Big-O & Asymptotic Analysis Big-O Big O : Implementation independent description of algorithm run time O for Omicron Technical: Function that by a constant factor, is
an upper bound for steps in particular algorithm Big-O Big O Practical def: Function that best describes how work grows in relation to problem size. Example:
n is a better match for the growth of 4n - 1 than n2 Functions Big-O Big O Practical def: Function that best describes how work grows
in relation to problem size. Other bounds Big Omega : function that is a lower bound by a constant factor Big Theta : function that can be either an upper bound or lower bound with right constant
Often mean Big-Theta when say Big-O Big-O Mathematical Definition Big O : Function that by a constant factor, is an upper bound for steps in particular algorithm if and only if There
f(n) is work done by your code g(n) is the function category you want to call it Algebra of Big-O Show: 4n 1 is O(n) Must pick c and k so that:
or 4n 1 Pick k = 0 and c = 5 Algebra of Big-O 4n 1 is O(n)
Important Idea 1 Constants dont matter in Big-O Can treat all constants as 1 Expression 5n 3n2 n2/20 log2(n) + 6
10 Category n : Linear n2 : Quadratic n2 : Quadratic log2(n) : Logarithmic 1 : Constant
Dominant Term Largest term drives growth curve n n2 + 20n n2 1
21 1 10 300
100 100 12,000 10,000 1000
1,020,000 1,000,000 10000 100,200,000
100,000,000 Example Show: 2n2 + 100n is O(n2) Must pick c and k so that: or 2n2 + 100n Pick k = 100 and c = 5
Example 2n2 + 100n is O(n2) Important Idea 2 For Big-O just report largest term Examples: 8n2 + 3n - 10 O(8n2) O(n2)
3log2n + 4n O(4n) O(n) 500 + 3n O(3n) O(n) BigO Limitations Asymptotic Analysis : focus on behavior for large values BigO Limitations BigO focuses on categories at scale
Results may not hold for small problem sizes BigO Limitations BigO focuses on categories at scale Other terms and constants may matter for algorithms in the same category Cases Algorithm may have different Big-O for
Best case Worst case Average case When in doubt, we describe the worst case
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