145 lines
3.5 KiB
Markdown
145 lines
3.5 KiB
Markdown
<!-- vim: set ts=2 sw=2 et tw=80: -->
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# Complexity
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General way to describe efficiency algorithms (linear vs exponential)
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indipendent from the computer architecture/speed.
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## The RAM - random-access machine
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Model of computer used in this course.
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Has random-access memory.
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### Basic types and basic operations
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Has basic types (like int, float, 64bit words). A basic step is an operation on
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a basic type (load, store, add, sub, ...). A branch is a basic step. Invoking a
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function and returning is a basic step as well, but the entire execution takes
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longer.
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Complexity is not measured by the input value but by the input size in bits.
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`Fibonacci(10)` in linear in `n` (size of the value) but exponential in `l`
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(number of bits in `n`, or size of the input).
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By default, WORST complexity is considered.
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## Donald Knuth's A-notation
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A(c) indicates a quantity that is absolutely at most c
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Antonio's weight = (pronounced "is") A(100)
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## (big-) O-notation
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f(n) = O(g(n))
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*Definition:* if f(n) is such that f(n) = k * A(g(n)) for all _n_ sufficiently
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large and for some constant k > 0, then we say that
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# Complexity notations (lecture 2019-02-26)
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## Characterizing unknown functions
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pi(n) = number of primes less than n
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## First approximation
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*Upper bound:* linear function
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pi(n) = O(n)
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*Lower bound:* constant function
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pi(n) = omega(1)
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*Non-trivial tight bound*:
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pi(n) = theta(n/log n)
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## Theta notation
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Given a functio ng(n), we define the __family__ of functions theta(g(n)) such
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that given a c_1, c_2 and an n_0, for all n >= n_0 g(n) is sandwiched between
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c_1g(n) and c_2g(n)
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## Big omega notation
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Omega(g(n)) is a family of functions such that there exists a c and an n_0 such
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that for all n>= n_0 g(n) dominates c\*g(n)
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## Big "oh" notation
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O(g(n)) is a family of functions such that there exists a c and an n_0 such
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that for all n>= n_0 g(n) is dominated by c\*g(n)
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## Small "oh" notation
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o(g(n)) is the family of functions O(g(n)) excluding all the functions in
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theta(g(n))
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## Small omega notation
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omega(g(n)) is the family of functions Omega(g(n)) excluding all the functions
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in theta(g(n))
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## Recap
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*asymptotically* = <=> theta(g(n))
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*asymptotically* < <=> o(g(n))
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*asymptotically* > <=> omega(g(n))
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*asymptotically* <= <=> O(g(n))
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*asymptotically* >= <=> Omega(g(n))
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# Insertion sort
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## Complexity
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- *Best case:* Linear (theta(n))
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- *Worst case:* Number of swaps = 1 + 2 + ... + n-1 = (n-1)n/2 = theta(n^2)
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- *Average case:* Number of swaps half of worst case = n(n-1)/4 = theta(n^2)
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## Correctness
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Proof sort of by induction.
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An algorithm is correct if given an input the output satisfies the conditions
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stated. The algorithm must terminate.
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### The loop invariant
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Invariant condition able to make a loop equivalent to a straight path in an
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execution graph.
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# Heaps and Heapsort
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A data structure is a way to structure data.
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A binary heap is like an array and can be of two types: max heap and min heap.
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## Interface of an heap
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- `Build_max_heap(A)` and rearranges a into a max-heap;
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- `Heap_insert(H, key)` inserts `key` in the heap;
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- `Heap_extract_max(H)` extracts the maximum `key`;
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- `H.heap_size` returns the size of the heap.
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A binary heap is like a binary tree mapped on an array:
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```
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1
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/ \
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/ \
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2 3
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/ \ / \
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4 5 6 7
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=> [1234567]
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```
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The parent position of `n` is the integer division of n by 2:
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```python
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def parent(x):
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return x // 2
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```
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The left of `n` is `n` times 2, and the right is `n` times 2 plus 1:
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```python
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def left(x):
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return x * 2
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def right(x):
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return x * 2 + 1
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```
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**Max heap property**: for all i > 1 A[parent(i)] >= A[i]
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