<!-- 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.

### The loop invariant

Invariant condition able to make a loop equivalent to a straight path in an
execution graph.

# Heaps and Heapsort

A data structure is a way to structure data.
A binary heap is like an array and can be of two types: max heap and min heap.

## Interface of an heap
- `Build_max_heap(A)` and rearranges a into a max-heap;
- `Heap_insert(H, key)` inserts `key` in the heap;
- `Heap_extract_max(H)` extracts the maximum `key`;
- `H.heap_size` returns the size of the heap.

A binary heap is like a binary tree mapped on an array:

```
    1
   / \
  /   \
 2     3
/ \   / \
4 5   6 7

=> [1234567]
```

The parent position of `n` is the integer division of n by 2:

```python
def parent(x):
  return x // 2
```

The left of `n` is `n` times 2, and the right is `n` times 2 plus 1:

```python
def left(x):
  return x * 2

def right(x):
  return x * 2 + 1
```

**Max heap property**: for all i > 1 A[parent(i)] >= A[i]

# Data structures

Way to organize information

A data structure has an interface (functions to work with the DS)

A data structure has data and meta-data (like size, length).

## Stack (LIFO)

### Operations:
- `push(S, x)` (put one element, move TOS)
- `pop(S)` (remove element in TOS, move TOS)
- `stack-empty(S)` (returns TRUE if stack is empty)

## Queue (FIFO)

## Structure

- Based on array
- `length`
- `head`
- `tail`

Queue has always 1 cell free to avoid confusion with full/empty

## Dictionary

Data structure for fast search

## Direct-access table

Implementation of dictionary