Download presentation

Presentation is loading. Please wait.

Published byGideon Northcutt Modified over 2 years ago

1
Data Structues and Algorithms Algorithms growth evaluation

2
Rate of growth Big theta The statement “f has the same growth rate as g.” A function f has the same growth rate as g (or f has the same order as g) if we can find a number m and two positive constants c and d such that c|g(n)| ≤ |f (n)| ≤ d|g(n)| for all n ≥ m. In this case we write f (n) = Θ (g(n)) and say that f (n) is big theta of g(n). It’s easy to verify that the relation “has the same growth rate as” is an equivalence relation. Proportionality: If two functions f and g are proportional, then f (n) = Θ (g(n)).

3
Rate of growth The Log Function. Recall that log functions with different bases are proportional. In other words, if we have two bases a > 1 and b > 1, then log a n = (log a b) (log b n) for all n > 0. So we can disregard the base of the log function when considering rates of growth. In other words, we have log a n = θ ( log b n )

4
Rate of growth Some approximations

5
Rate of growth A function f has a lower growth rate than g (or f has lower order than g) if In this case we write f (n) = o (g(n)) and say that f is little oh of g. We’ll show that log n = o (n). Since we can write log n = (log e)(loge n), it follows that the derivative of log n is (log e)(1/n). Therefore, we obtain the following equations:

6
Rate of growth Big O. Now let’s look at a notation that gives meaning to the statement “the growth rate of f is bounded above by the growth rate of g.” The standard notation to describe this situation is f(n) = O(g(n)), which we read as f (n ) is big oh of g (n ). The precise meaning of the notation f (n) = O (g (n) ) is given by the following definition. The notation f (n) = O (g (n) ) means that there are positive numbers c and m such that |f (n)| ≤ c |g (n)| for all n ≥ m.

7
Rate of growth Big O. Now let’s look at a notation that gives meaning to the statement “the growth rate of f is bounded above by the growth rate of g.” The standard notation to describe this situation is f(n) = O(g(n)), which we read as f (n ) is big oh of g (n ). The precise meaning of the notation f (n) = O (g (n) ) is given by the following definition. The notation f (n) = O (g (n) ) means that there are positive numbers c and m such that |f (n)| ≤ c |g (n)| for all n ≥ m.

8
Rate of growth Big Ω. Now let’s go the other way. We want a notation that gives meaning to the statement “the growth rate of f is bounded below by the growth rate of g.” The standard notation to describe this situation is f (n) = Ω ( g (n) ), which we can read as f (n ) is big omega of g (n ). The precise meaning of the notation f (n) = Ω ( g (n) ) is given by the following definition. The notation f (n) = Ω ( g (n) ) means that there are positive numbers c and m such that |f (n)| ≥ c |g (n)| for all n ≥ m.

9
Rate of growth The four symbols Θ, o, O, and Ω can also be used to represent terms within an expression. For example, the equation h (n) = 4n 3 + O (n 2 ) means that h (n) equals 4n 3 plus a term of order at most n 2. The four symbols Θ, o, O, and Ω can be formally defined to represent sets of functions: Θ (g) is the set of functions with the same order as g; o(g) is the set of functions with lower order than g; O(g) is the set of functions of order bounded above by that of g; Ω (g) is the set of functions of order bounded below by that of g.

10
Rate of growth When set representations are used, we can use an expression like f (n) ∈ Θ ( g (n) ) to mean that f has the same order as g. The set representations also give some nice relationships. For example, we have the following relationships, where the subset relation is proper. O ( g (n) ) ⊃ Θ ( g (n) ) ∪ o ( g (n) ), Θ ( g (n) ) = O ( g (n) ) ∩ Ω ( g (n) ).

Similar presentations

Presentation is loading. Please wait....

OK

Program Efficiency & Complexity Analysis

Program Efficiency & Complexity Analysis

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on different solid figures video Ppt on pre ignition symptoms Ppt on electricity and circuits bill Ppt on pricing policy of a company Ppt on leverages meaning Ppt on good manners for kindergarten Ppt on 7 wonders of the world 2013 Ppt on buddhism and jainism Ppt on different types of dogs Ppt on content addressable memory design