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# 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]

# Some 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

A way to implement a dictionary is a *Direct-access table*

### API of dictionary

- `Insert(D, k)` insert a ket `k` to dictionary `D`
- `Delete(D, k)` removes key `k`
- `Search(D, k)` tells whether `D` contains a key `k`

Many different implementations

# Direct-access tables

- universe of keys = {1,2,...,M}
- array `T` of size M
- each key has its own position in T

## The 'dumb' approach

```python

def Insert(D, x):
  D[x] == True

def Delete(D, x):
  D[x] == False

def Search(D, k):
  return D[x]

```

Everything is O(1), but memory complexity is awful.

Solve this by mapping keys to smaller keys using an hash function.

Table T is a table many times smaller than the universe and different by a small
constant factor from the set of keys stored S.

Instead of using keys directly, the index returned by the hash function is used.

**Pigeon hole problem:** More keys (pigeons) in the universe than holes (indexes
for table t)

## The solution: the *chained hash table*

For every cell in table T don't store True/False but a linked list of keys for
each index in the table. This is called *Chained hash table*.

```python

def Chained_hash_insert(T, k):
  return List_insert(T[hash(k)], k)

def Chained_hash_search(T, k):
  return List_search(T[hash(k)], k)

```

Elements are spreaded evenly across the table if the hash function is good.

*alpha* = *n / |T|*

is the average-case average length of the linked lists inside
the table (where n is the number of elements in the table and *|T|* is the size
of the table.

A good hash table implementation makes the complexity of *alpha* O(1).
If *n*, the number of elements that we want to store in the hash table, grows,
then *|T|* must also grow. *alpha* represents the time complexity of both
insertion and search.

## Growing a *chained hash table*

In order to grow a table, a new table must be created. The hash function (or
its range parameters) must be changed as well.

### Rehashing

*Rehashing* is the process of putting all the elements of the old table in the
new table according to the new hash function. The complexity is O(n), since
`Chained-hash-insert` is constant.

### Growing the table every time

If the table is grown by a constant factor every time the table *overflows*,
then the complexity of insertion is O(n^2) due to all the *rehashing* needed.

### Growing the table by doubling the size

If the table size is doubled when *overfloiwng*, then the complexity for
insertion becomes linear again by sacrificing some memory complexity.

# Binary search trees

Implementation of a dynamic set over a *totally ordered* (with an order relation
that has ...)

## Interface

- `Tree-Insert(T, k)` adds a key K to tree T;
- `Tree-Delete(T, k)` deletes the key K from tree T;
- `Tree-Search(T, k)` returns if key K is in the tree T;
- `Tree-Minimum(T)` finds the smallest element in the tree;
- `Tree-Maximum(T)` finds the biggest element in the tree;
- `Tree-successor(T, k)` and `Tree-predecessor(T, k)` find the next and
  previous position in the tree

## Height of a tree

The height of a tree is the maximum number of edges traversed from parent to
child in order to reach a leaf from the root of the tree.

## Rotation

```
                b
            /       \
          a           \
       /      \         \
     /          \         \
k <= a     a <= k <= b     k >= b
```

```
               a
           /       \
         /            b
       /          /       \
     /          /           \
k <= a     a <= k <= b     k >= b
```

# Red-Black trees

1. Every node has a color: red or black
2. The root is black
3. Every NULL leaf node is black
4. If a node is red, both of its children are black
5. The black height is the same for every branch in the tree

A red node is a way to strech the tree, but you cannot stretch it too far.

# B-trees

Disk access sucks (a billion times slower than registers). So, ignore the RAM
model and come up with a better solution.

If the number of keys per node is increased to k keys, the complexity of tree
search will be log_k+1(n). We can search between the keys linearly because we
have to wait for the disks. And log_1000() is much better than log_2().

## Implementation

- Minimum degree (min. num. of children in each node) defined as `t` (`t >= 2`)
- Every node other than the root must have at least `t - 1` keys
- Every node must contain at most `2t - 1` keys
- Every node has a boolean flag for leaf

### Insertion

- Read node from disk
- If full, split
- If leaf, insert in the right position
- If not leaf, recurse on the right branch based on the keys

### Search

- Read node from disk
- Linearly search between keys
  - If key found, return True
  - otherwise, if leaf, return False
  - otherwise, if not leaf, recurse on the right branch