R. Johnsonbaugh, Discrete Mathematics 5 th edition, 2001 Chapter 3 Algorithms

3.1 Introduction  An algorithm is a finite set of instructions with the following characteristics:  Precision: steps are precisely stated  Uniqueness: Results of each step of execution are uniquely defined. They depend only on inputs and results of preceding steps  Finiteness: the algorithm stops after finitely many steps

More characteristics of algorithms  Input: the algorithm receives input  Output: the algorithm produces output  Generality: the algorithm applies to various sets of inputs

Example: a simple algorithm Algorithm to find the largest of three numbers a, b, c: Assignment operator s := k means “copy the value of k into s”  1. x:= a  2. If b > x then x:= b  3. If c > x then x:= c A trace is a check of the algorithm for specific values of a, b and c

3.2 Notation for algorithms Pseudocode: Instructions given in a generic language similar to a computer language such as C++ or Pascal. Procedure If-then, action If-then-else begin Else Return While loop For loop End

3.3 The euclidean algorithm Divisors:  Given an integer n, we say that k is a divisor of n or k divides n, notation: k|n, if k is a positive integer n = kq for some integer q called the quotient.  A common divisor of two integers m and n is a positive integer k such that k|m and k|n.

Euclidean algorithm  Given two integers m and n, the gcd(m,n) or greatest common divisor of m and n is a common divisor k > 1 such that k is the largest of all common divisors of m and n. The Euclidean algorithm finds the gcd(m, n).  Theorem 3.3.6: If a is a nonnegative integer, b is a positive integer, and r = a mod b, then gcd(a,b) = gcd(b,r)  Example: if a = 120, b = 80, then r = 40 = 120 mod 80.  Thus, gcd(120,80) = gcd(80,40)

3.4 Recursive algorithms  A recursive procedure is a procedure that invokes itself  Example: given a positive integer n, factorial of n is defined as the product of n by all numbers less than n and greater than 0. Notation: n! = n(n-1)(n-2)…3.2.1  Observe that n! = n(n-1)! = n(n-1)(n-2)!, etc.  A recursive algorithm is an algorithm that contains a recursive procedure

Fibonacci sequence  Leonardo Fibonacci (Pisa, Italy, ca )  Fibonacci sequence f 1, f 2,… defined recursively as follows: f 1 = 1 f 2 = 2 f n = f n-1 + f n-2 for n > 3  First terms of the sequence are: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,…

3.5 Complexity of algorithms  Complexity: the amount of time and/or space needed to execute the algorithm.  Complexity depends on many factors: data representation type, kind of computer, computer language used, etc.

Types of complexity Best-case time = minimum time needed to execute the algorithm for inputs of size n Worst-case time = maximum time needed to execute the algorithm for inputs of size n Average-case time = average time needed

Order of an algorithm Z Let f and g be functions with domain Z + = {1, 2, 3,…}  f(n) = O(g(n)): f(n) is of order at most g(n)  if there exists a positive constant C 1 such that |f(n)| < C 1 |g(n)| for all but finitely many n  f(n) =  (g(n)): f(n) is of order at least g(n)  if there exists a positive constant C 2 such that |f(n)| > C 2 |g(n)| for all but finitely many n  f(n) =  (g(n)): f(n) is or order g(n) if it is O(g(n)) and  (g(n)).

3.6 Analysis of the Euclidean algorithm Theorem 3.6.1: Suppose that the pair a, b with a > b requires n >1 modulus operations when input to the Euclidean algorithm. Then a > f n+1 and b > f n+1, where {f n } is the Fibonacci sequence.

Number of operations Theorem 3.6.2: If integers in the range 0 to m, m > 8, not both zero, are input to the Euclidean algorithm, then the number of modulus operations required is at most log 3/2 (2m/3)

3.7 The RSA public-key cryptosystem  Cryptosystems: systems for secure communications  Used by government, industry, investigation agencies, etc.  Sender encrypts a message  Receiver decripts the message  RSA (Rivest, Shamir, Adleman) system  Messages are represented as numbers  Based on the fact that no efficient algorithm exists for factoring large digit integers in polynomial time O(n k ).