DSA/notes.md

<!-- vim: set ts=2 sw=2 et tw=80: -->

# 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))

# Insertion sort

## Complexity

- *Best case:* Linear (theta(n))
- *Worst case:* Number of swaps = 1 + 2 + ... + n-1 = (n-1)n/2 = theta(n^2)
- *Average case:* Number of swaps half of worst case = n(n-1)/4 = theta(n^2)

## Correctness

Proof sort of by induction.

An algorithm is correct if given an input the output satisfies the conditions
stated. The algorithm must terminate.