it is true for n1,2,,k, and

consider m(k1). - m(k1) 2m(k) 1 (use the recurrence relation)
- 2(2k-1) 1 2(k1) -1 ( we assume that m(k)

2k-1) - Since the conjecture is true for nk1, it is

true for all ngt1

Mathematical Induction

- Assume you want to prove some statement P, P(n)

is true for all n starting with n 1. The

Principle of Math Induction states that, to this

end, one should accomplish just two steps - 1). Prove that P(1) is true.
- 2). Assume that P(k) is true for some k. Derive

from here that P(k1) is also true. - look P(1) is true and implies P(2). Therefore

P(2) is true. But P(2) implies P(3). Therefore

P(3) is true which implies P(4) and so on.

Multiplying Rabbits (The Fibonacci Sequence)

- Facts about rabbits
- Rabbits never die
- A rabbit reaches sexual maturity exactly two

months after birth, that is, at the beginning of

its third month of life - Rabbits are always born in male-female pairs
- At the beginning of every month, each sexually

mature male-female pair gives birth to exactly

one male-female pair

Multiplying Rabbits (The Fibonacci Sequence)

- Problem
- How many pairs of rabbits are alive in month n?
- Month Month No calculation rabbit couples
- January 1 1 0 1
- February 2 1 0 1
- March 3 1 1 2
- April 4 2 1 3
- May 5 3 2 5
- June 6 5 3 8
- July 7 8 5 13
- August 8 13 8 21
- Recurrence relation
- rabbit(n) rabbit(n-1) rabbit(n-2)

A Poor Solution to a Simple Problem

- Fibonacci numbers
- First two numbers of sequence are 1 and 1
- Successive numbers are the sum of the previous

two - 1, 1, 2, 3, 5, 8, 13,
- This has a natural looking recursive solution
- Turns out to be a poor (inefficient) solution

A Poor Solution to a Simple Problem

- The recursive algorithm

Algorithm Fibonacci(n) if (n lt 1) return

1 else return Fibonacci(n-1) Fibonacci(n-2)

A Poor Solution to a Simple Problem

Time efficiency grows exponentially with n, which

is kn

int fib(int n) int fn1 f1 f2

1 for (int i 3 i lt n i) fi

fi-1 fi-2 return fn

Iterative solution is O(n)

The computation of the Fibonacci number F6 (a)

recursively (b) iteratively