Algorithm Analysis: common functions, asymptotic notation

2024-05-31 03:01:34

对于算法复杂度分析很多概念，自己理解的稀里糊涂的。或者好不容易弄明白了，隔一段时间又忘记了。特地整理记录在此

1. common functions used in analysis

constant, logarithmic, linear, N-log-N, quadratic, cubic, exponential

From best (least resource requirements ) to worst, the rankings are: O(1), O(logn), O(n), O(nlogn), O( n^2 ), O(n^3)O(2^n).

Graph: https://en.wikipedia.org/wiki/Analysis_of_algorithms

2.　asymptotic notations

Big O, Little o, Theta, Omega, Little w

3. involved math knowledge

3.1 exponent

8^2 could be called "8 to the power 2" or "8 to the second power", or simply "8 squared"

5^3 could be called "5 to the third power", "5 to the power 3" or simply "5 cubed"

2^4 could be called "2 to the fourth power" or "2 to the power 4" or simply "2 to the 4th"

a^n tells you to multiply n times a by itself, so there are n of those a'

3.2 laws of exponent

3.3 logarithm

A Logarithm says how many of one number to multiply to get another number. Example, what is x in log3(x) = 5

Start with:log3(x) = 5. We want to "undo" the log3 so we can get "x ="

Use the Exponential Function (on both sides):

And we know that , so:x = 35

Answer:x = 243

3.4 Series

3.4.1 等差数列 Arithmetic Series

To sum up the terms of this arithmetic sequence:

a + (a+d) + (a+2d) + (a+3d) + ...

use this formula:

3.4.2 等比数列 Geometric Series

a + ar + ar^2 + ... + ar^(n-1)

Each term is ar^k, where k starts at 0 and goes up to n-1

Use this formula:

a is the first term
r is the "common ratio" between terms
n is the number of terms

4. Reference

Algebra

Calculus

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