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MATEMATIK 4
INDUKTION 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:
MAT 4 – Kompleks Funktionsteori