MA/CSSE 473 Day 07 Euclid's Algorithm

MA/CSSE 473 Day 07 Student Questions Review topics not covered in class Euclid's algorithm (if there is time) extended Euclid's algorithm

REVIEW THREAD Quick look at review topics in textbook

Textbook Topics I Won't Cover in Class Chapter 1 topics that I will not discuss in detail unless you have questions. They should be review For some of them, there will be review problems in the homework –Sieve of Eratosthenes (all primes less than n) –Algorithm Specification, Design, Proof, Coding –Problem types : sorting, searching, string processing, graph problems, combinatorial problems, geometric problems, numerical problems –Data Structures: ArrayLists, LinkedLists, trees, search trees, sets, dictionaries,

Textbook Topics I Won't Cover* Chapter 2 –Empirical analysis of algorithms should be review –I believe that we have covered everything else in the chapter except amortized algorithms and recurrence relations –We will discuss amortized algorithms –Recurrence relations are covered in CSSE 230 and MA 375. We'll review particular types as we encounter them. *Unless you ask me to

Textbook Topics I Won't Cover* Chapter 3 - Review –Bubble sort, selection sort, and their analysis –Sequential search and simple string matching *Unless you ask me to

Textbook Topics I Won't Cover* Chapter 4 - Review –Mergesort, quicksort, and their analysis –Binary search –Binary Tree Traversal Orders (pre, post, in, level) *Unless you ask me to

Textbook Topics I Won't Cover* Chapter 5 - Review –Insertion Sort and its analysis –Search, insertion, delete in Binary Tree –AVL tree insertion and rebalance *Unless you ask me to

ARITHMETIC THREAD Euclid's Algorithm Heading toward Primality Testing

Euclid's Algorithm: the problem One of the oldest known algorithms (about 2500 years old) The problem: Find the greatest common divisor (gcd) of two non-negative integers a and b. The approach you learned in elementaryschool: –Completely factor each number –find common factors (with multiplicity) –multiply the common factors together to get the gcd Factoring is hard! Simpler approach is needed Q1

Euclid's Algorithm: the basis Based on the following rule: –If x and y are positive integers with x ≥ y, then gcd(x, y) = gcd(y, x mod y) Proof of Euclid's rule: –It suffices to show the simpler rule gcd(x, y) = gcd(y, x - y) since x mod y can be obtained from x and y by repeated subtraction –Any integer that divides both x and y must also divide x – y, so gcd(x, y) ≤ gcd(y, x – y) –Any integer that divides both y and x - y must also divide x, so gcd(y, x-y) ≤ gcd(y, x) –Putting these together, so gcd(y, x-y) = gcd(y, x)

Euclid's Algorithm: the algorithm Example: euclid(60, 36) Does the algorithm work? How efficient is it? Q2

Euclid's Algorithm: the analysis Lemma: If a ≥ b, then a % b < a/2 Proof –If b ≤ a/2, then a % b < b ≤ a/2 –If b > a/2, then a % b = a – b < a/2 Application –After two recursive calls, both a and b are less than half of what they were, (i.e. reduced by at least 1 bit) –Thus if a and b have n bits, at most 2n recursive calls are needed. –Each recursive call involves a division, Ѳ(n 2 ) –Entire algorithm is Ѳ(n 3 )

gcd and linear combinations Lemma: If d divides both a and b, and d = ax + by for some integers x and y, then d = gcd(a,b) Proof –By the first of the two conditions, d is a common divisor of a and b. It cannot exceed the greatest common divisor, so d ≤ gcd(a, b) –gcd(a, b) is a common divisor of a and b, so it must divide ax + by = d. Thus gcd(a, b) ≤ d –Putting these together, gcd(a, b) = d If we can supply the x and y as in the lemma, we know that d is the gcd. It turns out that a simple modification of Euclid's algorithm will calculate the x and y. Q3

Extended Euclid Algorithm Proof that it works –First, the number d it produces really is the gcd of a and b. We can just ignore the x and y values, and we have the same algorithm as before.

Extended Euclid Algorithm: proof Proof that it works –First, the number d it produces really is gcd (a,b) We can just ignore the x and y values, and we have the same algorithm as before. –We must show that the x and y it returns are such that ax + by =d. –We do that by induction on b. Q4

Proof that ax+by = d (induction on b) Base case: b=0 Then gcd(a,b) = a, and the algorithm produces x = 1, y = 0.  Induction step: b > 0. – It finds gcd(a, b) by calling euclidExtended(b, a%b) –Since a%b is smaller than b, by induction the x' and y' returned by the recursive call are such that gcd(b, a %b) = bx'+ (a%b)y' –We can write a % b as a –  a/b  * b –d = gcd(a, b) = gcd(b, a%b) = bx' + (a%b)y' = bx' +(a –  a/b  * b)y' = ay' +b(x' -  a/b  y') –Thus x = y' and y = x' -  a/b  y' are the numbers that make ax + by = d –This x and y are the numbers returned by the algorithm. Q4

Example: gcd (33, 14) 33 = 2* = 2 * = 1 * = 4 * 1 + 0, so gcd(33, 14) = 1. Now work backwards 1 = Substitute 4 = *5. 1 = 5 – (14 - 2*5) = 3* Substitute 5 = *14 1 = 3(33 - 2*14) -14 = 3 * 33 – 7 * 14 Thus x = 3 and y = -7 Done! Q5-6