By Laurence Boxer, Russ Miller
Equip your self for achievement with a state of the art method of algorithms on hand purely in Miller/Boxer's ALGORITHMS SEQUENTIAL AND PARALLEL: A UNIFIED process, 3E. This detailed and practical textual content supplies an creation to algorithms and paradigms for contemporary computing structures, integrating the learn of parallel and sequential algorithms inside a concentrated presentation. With a variety of functional workouts and interesting examples drawn from basic program domain names, this e-book prepares you to layout, examine, and enforce algorithms for contemporary computing platforms.
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Extra info for Algorithms Sequential & Parallel: A Unified Approach (3rd Edition)
This translates into a bound on the summation as b+1 ∫a b f (t)dt ≤ a f (i) ≤ i=a b ∫ a−1 f (t)dt. EXAMPLE As our final example of evaluating the asymptotic behavior of a summation by integrals, we consider the function n f (n) = a k p k=1 for p > 0. Recall that we showed earlier in this chapter that f (n) = a k p = Θ 1n p+12 . n k=1 However, the purpose of this example is to show how to obtain this result by the method we have been considering that relates summations to integrals. Recall that a function f is increasing if u < v ⇒ f (u) < f(v).
In the case where lim We now give some examples of how to determine asymptotic relationships based on taking limits of a quotient. EXAMPLE Let f (n) = n(n + 1) and g(n) = n2. 2 Then we can show that f (n) = Θ(g(n)) since lim n→ ∞ f (n) n2 + n = lim = g(n) n→ ∞ 2n2 (dividing both numerator and denominator by n2) 1+ lim n→ ∞ 2 1 n 1 = . 2 EXAMPLE If P(n) is a polynomial of degree d > 0, then P(n) = Θ(nd). This can be seen as d follows. The hypothesis implies P(n) = a ai ni for some set of coefficients 5ai6i=0 with ad ≠ 0.
Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Reference Guide xxix Common Summations n Sum of Constant Terms: a 1 = n i=1 Arithmetic Series: a ai = n c a1 + n i=1 (n − 1)d d , where ai+1 = ai + d, for some constant d.