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Lecture E Introduction to Algorithms

Lecture E Introduction to Algorithms. Unit E1 – Basic Algorithms . Lesson or Unit Topic or Objective. Demonstrate the notion of an algorithm using two classic ones. Computational problems. A computational problem specifies an input-output relationship What does the input look like?

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Lecture E Introduction to Algorithms

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  1. Lecture E Introduction to Algorithms Unit E1 – Basic Algorithms

  2. Lesson or Unit Topic or Objective Demonstrate the notion of an algorithm using two classic ones

  3. Computational problems • A computational problem specifies an input-output relationship • What does the input look like? • What should the output be for each input? • Example: • Input: an integer number N • Output: Is the number prime? • Example: • Input: A list of names of people • Output: The same list sorted alphabetically • Example: • Input: A picture in digital format • Output: An English description of what the picture shows

  4. Algorithms • An algorithm is an exact specification of how to solve a computational problem • An algorithm must specify every step completely, so a computer can implement it without any further “understanding” • An algorithm must work for all possible inputs of the problem. • Algorithms must be: • Correct: For each input produce an appropriate output • Efficient: run as quickly as possible, and use as little memory as possible – more about this later • There can be many different algorithms for each computational problem.

  5. Describing Algorithms • Algorithms can be implemented in any programming language • Usually we use “pseudo-code” to describe algorithms • In this course we will just describe algorithms in Java Testing whether input N is prime: For j = 2 .. N-1 If j|N Output “N is composite” and halt Output “N is prime”

  6. Greatest Common Divisor • The first algorithm “invented” in history was Euclid’s algorithm for finding the greatest common divisor (GCD) of two natural numbers • Definition: The GCD of two natural numbers x, y is the largest integer j that divides both (without remainder). I.e. j|x, j|y and j is the largest integer with this property. • The GCD Problem: • Input: natural numbers x, y • Output: GCD(x,y) – their GCD

  7. Euclid’s GCD Algorithm public staticint gcd(int x, int y) { while (y!=0) { int temp = x%y; x = y; y = temp; } return x; }

  8. Euclid’s GCD Algorithm – sample run while (y!=0) { int temp = x%y; x = y; y = temp; } Example: Computing GCD(48,120) temp x y After 0 rounds -- 72 120 After 1 round 72 120 72 After 2 rounds 48 72 48 After 3 rounds 24 48 24 After 4 rounds 0 24 0 Output: 24

  9. Correctness of Euclid’s Algorithm • Theorem: When Euclid’s GCD algorithm terminates, it returns the mathematical GCD of x and y. • Notation: Let g be the GCD of the original values of x and y. • Loop Invariant Lemma: For all k  0, The values of x, y after k rounds of the loop satisfy GCD(x,y)=g. • Proof of lemma: next slide. • Proof of Theorem: The method returns when y=0. By the loop invariant lemma, at this point GCD(x,y)=g. But GCD(x,0)=x for every integer x (since x|0 and x|x). Thus g=x, which is the value returned by the code. • Still Missing: The algorithm always terminates.

  10. Proof of Lemma • Loop Invariant Lemma: For all k  0, The values of x, y after k rounds of the loop satisfy GCD(x,y)=g. • Proof: By induction on k. • For k=0, x and y are the original values so clearly GCD(x,y)=g. • Induction step: Let x, y denote that values after k rounds and x’, y’ denote the values after k+1 rounds. We need to show that GCD(x,y)=GCD(x’,y’). According to the code: x’=y and y’=x%y, so the lemma follows from the following mathematical lemma. • Lemma: For all integers x, y: GCD(x, y) = GCD(x%y, y) • Proof: Let x=ay+b, where y>b 0. I.e. x%y=b. • (1) Since g|y, and g|x, we also have g|(x-ay), I.e. g|b. Thus GCD(b,y) g = GCD(x,y). • (2) Let g’=GCD(b,y), then g’|(x-ay) and g’|y, so we also have g’|x. Thus GCD(x,y)  g’=GCD(b,y).

  11. Termination of Euclid’s Algorithm • Why does this algorithm terminate? • After any iteration we have that x > y since the new value of y is the remainder of division by the new value of x. • In further iterations, we replace (x, y) with (y, x%y), and x%y < x, thus the numbers decrease in each iteration. • Formally, the value of xy decreases each iteration (except, maybe, the first one). When it reaches 0, the algorithm must terminate. public staticint gcd(int x, int y) { while (y!=0) { int temp = x%y; x = y; y = temp; } return x; }

  12. Square Roots • The problem we want to address is to compute the square root of a real number. • When working with real numbers, we can not have complete precision. • The inputs will be given in finite precision • The outputs should only be computed approximately • The square root problem: • Input: a positive real number x, and a precision requirement  • Output: a real number r such that |r-x|

  13. Square Root Algorithm public static double sqrt(double x, double epsilon){ double low = 0; double high = x>1 ? x : 1; while (high-low > epsilon) { double mid = (high+low)/2; if (mid*mid > x) high = mid; else low = mid; } return low; }

  14. Binary Search Algorithm – sample run while (high-low > epsilon) { double mid = (high+low)/2; if (mid*mid > x) high = mid; else low = mid; } Example: Computing sqrt(2) with precision 0.05: mid mid*mid low high After 0 rounds -- -- 0 2 After 1 round 1 1 1 2 After 2 rounds 1.5 2.25 1 1.5 After 3 rounds 1.25 1.56.. 1.25 1.5 After 4 rounds 1.37.. 1.89.. 1.37.. 1.5 After 5 rounds 1.43.. 2.06.. 1.37.. 1.43.. After 6 rounds 1.40.. 1.97.. 1.40.. 1.43.. Output: 1.40…

  15. Correctness of Binary Search Algorithm • Theorem: When the algorithm terminates it returns a value r that satisfies |r-x|. • Loop invariant lemma:For all k  0, The values of low, high after k rounds of the loop satisfy: low  x  high. • Proof of Lemma: • For k=0, clearly low=0  x  high=max(x,1). • Induction step: The code only sets low=mid if mid  x, and only sets high=mid if mid>x. • Proof of Theorem: The algorithm terminates when high-low, and returns low. At this point, by the lemma: low  x  high  low+. Thus |low-x|. • Missing Part: Does the algorithm always terminate? How Fast? We will deal with this later.

  16. In General… • This type of binary search can be used to find the roots of any continuous function f. • Mean Value Theorem: if f(low)<0 and f(high)>0 then for some low<x<high, f(x)=0. • In our case, to find 2, we solved

  17. Lecture E Introduction to Algorithms Unit E1 – Basic Algorithms

  18. Lecture E Introduction to Algorithms Unit E2 – Running Time Analysis

  19. Lesson or Unit Topic or Objective Analysis of Running Times of Algorithms

  20. How fast will your program run? • The running time of your program will depend upon: • The algorithm • The input • Your implementation of the algorithm in a programming language • The compiler you use • The OS on your computer • Your computer hardware • Maybe other things: temperature outside; other programs on your computer; … • Our Motivation: analyze the running time of an algorithm as a function of only simple parameters of the input.

  21. Basic idea: counting operations • Each algorithm performs a sequence of basic operations: • Arithmetic: (low + high)/2 • Comparison: if ( x > 0 ) … • Assignment: temp = x • Branching: while ( true ) { … } • … • Idea: count the number of basic operations performed on the input. • Difficulties: • Which operations are basic? • Not all operations take the same amount of time. • Operations take different times with different hardware or compilers

  22. Testing operation times on your system import java.util.*; public class PerformanceEvaluation { public static void main(String[] args) { int i=0; double d = 1.618; SimpleObject o = new SimpleObject(); finalint numLoops = 1000000; long startTime = System.currentTimeMillis();; for (i=0 ; i<numLoops ; i++){ // put here a command to be timed } long endTime = System.currentTimeMillis(); long duration = endTime - startTime; double iterationTime = (double)duration / numLoops; System.out.println("duration: "+duration); System.out.println("sec/iter: "+iterationTime); }} class SimpleObject { private int x=0; public void m() { x++; } }

  23. Sample running times of basic Java operations Operation Loop Body nSec/iteration Sys1 Sys2 Sys1: PII, 333MHz, jdk1.1.8, -nojit Sys2: PIII, 500MHz, jdk1.3.1

  24. Asymptotic running times • Operation counts are only problematic in terms of constant factors. • The general form of the function describing the running time is invariant over hardware, languages or compilers! • Running time is “about” . • We use “Big-O” notation, and say that the running time is O( ) public static int myMethod(int N){ int sq = 0; for(int j=0; j<N ; j++) for(int k=0; k<N ; k++) sq++; return sq; }

  25. Asymptotic behavior of functions

  26. Mathematical Formalization • Definition: Let f and g be functions from the natural numbers to the natural numbers. We write f=O(g) if there exists a constant c such that for all n: f(n)  cg(n). f=O(g) c n: f(n)  cg(n) • This is a mathematically formal way of ignoring constant factors, and looking only at the “shape” of the function. • f=O(g) should be considered as saying that “f is at most g, up to constant factors”. • We usually will have f be the running time of an algorithm and g a nicely written function. E.g. The running time of the previous algorithm was O(N^2).

  27. Asymptotic analysis of algorithms • We usually embark on an asymptotic worst case analysis of the running time of the algorithm. • Asymptotic: • Formal, exact, depends only on the algorithm • Ignores constants • Applicable mostly for large input sizes • Worst Case: • Bounds on running time must hold for all inputs. • Thus the analysis considers the worst-case input. • Sometimes the “average” performance can be much better • Real-life inputs are rarely “average” in any formal sense

  28. The running time of Euclid’s GCD Algorithm • How fast does Euclid’s algorithm terminate? • After the first iteration we have that x > y. In each iteration, we replace (x, y) with (y, x%y). • In an iteration where x>1.5y then x%y < y < 2x/3. • In an iteration where x  1.5y then x%y  y/2 < 2x/3. • Thus, the value of xy decreases by a factor of at least 2/3 each iteration (except, maybe, the first one). public staticint gcd(int x, int y) { while (y!=0) { int temp = x%y; x = y; y = temp; } return x; }

  29. The running time of Euclid’s Algorithm • Theorem: Euclid’s GCD algorithm runs it time O(N), where N is the input length (N=log2x + log2y). • Proof: • Every iteration of the loop (except maybe the first) the value of xy decreases by a factor of at least 2/3. Thus after k+1 iterations the value of xy is at most the original value. • Thus the algorithm must terminate when k satisfies: (for the original values of x, y). • Thus the algorithm runs for at most iterations. • Each iteration has only a constant L number of operations, thus the total number of operations is at most • Formally, • Thus the running time is O(N).

  30. Running time of Square root algorithm • The value of (high-low) decreases by a factor of exactly 2 each iteration. It starts at max(x,1), and the algorithm terminates when it goes below . • Thus the number of iterations is at most • The running time is public static double sqrt(double x, double epsilon){ double low = 0; double high = x>1 ? x : 1; while (high-low > epsilon) { double mid = (high+low)/2; if (mid*mid > x) high = mid; else low = mid; } return low; }

  31. Newton-Raphson Algorithm public static double sqrt(double x, double epsilon){ double r = 1; while ( Math.abs(r - x/r) > epsilon) r = (r + x/r)/2; return r; }

  32. Newton-Raphson – sample run while ( Math.abs(r - x/r) > epsilon) r = (r + x/r)/2; Example: Computing sqrt(2) with precision 0.01: r x/r After 0 rounds 1 2 After 1 round 1.5 1.33.. After 2 rounds 1.41.. 1.41.. Output: 1.41…

  33. Analysis of Running Time • Correctness is clear since for every r the square root of x is between and r and x/r. • Here we will analyze the running time only for 1<x<2 • Denote: • Thus , where after n loops • At the beginning , and • In general we have that • At the end it suffices that , since • Thus the algorithm terminates when

  34. In General… • The Newton-Raphson method can be used to find the roots of any differentiable function f. • In our case, to find 2, we solved • So,

  35. Lecture E Introduction to Algorithms Unit E2 – Running Time Analysis

  36. Lecture E Introduction to Algorithms Unit E3 - Recursion

  37. Lesson or Unit Topic or Objective Using and understanding Recursion

  38. Designing Algorithms • There is no single recipe for inventing algorithms • There are basic rules: • Understand your problem well – may require much mathematical analysis! • Use existing algorithms (reduction) or algorithmic ideas • There is a single basic algorithmic technique: Divide and Conquer • In its simplest (and most useful) form it is simple induction • In order to solve a problem, solve a similar problem of smaller size • The key conceptual idea: • Think only about how to use the smaller solution to get the larger one • Do not worry about how to solve to smaller problem (it will be solved using an even smaller one)

  39. Recursion • A recursive method is a method that contains a call to itself • Technically: • All modern computing languages allow writing methods that call themselves • We will discuss how this is implemented later • Conceptually: • This allows programming in a style that reflects divide-n-conquer algorithmic thinking • At the beginning recursive programs are confusing – after a while they become clearer than non-recursive variants

  40. Factorial public static void Factorial { public static void main() { System.out.println(“5!=“ + factorial(5)); } public static long factorial(int n){ if (n == 0) return 1; else return n * factorial(n-1); } }

  41. Elements of a recursive program • Basis: a case that can be answered without using further recursive calls • In our case: if (n==0) return 1; • Creating the smaller problem, and invoking a recursive call on it • In our case: factorial(n-1) • Finishing to solve the original problem • In our case: return n * /*solution of recursive call*/

  42. Tracing the factorial method System.out.println(“5!=“ + factorial(5)) 5 * factorial(4) 4 * factorial(3) 3 * factorial(2) 2 * factorial(1) 1 * factorial(0) return 1 return 1 return 2 return 6 return 24 return 120

  43. Correctness of factorial method • Theorem: For every positive integer n, factorial(n) returns the value n!. • Proof: By induction on n: • Basis: for n=0, factorial(0) returns 1=0!. • Induction step: When called on n>1, factorial calls factorial(n-1), which by the induction hypothesis returns (n-1)!. The returned value is thus n*(n-1)!=n!.

  44. Raising to power – take 1 public static double power(double x, long n) { if (n == 0) return 1.0; return x * power(x, n-1); }

  45. Running time analysis • Simplest way to calculate the running time of a recursive program is to add up the running times of the separate levels of recursion. • In the case of the power method: • There are n+1 levels of recursion • power(x,n), power(x,n-1), power(x, n-2), … power(x,0) • Each level takes O(1) steps • Total time = O(n)

  46. Raising to power – take 2 public static double power(double x, long n) { if (n == 0) return 1.0; if (n%2 == 0) { double t = power(x, n/2); return t*t; } return x * power(x, n-1); }

  47. Analysis • Theorem: For any x and positive integer n, the power method returns . • Proof: by complete induction on n. • Basis: For n=0, we return 1. • If n is even, we return power(x,n/2)*power(x,n/2). By the induction hypothesis power(x,n/2) returns , so we return . • If n is odd, we return x*power(x,n-1). By the induction hypothesis power(x,n-1) returns , so we return . • The running time is now O(log n): • After 2 levels of recursion n has decreased by a factor of at least two (since either n or n-1 is even, in which case the recursive call is with n/2) • Thus we reach n==0 after at most 2log2n levels of recursion • Each level still takes O(1) time.

  48. Reverse public class Reverse { static InputRequestor in = new InputRequestor(): public static void main(String[] args) { printReverse(); } public static void printReverse() { int j = in.requestInt(“Enter another Number”+ ” (0 for end of list):”); if (j!=0){ printReverse(); System.out.println(j); } } }

  49. Recursive Definitions • Many things are defined recursively. • Fibonaci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, … • fn = fn-1 + fn-2 • Arithmetic Expressions. E.g. 2+3*(5+(3-4)) • A number is an expression • For any expression E: (E) is an expression • For any two expressions E1, E2: E1+E2, E1-E2, E1*E2, E1/E2 are expressions • Fractals • In such cases recursive algorithms are very natural

  50. Fibonaci Numbers public class Fibonaci { public static void main(String[] args) { for(int j = 1 ; j<20 ; j++) System.out.println(fib(j)); } public static int fib (int n) { if (n <= 1) return 1; return fib(n-1) + fib(n-2); } }

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