1 Gentle Introduction to Programming Session 4: Arrays, Sorting, Efficiency

2 Admin. Upcoming Tuesday – practical session in the lab Please fix all your account issues before it and make sure you can write Scala programs in Eclipse

3 Review Review on Functions Higher order functions Recursion Solving big instances using the solution to smaller instances Solving directly the base cases Tower of Hanoi Recursion and efficiency (Fibonacci) Importance of correct base-cases (Odd-Even)

4 Today Home work review Guest lecture by Prof. Benny Chor Recursive vs. Iterative Arrays Sorting, searching and time-complexity analysis Home work

5 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 x 0 =1, the following series converges to sqrt(n):

6 Algorithm x = 2g = 1 x/g = 2g = ½ (1+ 2) = 1.5 x/g = 4/3g = ½ (3/2 + 4/3) = 17/12 = x/g = 24/17g = ½ (17/ /17) = 577/408 = 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

7 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

8 Solution ApproxSQRT.scala g – x/g תיקון לשנה הבאה !!

9 Exercise 2 Write a function that uses the formula :  2 /6=1/1+1/4+1/9+1/16+…+1/n 2 (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:

10 Solution ApproxPI.scala

11 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

12 Usage Example // the function you want to derive def f(x : Double) : Double = x * x // resolution/accuracy/granularity of derivative val dx = /* this is the invokation of the function you are supposed to implement*/ val df = deriv(f,dx) // print the derivative of f(x) at x = 10 println(df(10))

13 Solution Derivative.scala toDouble תיקון לשנה הבאה !!

14 Recursion - 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?

15 Solution Product.scala

16 Recursion - Exercise 2 Given the following iterative version of sum- of-digits calculation Write the recursive definition

17 Solution SumDigits.scala

18 Today Home work review Guest lecture by Prof. Benny Chor Recursive vs. Iterative Arrays Sorting, searching and time-complexity analysis Home work

19 Compute a b Iteratively Operationally: Halting condition: product  product * a counter  counter - 1 counter = 0 a b = a 2 *a*…*a = a 3 *…*a Which is: a b = a * a * a*…*a b

20 Compute a b Iteratively

21 Compute a b (Recursive Approach) Recursive call: a b = a * a (b-1) Base case: a 0 = 1

22 Compute a b (Iterative Approach) How then, do the two procedures differ?

23 Recursive Process powRec(3,4) 3 * powRec(3,3) 3 * (3 * powRec(3,2)) 3 * (3 * (3 * powRec(3,1))) 3 * (3 * (3 * (3 * powRec(3,0)))) 3 * (3 * (3 * (3 * 1))) 3 * (3 * (3 * 3)) 3 * (3 * 9) a * 27 81

24 Iterative Process powTailRec(3,4) pow1(3,4,1) pow1(3,3,3) pow1(3,2,9) pow1(3,1,27) pow1(3,0,81) 81

25 powRec(3,4) 3 * powRec(3,3) 3 * (3 * powRec(3,2)) 3 * (3 * (3 * powRec(3,1))) 3 * (3 * (3 * (3 * powRec(3,0)))) 3 * (3 * (3 * (3 * 1))) 3 * (3 * (3 * 3)) 3 * (3 * 9) a * The Difference Growing amount of space Constant amount of space powTailRec(3,4) pow1(3,4,1) pow1(3,3,3) pow1(3,2,9) pow1(3,1,27) pow1(3,0,81) 81

26 Why More Space? Operation pending No pending operations

27 Summary Recursive process num of deferred operations “grows proportional to b” Iterative process num of deferred operations stays “constant” (actually it’s zero) Can we better quantify these observations?

28 Order of Growth: Recursive Process powRec(3,5) 3 * powRec(3,4) 3 * (3 * powRec(3,3)) 3 * (3 * (3 * powRec(3,2))) 3 * (3 * (3 * (3 * powRec(3,1)))) 3 * (3 * (3 * (3 * (3 * powRec(3,0))))) 3 * (3 * (3 * (3 * (3 * 1)))) 3 * (3 * (3 * (3 * 3))) 3 * (3 * (3 * 9)) 3 * (3 * 27) a * powRec(3,4) 3 * powRec(3,3) 3 * (3 * powRec(3,2)) 3 * (3 * (3 * powRec(3,1))) 3 * (3 * (3 * (3 * powRec(3,0)))) 3 * (3 * (3 * (3 * 1))) 3 * (3 * (3 * 3)) 3 * (3 * 9) a * Dependent on b

29 Order of Growth: Iterative Process powTailRec(3,5) pow1(3,5,1) pow1(3,4,3) pow1(3,3,9) pow1(3,2,27) pow1(3,1,81) pow1(3,0,243) 243 powTailRec(3,4) pow1(3,4,1) pow1(3,3,3) pow1(3,2,9) pow1(3,1,27) pow1(3,0,81) 81 Same constant, independent of b

30 “Tail” Recursion in Scala Scala compiler translate tail-recursion to iterative execution Thus, the functions-stack is not growing

31 Today Home work review Guest lecture by Prof. Benny Chor Recursive vs. Iterative Arrays Sorting, searching and time-complexity analysis Home work

32 Arrays Array: sequential block of memory that holds variables of the same type Array can be declared for any type Example: new Array[Int](3) Examples: list of students’ marks series of numbers entered by user vectors matrices

33 Arrays in Memory Sequence of variables of specified type The array variable itself holds the address in memory of beginning of sequence Example: val s = new Array[Double](10) The k-th element of array A is specified by A[k-1] (0 based) s ……

34 Example - Initialization

35 Arrays in Memory Access array’s content Change array’s content

36 Example Array.scala

37 foreach, filter Iterates over arrays (and other containers) foreach – for each element apply a given function filter – create a new container containing only elements that pass the given Boolean-function

38 Example – Print Arrays PrintArray.scala

39 Example – filter

40 Example – Find Minimum FindMin.scala

41 Arrays as Function Arguments Functions can accept arrays as arguments Example: Within the function, arr is accessed in the usual way Changes to the array in the function change the original array! (why?)

42 Example ArraySum.scala

43 Example MultAll.scala

44 Efficient Factorial Write a program that repeatedly receives a natural number as its input and returns its factorial The maximal number that will be given is known to be 1000 The program should be efficient: Do not calculate 1*2*…*n for every n from the beginning on every input Do not calculate values that were not requested

45 Solution: Main Idea We can keep in an array the results When given a new n If calculated before – return it Otherwise, get a better start

46 Solution EfficientFact.scala

47 Solution (main) EfficientFact.scala

48 Multi-Dimensional Arrays Array of arrays: val arr = Array[Array[Int]](3,2) Means an array of 3 integer arrays, each of length 2 Access: j th element of the i th array is a[i][j]

49 References to Arrays What is going on here?

50 In Memory 321 x1 x2 100

51 Today Home work review Guest lecture by Prof. Benny Chor Recursive vs. Iterative Arrays Sorting, searching and time-complexity analysis Home work

52 Sort We would like to sort the elements in an array in an ascending order sort

53 What is Sorting Good For? Finding a number in an array Consider a large array (of length n) and multiple queries on whether a given number exists in the array (and what is its position in it) Naive solution: given a number, traverse the array and search for it Not efficient ~ n/2 steps for each search operation Can we do better? Sort the array as a preliminary step. Now search can be performed much faster!

54 Binary Search Input: A sorted array of integers A An integer query q Output: -1 if q is not a member of A The index of q in A otherwise Algorithm: Check the middle element of A If it is equal to q, return its index If it is >= q, search for q in A[0,…,middle-1] If it is < q, search for q in A[middle+1,...,end]

55 Example index value

56 Today Home work review Guest lecture by Prof. Benny Chor Recursive vs. Iterative Arrays Sorting, searching and time-complexity analysis Home work

57 Exercise 1 Write a program that gets 10 numbers from the user. It then accepts another number and checks to see if that number was one of the previous ones. Example 1: Please enter 10 numbers: Please enter a number to search for: 8 Found it! Example 2: Please enter 10 numbers: Please enter a number to search for: 30 Sorry, it’s not there

58 Exercise 2 Implement a function that accepts two integer arrays and returns true if they are equal, false otherwise. The arrays are of the same size Write a program that accepts two arrays of integers from the user and checks for equality