MAT 4 – Kompleks Funktionsteori

MATEMATIK 4INDUKTION OG REKURSION

MM 1.5

MM 1.5: Kompleksitet

Topics: Computational complexity

Big O notation

Complexity and recursion

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?

MAT 4 – Kompleks Funktionsteori

Fra gamle eksamenationopgaver

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

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

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

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 ( n2 operations) + summation (n times addition of n-digit numbers n2 operations)

• Complexity is quadratic

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

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

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…

MAT 4 – Kompleks Funktionsteori

Exponential, polynomial, logarithmic functions

• Exponential growth an

• Polynomial growth na

• 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.

MAT 4 – Kompleks Funktionsteori

MAT 4 – Kompleks Funktionsteori

Complexity and recursion

• Example: factorial function is defined recursively

• t(n) is computational complexity:

• Example: Fibonacci numbers

• Computational complexity: