1. MAT 4 – Kompleks Funktionsteori MATEMATIK 4 INDUKTION OG REKURSION MM 1.5 MM 1.5: Kompleksitet Topics: Computational complexity Big O notation Complexity and recursion

2. MAT 4 – Kompleks Funktionsteori What should we learn today? What is computational complexity of algorithms? When the growth of one function is higher (the same/ lower) than the growth of another function? How we can estimate complexity of functions that are defined recursively? What type of problems will be at the exam?

3. MAT 4 – Kompleks Funktionsteori Fra gamle eksamenationopgaver

4. MAT 4 – Kompleks Funktionsteori Computational complexity theory Algorithm: a detailed ”step by step” description of a method Complexity theory investigates the problems related to the amount of resources required for the execution of algorithms (e.g. execution time) Complexity theory is also dealing with the scalability of computational problems and algorithms –As the size of the input to an algorithm increases, how do the running time and memory requirements of the algorithm change

5. MAT 4 – Kompleks Funktionsteori Time complexity The time complexity of a problem is the number of steps it takes to solve the problem. It is a function of the size of the input. Example. It takes 1 min to mow a lawn of 10 m^2. 2 min – 20 m^2 … N min – N x 10 m^2 The time complexity is linear

6. MAT 4 – Kompleks Funktionsteori Addition Example. Addition of 2 numbers with n digits We perform n *simple* operations of type a+b+m (m is carry) Assume that the time needed to perform a simple operation is s sec; the time needed to write down the last carry is t sec  Total time required is sn+t  complexity is linear with respect to the number of digits

7. MAT 4 – Kompleks Funktionsteori Multiplication Example 1: multiplication of n-digit number with 1-digit number The number of *simple* operations (ab+m) is n Complexity is linear Example 2: multiplication of two n-digit numbers consists of n multiplications of 1-digit and n-digit numbers (  n 2 operations) + summation (n times addition of n-digit numbers  n 2 operations) Complexity is quadratic

8. MAT 4 – Kompleks Funktionsteori Big-O notation This notation is used to describe how the size of the input data affects algorithm’s usage of resources (e.g. computational time) Q: what is faster: to add or to multiply? Definition. f(n) has the higher order growth then g(n), if the ratio f(n)/g(n) goes to infinity as n goes to infinity. Note: both f(n) and g(n) are functions taking on positive values and they are increasing functions starting from a certain point. Example: Example: polynomials of degree m and k

9. MAT 4 – Kompleks Funktionsteori Big-O notation Definition. f and g has the same order growth, if their ratio f/g goes to a positive constant when n goes to infinity. Example. Polynomials of the same degree

10. MAT 4 – Kompleks Funktionsteori Big-O notation Definition. Let f(n) be a positive, increasing function starting from a certain point. O(f) is a collection (set) of functions that exhibit a growth that is limited to the growth of f(n) (growth of smaller order or the same order as f(n) ) Examples, propositions, properties…

11. MAT 4 – Kompleks Funktionsteori Exponential, polynomial, logarithmic functions Exponential growth  a n Polynomial growth  n a Logarithmic growth (logarithmic function is inverse to exponential function )  log n Proposition. –Exponential function has a higher order growth than any polynomial function. –Polynomial function has a higher order growth than a logarithmic function.

12. MAT 4 – Kompleks Funktionsteori

13. Complexity and recursion Example: factorial function is defined recursively t(n) is computational complexity: Example: Fibonacci numbers Computational complexity: