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RecursionPurpose: There are situations

where iterative algorithms become too complicated and a more elegant solution would be helpful. You can design an algorithm where it performs a simple task and that task it repeatedly called.

This process is called recursion.

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Resources:

Java Essentials Chapter 17 p.667

Java Essentials Study Guide Chapter 14 p.221

Barrons AP Java Chapter 17 p.219

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Handouts: Fibonacci.java Factorial.java combo.java M/C Questions

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Recursive Functions:

Functions whose code includes calls to itself

Possible because successive function calls create instances of the “stateful” properties of the function

A frame on the stackThe function code is shared

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Recursive Functions:

The system implements a function call the same way : whether the function calls itself or another function.

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3 Rules:

find out how to take just 1 step

break each journey into 1 step plus a smaller journey (work towards the base case)

know when to stop --- BASE CASE

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Base Case:

The “regular line”

The lowest level where no recursive calls are made

The place where no recursive calls are made

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Recursive Case:

A repeat command

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Recursion & Iteration:

Iteration explicitly uses a repetition structure

(loops)

Recursion achieves repetition through repeated function calls

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Recursion & Iteration...

Both involve a termination test:Iteration ends when loop fails

Recursion ends when a base case is recognized ( a simpler version of the original problem is produced until the base case is reached)

Both can occur indefinitely if not designed correctly

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When NOT to use Recursion: functions declaring large local arrays

functions that manipulate static vars, global vars or arrays

if performance is vital when dealing with linear structures and processes (simple iterations are best)

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Recursion Drawbacks:

It repeatedly invokes function calls onto the stack

Expensive in CPU time and memory

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When Recursion is Best Used:

Used best when it significantly simplifies the code without excessive performance loss.

Useful for dealing with nested structures or branching processes

Used in traversing tree structures

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Sample Problem:A recursive algorithm for painting a

square.

Given a squareIf the length is less than 2 ft, stop Divide the square into 4 equal size squaresPaint 1 of these small squaresRepeat from the top for each of the 3 unpainted squaresSquare of 16 feet (256 square ft)

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Sample Problem…

How Many Squares Created & Painted...

In the 1st pass ?In the 2nd pass ?In the 3rd pass ?In the 4th pass ?What is the TOTAL SQUARES Painted?Take some time now to try this...

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Sample Problem:

1st pass, four 8” squares 1p 3 up2nd , 12 4” squares 3p 9 up3rd, 36 2” squares, 9p 27up4th, 108 1” squares, 27p, 81up

total painted = 1+ 3 + 9 + 27 = 40

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SIGMA Example

Use the following function shell to code for a recursive algorithm that SUMS INTEGERS FROM 1 to N

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SIGMA Example

Int Sigma (int n){

}

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Int Sigma (int n){if (n <= 1)

return n; else

return n + Sigma(n-1);}

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TPS:Convert and Print a Decimal

number (100) to a binary number (base 2)

100 / 2 = 50 remainder 050 / 2 = 25 remainder 025 / 2 = 12 remainder 112 / 2 = 6 remainder 06 / 2 = 3 remainder 03 / 2= 1 remainder 11 / 2 = 0 remainder 1

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Reading remainders in reverse order gives result: 1100100 the binary for 100

Write a Recursive solution:Identify the first stepBreak down step into a smaller versionKnow when to stop --- the base case (quotient <= 0)The last remainder must be the first printed

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Here is the Shell Code:public static void convertToBinary(int dec){

}

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ANS: Java Essentials Study Guide Chapter 14 p.222public static void convertToBinary(int dec){

int quotient = decimalNum / 2;int remainder = decimalNum % 2;

if (quotient > 0){

convertToBinary(quotient); // smaller version

}System.out.println(remainder);

// base case}

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FIBONACCI Example (handout)

COMBO Example (handout)

REVERSE A SENTENCE

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EFFICIENCY“a measure of the runtime usage of computational processes”

Select an instruction in the algorithm that executes more/less based on the size of the data (this process dominates the work)

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EFFICIENCY...

Linear Behavior -- Number of instructions executed INCREASES PROPORTIONAL to the size of the data

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EFFICIENCY...

Quadratic Behavior -- Number of instructions executed INCREASES PROPORTIONAL to the size of the data SQUARED

Least Efficient of these

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EFFICIENCY...

Logarithmic Behavior -- LOG 2 N

MOST Efficient

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EFFICIENCY… Illustration

Data Log2 Linear N log N Quadratic

1 1 1 1 110 4 10 40 100100 7 100 700

10,0001,000 10 1,000 10,000

1,000,00010,000 14 10,000 140,000 100M

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Efficiency…

Exponential Growth O(a^n)

Recursive functions, like FIBONACCI, have an exponential order of growth

Lets Run Fibonacci Recursively…..\recursion programs\fibonacci.exe

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Linked Lists as Recursive Linked Lists as Recursive Data StructuresData Structures

Given A Simple Structure, Given A Simple Structure, how Can we design a how Can we design a recursive function that recursive function that traverses a linked list ?traverses a linked list ?

(This will be discussed during our lecture on Linked Lists)

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TPS (OPTIONAL): Permutations of a String (Java Essentials ch 17 p.672 - 676)

Design a class that lists all the permutations of a string

For example, the string “eat” has 6 permutations:

eatetaaetateteatae

Use the text for approach and code solution

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TPS (OPTIONAL): GasPump (Java essentials Study Guide

Vhapter 14 p.223-224)

Write a recursive class that simulates the spinning of the digits on a gas pump

Use the text for approach and code solution

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Tips for the AP Exam:

The AP Exam includes M/C questions that give a recursive algorithm and then ask about that algorithm

You will need to be able to TRACE through a recursive process to obtain the answer

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Tips for the AP Exam:

Never use a while statement when an if statement should be used to check for a base case in a recursive algorithm

Avoid infinite recursion by coding for a base case that WILL be reached

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Tips for the AP Exam:

Unless the solution requires you to write a recursive solution OR the problem is stated in a recursive nature, code solutions ITERATIVLY

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Tips for the AP Exam:

When there is ONE recursive call from “return”, use the STACK METHOD to help you resolve the code

Example, Factorial

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Tips for the AP Exam:

When there are TWO recursive calls from “return”, use the BINARY TREE METHOD to help you resolve the code

Example, Combo & Fibonacci

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LAB:

Factorial

Making Change

Multiple Choice Questions