1 gentle introduction to programming session 3: higher order functions, recursion

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Gentle Introduction to Programming

Session 3: Higher Order Functions, Recursion

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Admin• Sorry about the technical problems in the practical session (11-13)

• Your feedback about it?• Scala / Eclipse installation status?

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How to Find Mistakes in Your Code• Compiler Errors• Runtime errors• What are you expected to do before contacting me

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Today• Review• Higher order functions• Debugger• Home work review• Home work• Recursion

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Review• More Scala:

• var / val• Strings & API• Defining variables (type inference)• Variable scope• Importance of style

• Functions• Why do we need them• Types: methods, local (nested), literal, valu

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Review (Cont.)

• Odd’s talk (bioinformatics)• Practical session:

• Account• Linux• Interpreter / Eclipse• Understanding the compiler• Command line arguments• Homework – practice shell commands

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Why Functions?• Why do we need functions?

• Code reusability• Modularity• Abstraction

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Function Definition in Scala

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Example

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There are many ways to define functions in Scala

FuncDef.scala

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Example

((x : Int) => x * 2)(a)

((x : Int) => x * 2)(5)

((x : Int) => x * 2)5 5

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Closures

(x : Int) => x + more // how much more?

free variable

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Closures (Cont.)

What happens if we change “more”?

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Can procedures get and return procedures?

In Scala a function can:

• Return a function as its return value,

• Receive functions as arguments.Why is this useful?

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Consider the Following Three Sums•1 + 2 + … + 100 = (100 * 101)/2

•1 + 4 + 9 + … + 1002 = (100 * 101 * 102)/6

•1 + 1/32 + 1/52 + … + 1/1012 ~ 2/8

In mathematics they are all captured by the notion of a sum

100

1

2

k

k

100

1k

k

101

,1

2

oddk

k

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In Scala

101

,1

2

oddk

k

100

1k

k

100

1

2

k

k

Generalization:

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The “Sum” Function

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Usage (HigherOrderFunc.scala)

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,1

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oddk

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1k

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100

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2

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k

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How Does it Work?

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1k

k

def f1(start : Int, end : Int) : Double = {

sum((x:Int)=>x,start,(x:Int)=>x+1,end)

}

f1(1,100)

f1(50,60)

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How Does it Work?

100

1k

k

sum += ((x:Int) => x)(i)

i = ((x:Int) => x+1)(i)

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Integration as a Function

Integration under a curve f is approximated by

dx (f(a) + f(a + dx) + f(a + 2dx) + … + f(b))

a bdx

f

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Integration as a Function (Cont.)

a bdx

f

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arctan(a) = ∫(1/(1+y2))dy0

a

Integration.scala

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Function as a Contract

• If the operands have the specified types,the procedure will result in a value of the specified type• otherwise, its behavior is undefined (maybe an error,

maybe random behavior)

A contract between the caller and the procedure.• Caller responsible for argument number and types

• function responsible to deliver correct result

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Example

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The Functions “Stack”

g()->h()

f()->g()

main-> f()

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The Debugger

• Some programs may compile correctly, yet not produce the desirable results

• These programs are valid and correct Scala programs, yet not the programs we meant to write!

• The debugger can be used to follow the program step by step and may help detecting bugs in an already compiled program

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Debugger – Add Breakpoint

• Right click on the desired line

• “Toggle Breakpoint”

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Debugger – Start Debugging

breakpoint

debug

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Debugger – Debug Perspective

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Debugger – Debugging

Current state

Current location

Back to Scala perspective

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Exercise – Integer Division

• Input:• Two integers – A and B

• Output:• How many times A contains B (it is the result of

the integer division A/B)

• Do not use the operators ‘/’, ‘*’

• Solution:

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Integer Division Solution

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Exercise – Power of Two

• Input: integer A

• Output: is there an integer N such that A == 2^N?

• Solution:

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Power of Two Solution

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Write a program that prints an upside-down half triangle of *.

The height of the pyramid is the input.

*****

*****

****

*

Exercise – Triangle Printout

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Triangle Printout Solution

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Approximating Square Root

• Given integer x > 0, how to find an approximation of x?

• Newton was the first to notice that for any positive n, and when x0=1, the following series converges to sqrt(n):

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Algorithm

x = 2 g = 1x/g = 2 g = ½ (1+ 2) = 1.5

x/g = 4/3 g = ½ (3/2 + 4/3) = 17/12 = 1.416666

x/g = 24/17 g = ½ (17/12 + 24/17) = 577/408 = 1.4142156

• Algorithm:• Make a guess g• Improve the guess by averaging g, x/g• Keep improving the guess until it is good enough

• Example: calculate the square root of 2

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Exercise 1

• Write a program that receives from the user:• An integer x > 0• A double representing the approximation’s accuracy

• Calculates an approximation of x within the required accuracy

Use the function Math.abs(x) which returns the absolute value of x

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Exercise 2

Write a function that uses the formula :2/6=1/1+1/4+1/9+1/16+…+1/n2 (where n goes to infinity)

in order to approximate . The function should accept an argument n which determines the number of terms in the formula. It should return the approximation of

Write a program that gets an integer n from the user, and approximate using n terms of the above formula• Use a nested function sum, with signature:

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Exercise 3

• Write a function deriv: • Input: a function f (Double=>Double), accuracy dx• Output: function df (Double=>Double), the derivative

of f. • df (x) = (f(x+dx) – f(x))/dx• Define a few simple functions (linear, square) and

calculate their derivative at various points received by command line arguments

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For next Year – toy example of a function that returns a function (to

help with the solution of ex#3)

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The Functions “Stack”

g()->h()

f()->g()

main-> f()

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Recursive Functions

• How to create a process of unbounded length?

• Needed to solve more complicated problems

• Tower of Hanoi

• Start with a simple example

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Tower of Hanoi

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Tower of Hanoi

• How to solve for arbitrary number of disks n?• Let’s name the pegs: source, target, spare• All n disks are initially placed on source peg• Solution:

1. Move n-1 upper disks from source to spare

2. More lower (bigger) disk from source to target

3. Move n-1 disks from spare to target (smaller. Easier?)• Base condition: a single disk

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Factorial

• As we saw, n! = 1*2*3*… *(n-1)*n

• Thus, we can also define factorial the following way: • 0! = 1 • n! = n*(n-1)! for n>0

(n-1)! * n

RecursiveDefinition

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A Recursive Definition

• Functions can also call themselves!• However, usually not with the same parameters

• Some functions can be defined using smaller occurrences of themselves

• Every recursive function has a “termination condition”. The function stops calling itself when it is satisfied

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Example - Factorial

Termination Condition

Recursive Calll

Factorial.scala

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Example - factorial

factRec(1)

factRec(2)->2*factRec (1)

factRec(3)->3*factRec (2)

main-> factRec (3)

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Recursive factorial – step by step

FactRec(4)n

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Returns…

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FactRec(4)n

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Returns…4*…

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FactRec(4)n

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Returns…

FactRec(3)n

3

Returns…

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FactRec(4)n

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Returns…

FactRec(3)n

3

Returns…

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FactRec(4)n

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Returns…

FactRec(3)n

3

Returns…3*…

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FactRec(4)n

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Returns…

FactRec(3)n

3

Returns…

FactRec(2)n

2

Returns…

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FactRec(4)n

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Returns…

FactRec(3)n

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Returns…

FactRec(2)n

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Returns…

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FactRec(4)n

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Returns…

FactRec(3)n

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Returns…

FactRec(2)n

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Returns…2*…

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FactRec(4)n

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Returns…

FactRec(3)n

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Returns…

FactRec(2)n

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Returns…

FactRec(1)n

1

Returns…

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FactRec(4)n

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Returns…

FactRec(3)n

3

Returns…

FactRec(2)n

2

Returns…

FactRec(1)n

1

Returns…

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FactRec(4)n

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Returns…

FactRec(3)n

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Returns…

FactRec(2)n

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Returns…

FactRec(1)n

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Returns… 1

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FactRec(4)n

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Returns…

FactRec(3)n

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Returns…

FactRec(2)n

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Returns…2*1

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FactRec(4)n

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Returns…

FactRec(3)n

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Returns…3*2

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FactRec(4)n

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Returns…4*6

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General Form of Recursive Algorithms

• Base case: small (non-decomposable) problem• Recursive case: larger (decomposable) problem• At least one base case, and at least one recursive

case.

test + base case recursive case

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Short Summary

• Design a recursive algorithm by1. Solving big instances using the solution to smaller

instances

2. Solving directly the base cases

• Recursive algorithms have1. test

2. recursive case(s)

3. base case(s)

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More Examples

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Example - Modulo

• Given the following iterative version of modulo calculation

Find the recursive definition

Modulo.scala

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Solution

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Fibonacci

• “Naturally” recursive• Because the nth term is the sum of the (n-1)th and

(n-2)th terms• Therefore, the recursive definition is:

• fib(1) = 0 ; fib(2) = 1 /* base */• fib(n) = fib(n-1) + fib(n-2)

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Or, in Scala…Fibonacci.scala

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Recursion and Efficiency

• The recursive form, however elegant, may be much less efficient

• The number of redundant calls grow exponentially!

Fib(5)

Fib(4) Fib(3)

Fib(3) Fib(2) Fib(2) Fib(1)

Fib(2)

Fib(1)

Fib(6)

Fib(4)

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Efficient Recursive Fibonacci

• Pass the values that were already calculated to the recursive function

• This technique is called Memoization

• How would we keep the values that were already calculated?

Fib(5)

Fib(4) Fib(3)

Fib(3) Fib(2) Fib(2) Fib(1)

Fib(2)

Fib(1)

Fib(6)

Fib(4)

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Odd-Even

• Given a function ‘odd(n : Int) : Boolean’ • Return true on n that are odd, false on even n

• Write a function ‘even(n : Int) : Boolean’ that:• Return true on n that are even, false on odd n

• This is easy

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Odd-EvenEvenOdd.scala

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More Home Work (Recursion)

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Exercise 1

• Write a program that receives two non-negative integers and computes their product recursively

• Hint: Notice that the product a*b is actually a+a+…+a (b times). How does this help us define the problem recursively?

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Exercise 2

• Given the following iterative version of sum-of-digits calculation

Write the recursive definition

1 gentle introduction to programming session 3: higher order functions, recursion

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