- Data Structures Overview
- Linear Structures, Trees, Hash Tables, Others

- Algorithms Overview
- Algorithms Complexity
- Time and Memory Complexity
- Mean, Average and Worst Case
- Asymptotic Notation
`O(g)`

“In computer science, a *data structure* is a particular way of storing and organizing data in a computer so that it can be used efficiently.”

-- Wikipedia

- Examples of data structures:
`Person`

structure (first name + last name + age)- Array of integers –
`int[]`

- List of strings –
`List`

- Queue of people –
`Queue`

`Data structures`

and`algorithms`

are the foundation of computer programming- Algorithmic thinking, problem solving and data structures are vital for software engineers
- All .NET developers should know when to use
`T[]`

,`LinkedList`

,`List`

,`Stack`

,`Queue`

,`Dictionary<K,T>`

,`HashSet`

,`SortedDictionary<K,T>`

and`SortedSet`

- All .NET developers should know when to use
`Computational complexity`

is important for algorithm design and efficient programming

- Primitive data types
- Numbers:
`int`

,`float`

,`decimal`

, … - Text data:
`char`

,`string`

, …

- Numbers:
- Simple structures
- A group of fields stored together
- E.g.
`DateTime`

,`Point`

,`Rectangle`

, …

- Collections
- A set of elements (of the same type)
- E.g. array, list, stack, tree, hash-table, …

`An Abstract Data Type (ADT)`

is- A data type together with the operations, whose properties are specified independently of any particular implementation
- ADT are set of definitions of operations
- Like the interfaces in C#

- ADT can have multiple different
`implementations`

- Different implementations can have different efficiency, inner logic and resource needs

- Linear structures
- Lists: fixed size and variable size
- Stacks: LIFO (Last In First Out) structure
- Queues: FIFO (First In First Out) structure

- Trees and tree-like structures
- Binary, ordered search trees, balanced, B-trees, etc.

- Dictionaries (maps)
- Contain pairs (key, value)
- Hash tables: use hash functions to search/insert

- Sets and bags
- Set – collection of unique elements
- Bag – collection of non-unique elements

- Ordered sets, bags and dictionaries
- Priority queues / heaps
- Special tree structures
- Suffix tree, interval tree, index tree, trie

- Graphs
- Directed / undirected
- Weighted / un-weighted
- Connected / non-connected, …

"In mathematics and computer science, an *algorithm* is a step-by-step procedure for calculations. An algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function.”

-- Wikipedia

- The term “algorithm” comes from the
- Derived from
`Muḥammad Al-Khwārizmī`

, a Persian mathematician and astronomer- An algorithm for solving quadratic equations

- Derived from

- Algorithms are fundamental in programming
- Imperative (traditional) programming means to
`describe in formal`

steps how to do something - Algorithm == sequence of operations (steps)
- Can include branches (conditional blocks) and repeated logic (loops)

- Imperative (traditional) programming means to
- Algorithmic thinking (mathematical thinking, logical thinking, engineering thinking)
- Ability to decompose the problems into formal sequences of steps (algorithms)

- Algorithms can be expressed in pseudocode, through flowcharts or program code

```
BFS(node)
{
queue <- node
while queue not empty
v <- queue
print v
for each child c of v
queue <- c
}
```

- Sorting and searching
- Dynamic programming
- Graph algorithms
- DFS and BFS traversals

- Combinatorial algorithms
- Recursive algorithms

- Other algorithms
- Greedy algorithms, computational geometry, randomized algorithms, genetic algorithms

- Why we should analyze algorithms?
- Predict the resources the algorithm requires
- Computational time (CPU consumption)
- Memory space (RAM consumption)
- Communication bandwidth consumption

- The
`running time`

of an algorithm is:- The total number of primitive operations executed (machine independent steps)
- Also known as
`algorithm complexity`

- Predict the resources the algorithm requires

- What to measure?
- CPU Time
- Memory
- Number of steps
- Number of particular operations
- Number of disk operations
- Number of network packets

- Asymptotic complexity

`Worst-case`

- An upper bound on the running time for any input of given size

`Average-case`

- Assume all inputs of a given size are equally likely

`Best-case`

- The lower bound on the running time (the optimal case)

- Sequential search in a list of size
`n`

- Worst-case:
`n`

comparisons

- Best-case:
`1`

comparison

- Average-case:
`n/2`

comparisons

- Worst-case:
- The algorithm runs in linear time
- Linear number of operations

`Algorithm complexity`

is a rough estimation of the number of steps performed by given computation depending on the size of the input data- Measured through
`asymptotic notation`

`O(g)`

where`g`

is a function of the input data size

- Examples:
- Linear complexity
`O(n)`

– all elements are processed once (or constant number of times) - Quadratic complexity
`O(n`

^{2}`)`

– each of the elements is processed`n`

times

- Linear complexity

- Measured through

- Asymptotic upper bound
- O-notation (Big O notation)

- For given function
`g(n)`

, we denote by`O(g(n))`

the set of functions that are different than`g(n)`

by a constant

`O(g(n))`

= {`f(n)`

: there exist positive constants `c`

and `n`

`f(n) <= c*g(n)`

for all `n>=n`

- Examples:
- 3 * n
^{2}+ n/2 + 12 ∈ O(n^{2}) - 4
*n*log_{2}(3*n+1) + 2*n-1 ∈ O(n * log n)

- 3 * n

Complexity | Notation | Description |
---|---|---|

constant | O(1) | Constant number of operations, not depending on the input data size, e.g. n = 1 000 000 → 1-2 operations |

logarithmic | O(log n) | Number of operations propor-tional of log2(n) where n is the size of the input data, e.g. n = 1 000 000 000 → 30 operations |

linear | O(n) | Number of operations proportional to the input data size, e.g. n = 10 000 → 5 000 operations |

Complexity | Notation | Description |
---|---|---|

quadratic | O(n^{2}) |
Number of operations proportional to the square of the size of the input data, e.g. n = 500 → 250 000 operations |

cubic | O(n^{3}) |
Number of operations propor-tional to the cube of the size of the input data, e.g. n = 200 → 8 000 000 operations |

exponential | O(2^{n}),O(k ^{n}),O(n!) |
Exponential number of operations, fast growing, e.g. n = 20 → 1 048 576 operations |

Complexity | 10 | 20 | 50 | 100 | 1000 | 10000 | 100000 |
---|---|---|---|---|---|---|---|

O(1) | < 1 s | < 1 s | < 1 s | < 1 s | < 1 s | < 1 s | < 1 s |

O(log(n)) | < 1 s | < 1 s | < 1 s | < 1 s | < 1 s | < 1 s | < 1 s |

O(n) | < 1 s | < 1 s | < 1 s | < 1 s | < 1 s | < 1 s | < 1 s |

O(n*log(n)) | < 1 s | < 1 s | < 1 s | < 1 s | < 1 s | < 1 s | < 1 s |

O(n^{2}) |
< 1 s | < 1 s | < 1 s | < 1 s | < 1 s | 2 s | 3-4 min |

O(n^{3}) |
< 1 s | < 1 s | < 1 s | < 1 s | 20 s | 5 hours | 231 days |

O(2^{n}) |
< 1 s | < 1 s | 260 days | hangs | hangs | hangs | hangs |

O(n!) | < 1 s | hangs | hangs | hangs | hangs | hangs | hangs |

O(n^{n}) |
3-4 min | hangs | hangs | hangs | hangs | hangs | hangs |

- Complexity can be expressed as formula on multiple variables, e.g.
- Algorithm filling a matrix of size [
`n`

x`m`

] with the natural numbers 1, 2, … will run in`O(n*m)`

- A traversal of graph with
`n`

vertices and`m`

edges will run in`O(n + m)`

- Algorithm filling a matrix of size [
- Memory consumption should also be considered, for example:
- Running time
`O(n)`

& memory requirement`O(n`

^{2}`)`

- n = 50 000 →
`OutOfMemoryException`

- Running time

- Sometimes a linear algorithm could be slower than quadratic algorithm
- The hidden constant could be significant

- Example:
- Algorithm A makes:
`100*n`

steps →`O(n)`

- Algorithm B makes:
`n*n/2`

steps →`O(n`

^{2}`)`

- For
`n < 200`

the algorithm B is faster

- Algorithm A makes:
- Real-world example:
- Insertion sort is faster than quicksort for n <= 16

- A
`polynomial-time`

algorithm is one whose worst-case time complexity is bounded above by a polynomial function of its input size

`W(n) ∈ O(p(n))`

- Examples:
- Polynomial-time: log(n), n
^{2}, 3n^{3}+ 4n, 2 * n log(n) - Non polynomial-time : 2
^{n}, 3^{n}, n^{k}, n! - Non-polynomial algorithms hang for large input data sets

- Polynomial-time: log(n), n

- Computational complexity theory divides the computational problems into several classes:

```
int FindMaxElement(int[] array)
{
int max = array[0];
for (int i = 0; i < array.length; i++)
{
if (array[i] > max)
{
max = array[i];
}
}
return max;
}
```

- Runs in
`O(n)`

where`n`

is the size of the array

- The number of elementary steps is
`~n`

```
long FindInversions(int[] array)
{
long inversions = 0;
for (int i = 0; i < array.Length; i++)
for (int j = i + 1; j < array.Length; i++)
if (array[i] > array[j])
inversions++;
return inversions;
}
```

- Runs in
`O(n`

^{2}`)`

where`n`

is the size of the array

- The number of elementary steps is
`~n*(n+1)/2`

```
decimal Sum3(int n)
{
decimal sum = 0;
for (int a = 0; a < n; a++)
for (int b = 0; b < n; b++)
for (int c = 0; c < n; c++)
sum += a * b * c;
return sum;
}
```

- Runs in cubic time
`O(n`

^{3}`)`

- The number of elementary steps is
`~n`

^{3}

```
long SumMN(int n, int m)
{
long sum = 0;
for (int x = 0; x < n; x++)
for (int y = 0; y < m; y++)
sum += x * y;
return sum;
}
```

- Runs in quadratic time
`O(n*m)`

- The number of elementary steps is
`~n*m`

```
long SumMN(int n, int m)
{
long sum = 0;
for (int x = 0; x < n; x++)
for (int y = 0; y < m; y++)
if (x == y)
for (int i = 0; i < n; i++)
sum += i * x * y;
return sum;
}
```

- Runs in quadratic time
`O(n*m)`

- The number of elementary steps is
`~n*m + min(m,n)*n`

```
decimal Calculation(int n)
{
decimal result = 0;
for (int i = 0; i < (1 << n); i++)
result += i;
return result;
}
```

- Runs in exponential time
`O(2`

^{n}`)`

- The number of elementary steps is
`~2`

^{n}

```
decimal Factorial(int n)
{
if (n == 0)
return 1;
else
return n * Factorial(n-1);
}
```

- Runs in linear time
`O(n)`

- The number of elementary steps is
`~n`

```
decimal Fibonacci(int n)
{
if (n == 0)
return 1;
else if (n == 1)
return 1;
else
return Fibonacci(n-1) + Fibonacci(n-2);
}
```

- Runs in exponential time
`O(2`

^{n}`)`

- The number of elementary steps is
`~Fib(n+1)`

where`Fib(k)`

is the`k`

-th Fibonacci’s number

- Data structures organize data for efficient use
- ADT describe a set of operations
- Collections hold a group of elements

- Algorithms are sequences of steps for performing or calculating something
- Algorithm complexity is rough estimation of the number of steps performed by given computation
- Complexity can be logarithmic, linear, n log n, square, cubic, exponential, etc.
- Allows to estimating the speed of given code before its execution

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