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Theorem A.2 Laws of Logarithms 1. log b (b y) = y 2. b logb x = x 3. log b uv = log b u + log b v 4. log b u/v = log b u log b v 5. log b u v = v log b u 6. log b x = (log c x)/(log c b) = (log b c) (log c x) 7. For a positive integer n, lg n + 1 = lgn + 1. EXAMPLE A.1 Applying the Laws of Logarithms
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log 2 256 = log 2 (28) = 8 log 2 1000 = (log 10 1000)/(log 10 2) = 3/0.30103 = 9.966 log 2 1,000,000,000,000 = log 2 10004 = 4(log 2 1000) = 4(9.966) = 39.86 (ln n)/(lg n) = (log e n)/(log 2 n) = log e 2 = ln 2 = 0.693147, for any n > 1
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ASYMPTOTIC COMPLEXITY CLASSES In computer science, algorithms are classified by their complexity functions. These are functions that describe an algorithm s running time relative to the size of the problem. For example, the Bubble Sort belongs to the complexity class (n2). This means that if the Bubble Sort takes T milliseconds to sort an array of n elements, then it will take about 4T milliseconds to sort an array of 2n elements because (2n)2 = 4n2. The symbol () is one of five symbols used to describe complexity functions. They all can be defined in terms of the ratios of f(n) and g(n), where f(n) is the algorithm s timing function and g(n) is a characterizing function such as lg n or n2. For a given function g(n), the five asymptotic complexity classes are O(g(n)) = { f(n) S | f(n)/g(n) is bounded } (g(n)) = { f(n) S | g(n)/f(n) is bounded } (g(n)) = { f(n) S | f(n)/g(n) is bounded and g(n)/f(n) is bounded } o(g(n)) = { f(n) S | f(n)/g(n) 0 as n } (g(n)) = { f(n) S | g(n)/f(n) 0 as n } These definitions assume that f(n) and g(n) that are positive ascending functions. As sets of functions, o(g) O(g) (g) (g) (g) = O(g) (g) EXAMPLE A.2 Asymptotic Growth Classes
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For every k > 0, nk = o(2n ), because nk/2n 0. k = o(n), because (lgn)k/n 0. For every k > 0, (lgn) For every base b > 1, log b n = (lgn), because log b n/lg n = log b 2. 0. The factorial numbers n! = (2n ), because 2n/n!
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The five complexity classes can be imprecisely described by these phrases: f(n) = o(g(n)) means that f(n) grows more slowly than g(n). f(n) = O(g(n)) means that f(n) grows more slowly or at the same rate as g(n). f(n) = (g(n)) means that f(n) grows at the same rate g(n). f(n) = (g(n)) means that f(n) grows faster or at the same rate as g(n). f(n) = (g(n)) means that f(n) grows faster than g(n). EXAMPLE A.3 Asymptotic Growth Classes
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250 lgn = o(n), because 250 lgn grows more slowly than n. 0.086 n lgn = (n), because 0.086 n lgn grows faster than n.
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Keep in mind that these functions f(n), g(n), and so on, are usually used to describe how long it takes to run an algorithm. So if f(n) grows more slowly than g(n), then the algorithm with complexity f(n) is generally faster than the algorithm with complexity g(n). Less time is better. THE FIRST PRINCIPLE OF MATHEMATICAL INDUCTION The First Principle of Mathematical Induction, also called weak induction, is often used to prove formulas about positive integers. Theorem A.3 The First Principle of Mathematical Induction If {P1 , P2, P3 , . . . } is a sequence of statements such that: P1 is true. Each statement Pn can be deduced from its predecessor Pn 1 . Then all of the statements P1, P2 , P3, . . . are true. EXAMPLE A.4 Weak Induction
Suppose we want to prove that the inequality 2n (n + 1)! is true for every n 1. Then the sequence of statements is 21 2! P1: P2 : 22 3! 23 4! P3 : etc. The first few statements can be verified explicitly: 21 = 2 2 = 2! 22 = 4 6 = 3! 23 = 8 24 = 4! In particular, P1 is true, satisfying the first of the two requirements for weak induction. This is called the base of the induction. To verify the second requirement, we have to show that each statement Pn can be deduced from its predecessor Pn 1. So we examine the two general statements Pn 1 and Pn , and look for a connection: Pn 1: 2n 1 n! Pn : 2n (n + 1)! To derive Pn from Pn 1, we note that 2n = (2)(2n 1) and (n + 1)! = (n + 1)(n!). Thus, if we assume that Pn 1 is true, then we have 2n = (2)(2n 1) (2)(n!) (n + 1)(n!) = (n + 1)!, because n + 1 > 2. Verifying the second requirement of mathematical induction is called the inductive step.
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