Lesson Objectives Aims Understand the following: The big O notation

Big O Notation Big O is the “Order” of magnitude for an algorithm What does that mean? It’s an estimate of the time an algorithm will take for a given data set. It can be used to measure its efficiency

Big O Represented as O(function of N) N = the size of the input data set O = generally the worst case scenario

Big O Algorithms, and indeed algorithmic complexity, are divided in to types: Constant Linear Polynomial Exponential Logarithmic

Big O notation is used to show: Key Points Big O notation is used to show: The time algorithms take Or the space they need increases as the size of the data set they operate on increases

Calculating big O We can express the complexity of an algorithm in an equation We then simplify the equation to express the term in one of the big O forms (linear, exponential etc)

Example 7n3 + n2 + 4n +1 as n increases - what happens to the different terms of this equation? The larger n gets, the less an impact n2 + 4n +1 has on the total compared to n3

To calculate Big O for a given expression: Remove all terms except the one with the largest exponent Remove any constant factors

Questions

1 = O(n4) = Polynomial 2 = O(n) = linear complexity

Questions

3 = O(n2) = Quadratic 4 = O(10) = Constant

Constant complexity O(1) O(1) Algorithms take the same time to run regardless of the size of the dataset Example: push an item to the stack

Linear complexity O(n) The time taken to solve an O(n) algorithms increases at the same rate as the input size If the input dataset doubles in size, the time taken doubles Example: find an element using linear search

Polynomial complexity O(nk) where k is a positive integer The complexity varies depending on the power of the function. If k = 0: n^0 = 1 constant complexity If k = 1 linear complexity If K = 2 quadratic complexity If K = 3 cubic complexity

Quadratic complexity O(n2) Time taken rapidly increases with the size of the data set example: bubble sort

Exponential complexity Instead of O(nk), it is O(kn) Time taken rises at a much faster rate than polynomial complexity The complexity is to the power of the number of elements Does not scale well

Logarithmic Complexity O(log n) Start off expensive As the data set grows, the complexity decreases and eventually levels out A good situation to be in E.g. Binary Search.

Comparison

Reading https://rob-bell.net/2009/06/a-beginners-guide-to-big-o-notation/ https://justin.abrah.ms/computer-science/big-o-notation-explained.html Shows in the context of code

Review/Success Criteria You should know: The