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R. Johnsonbaugh, Discrete Mathematics 5 th edition, 2001

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

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R. Johnsonbaugh, Discrete Mathematics 5 th edition, 2001

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  1. R. Johnsonbaugh,Discrete Mathematics5th edition, 2001 Chapter 3 Algorithms

  2. 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

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

  4. 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

  5. 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 3.2 Notation for algorithms

  6. 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.

  7. 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)

  8. 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

  9. Fibonacci sequence • Leonardo Fibonacci (Pisa, Italy, ca. 1170-1250) • Fibonacci sequence f1, f2,… defined recursively as follows: f1 = 1 f2 = 2 fn = fn-1 + fn-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,…

  10. 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. 3.5 Complexity of algorithms

  11. 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

  12. Order of an algorithm 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 C1 such that |f(n)| < C1|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 C2 such that |f(n)| > C2|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)).

  13. 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 > fn+1 and b > fn+1, where {fn} is the Fibonacci sequence. 3.6 Analysis of the Euclidean algorithm

  14. 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)

  15. 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(nk).

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