2.2 KiB
Complexity
General way to describe efficiency algorithms (linear vs exponential) indipendent from the computer architecture/speed.
The RAM - random-access machine
Model of computer used in this course.
Has random-access memory.
Basic types and basic operations
Has basic types (like int, float, 64bit words). A basic step is an operation on a basic type (load, store, add, sub, ...). A branch is a basic step. Invoking a function and returning is a basic step as well, but the entire execution takes longer.
Complexity is not measured by the input value but by the input size in bits.
Fibonacci(10) in linear in n (size of the value) but exponential in l
(number of bits in n, or size of the input).
By default, WORST complexity is considered.
Donald Knuth's A-notation
A(c) indicates a quantity that is absolutely at most c
Antonio's weight = (pronounced "is") A(100)
(big-) O-notation
f(n) = O(g(n))
Definition: if f(n) is such that f(n) = k * A(g(n)) for all n sufficiently large and for some constant k > 0, then we say that
Complexity notations (lecture 2019-02-26)
Characterizing unknown functions
pi(n) = number of primes less than n
First approximation
Upper bound: linear function
pi(n) = O(n)
Lower bound: constant function
pi(n) = omega(1)
Non-trivial tight bound:
pi(n) = theta(n/log n)
Theta notation
Given a functio ng(n), we define the family of functions theta(g(n)) such that given a c_1, c_2 and an n_0, for all n >= n_0 g(n) is sandwiched between c_1g(n) and c_2g(n)
Big omega notation
Omega(g(n)) is a family of functions such that there exists a c and an n_0 such that for all n>= n_0 g(n) dominates c*g(n)
Big "oh" notation
O(g(n)) is a family of functions such that there exists a c and an n_0 such that for all n>= n_0 g(n) is dominated by c*g(n)
Small "oh" notation
o(g(n)) is the family of functions O(g(n)) excluding all the functions in theta(g(n))
Small omega notation
omega(g(n)) is the family of functions Omega(g(n)) excluding all the functions in theta(g(n))
Recap
asymptotically = <=> theta(g(n)) asymptotically < <=> o(g(n)) asymptotically > <=> omega(g(n)) asymptotically <= <=> O(g(n)) asymptotically >= <=> Omega(g(n))