# algorithmic

A collection of useful algorithms keeping performance and flexibility on mind.

## Usage

The following import will give you access to all the algorithms in this package.

```
import 'package:algorithmic/algorithmic.dart';
```

You can also import these algorithms separately:

Libraries | Imported By |
---|---|

Search algorithms | `'package:algorithmic/searching.dart'` |

Sort algorithms | `'package:algorithmic/sorting.dart'` |

String algorithms | `'package:algorithmic/string.dart'` |

## Benchmarks

To run benchmark on your own machine:

```
$ dart run benchmark
```

You can check the benchmark.log file for the benchmark results.

## Searching algorithms

Index searching algorithms attempt to find the index of an item from a list.

### Binary Search

A faster searching algorithm for sorted list of items. It divides the list into two parts and discard one based on the middle value of them. This requires the list to be sorted in an increasing order.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`lowerBound()` |
O(`log` n) |
✔️ | ✔️ | 0.0.3 |

`upperBound()` |
O(`log` n) |
✔️ | ✔️ | 0.0.3 |

`binarySearch()` |
O(`log` n) |
✔️ | ✔️ | 0.0.3 |

`binarySearchUpper()` |
O(`log` n) |
✔️ | ✔️ | 0.0.6 |

`binarySearchQuick()` |
O(`log` n) |
✔️ | ✔️ | 0.0.6 |

### Linear Search

A general searching algorithm for any kind of list. It tests every elements on the list one by one.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`linearSearch()` |
O(n) |
✔️ | ✔️ | 0.0.1 |

`linearSearchBy()` |
O(n) |
✔️ | ✔️ | 0.0.4 |

`linearSearchReversed()` |
O(n) |
✔️ | ✔️ | 0.0.1 |

`linearSearchReversedBy()` |
O(n) |
✔️ | ✔️ | 0.0.4 |

## Sorting algorithms

Sorting algorithms puts a list of items into an increasing order.

### Bubble Sort

Bubble sort performs sorting by repeatedly stepping through the list, comparing adjacent elements and swapping them if they are in the wrong order.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`bubbleSort()` |
O(n²) |
✔️ | ✔️ | 0.0.5 |

### Comb Sort

Comb sort improves bubble sort by eliminating small values near the end of the list, since they slows down bubble sort. It has `O(n)`

time complexity for an already sorted list.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`combSort()` |
O(n²) |
✔️ | ✔️ | 0.0.8 |

### Cocktail Shaker Sort

Cocktail shaker sort extends bubble sort by operating in two directions. It has `O(n)`

time complexity for an already sorted list.

This algorithm is known by many other names, e.g.: bidirectional bubble sort, cocktail sort, shaker sort, ripple sort, shuffle sort, shuttle sort etc.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`cocktailShakerSort()` |
O(n²) |
✔️ | ✔️ | 0.0.8 |

### Selection Sort

Selection sort algorithm performs sorting by finding the minimum element from the unordered range and putting it at the beginning in each iteration.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`selectionSort()` |
O(n²) |
✔️ | ✔️ | 0.0.5 |

### Insertion Sort

Insertion sort sorts splits the list into two parts: the left part is ordered and the right part is unordered. In each iteration, it removes the first item from right part, and insert it into the left part maintaining the increasing order.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`insertionSort()` |
O(n²) |
✔️ | ✔️ | 0.0.5 |

### Gnome Sort

Gnome sort is a variation to the insertion sort which uses a much simpler implementation. It has `O(n)`

time complexity for an already sorted list.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`gnomeSort()` |
O(n²) |
✔️ | ✔️ | 0.0.5 |

### Quick Sort

Quicksort is an in-place sorting algorithm that works by selecting a *pivot* element and partitioning the list surrounding it. There are several schemes for selecting the pivot. The sorting performance varies for different schemes.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`quickSortLomuto()` |
O(n `log` n) |
✔️ | ✔️ | 0.0.6 |

`quickSortHaore()` |
O(n `log` n) |
✔️ | ✔️ | 0.0.7 |

### Merge Sort

Merge sort is a divide and conquer based algorithm, which can handle very large arrays. Unlike quick sort, it promises `O(n log n)`

performance in worst case and provides stable sort. But it requires equal amount of memory as the length of the array and the implementation runs slower compared than quick sort.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`mergeSort()` |
O(n `log` n) |
✔️ | ✔️ | 0.0.8 |

### Counting Sort

Counting sort is a specialized sorting algorithm to sort integers of ranges in linear time. It counts the frequencies of the numbers appearing in an array, and uses it to reconstruct a sorted list.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`countingSort()` |
O(n + k) |
✔️ | ✔️ | 0.0.9 |

`countingSortOf()` |
O(n + k) |
✔️ | ✔️ | 0.0.9 |

Here,

`k`

is the range of numbers

### Radix Sort

Radix sort can sort a range of integers without using any comparisons. It is a special form of bucket sort, which distribute elements into buckets according to their radix.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`radixSort()` |
O(n `log_b` n) |
✔️ | ✔️ | 0.0.9 |

`radixSortOf()` |
O(n `log_b` n) |
✔️ | ✔️ | 0.0.9 |

Here,

`b`

is the radix

## String Algorithms

### String Metrics

#### Levenshtein Distance

Levenshtein distance returns the minimum number of ediit operations to transform one string (or array) to another. The permitted operations here are *insertion*, *deletion* and *substitution*. The Levenshtein distance between **ABCD** and **BED** is `2`

.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`levenshteinDistance()` |
O(n m) |
✔️ | ✔️ | 0.0.9 |

`levenshteinDistanceOf()` |
O(n m) |
✔️ | ✔️ | 0.0.9 |

Here,

`n`

and`m`

are the length of first and second string respectively.

#### Damerau-Levenshtein Distance

Damerau–Levenshtein distance is a variation of Lavenshtein distance which permits *transposition* of two adjacent characters along with *insertion*, *deletion* and *substitution*. e.g.: The Damerau-Levenshtein distance between **CA** and **ABC** is `2`

.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`damerauLevenshteinDistance()` |
O(n m `log` k) |
✔️ | ✔️ | 0.0.10 |

`damerauLevenshteinDistanceOf()` |
O(n m `log` k) |
✔️ | ✔️ | 0.0.10 |

Here,

`n`

and`m`

are the length of first and second string respectively and`k`

is number of unique characters appearing in both string.

#### Restricted Damerau-Levenshtein Distance / Optimal String Alignment Distance

The Damerau–Levenshtein distance can be restricted with a condition that no substring can be edited more than once, which simplifies the implementation of transposition lookup. This distance is also known as *Optimal String Alignment* distance. e.g.: The restricted Damerau-Levenshtein edit distance between **CA** and **ABC** is `3`

.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`restrictedDamerauDistance()` |
O(n m) |
✔️ | ✔️ | 0.0.10 |

`restrictedDamerauDistanceOf()` |
O(n m) |
✔️ | ✔️ | 0.0.10 |

Here,

`n`

and`m`

are the length of first and second string respectively.

#### Hamming Distance

Hamming distance between two equal-length strings is the number of positions at which the characters differs. In this implementation, if two strings are not equal, the shorter string is padded to match the longer ones.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`hammingDistance()` |
O(n) |
✔️ | ➖ | 0.0.10 |

`hammingDistanceOf()` |
O(n) |
✔️ | ➖ | 0.0.10 |

#### Lee Distance

Lee distance can only be calculated between two equal-length strings.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`leeDistance()` |
O(n) |
✔️ | ➖ | 0.0.10 |

`leeDistanceOf()` |
O(n) |
✔️ | ➖ | 0.0.10 |

#### Longest Common Subsequence Length

Longest Common Subsequence is the longest of the common subsequences of two strings.

Longest common substrings and longest common subsequences are not the same. Unlike substrings, subsequences are not required to occupy consecutive positions.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`longestCommonSubsequenceLength()` |
O(n m) |
✔️ | ✔️ | 0.0.10 |

`longestCommonSubsequenceLengthOf()` |
O(n m) |
✔️ | ✔️ | 0.0.10 |

Here,

`n`

and`m`

are the length of first and second string respectively.

#### Tversky Index

Tversky similarity index an asymmetric similarity measure between sets that compares a variant with a prototype.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`tverskyIndex()` |
O(n `log` n) |
✔️ | ✔️ | 0.0.10 |

`tverskyIndexOf()` |
O(n `log` n) |
✔️ | ✔️ | 0.0.10 |

#### Dice coefficient / Sørensen index

Sørensen–Dice coefficient is a metric used to measure similarity between two samples.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`diceIndex()` |
O(n `log` n) |
✔️ | ✔️ | 0.0.10 |

`diceIndexOf()` |
O(n `log` n) |
✔️ | ✔️ | 0.0.10 |

#### Jaccard index / Tanimoto coefficient

Jaccard index is a metric used to measure similarity of two samples sets.

The complement of it is *Jaccard distance* which measures the total number of items that is present in one list but not the other.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`jaccardIndex()` |
O(n `log` n) |
✔️ | ✔️ | 0.0.10 |

`jaccardIndexOf()` |
O(n `log` n) |
✔️ | ✔️ | 0.0.10 |

`jaccardDistance()` |
O(n `log` n) |
✔️ | ✔️ | 0.0.10 |

`jaccardDistanceOf()` |
O(n `log` n) |
✔️ | ✔️ | 0.0.10 |

#### Jaro similarity & Jaro-Winkler distance

The Jaro similarity between two strings is the weighted sum of percentage of matched characters from each string and transposed characters. Winkler increased this measure for matching initial characters.

Functions | Performance | Tests | Benchmark | Since |
---|---|---|---|---|

`jaroSimilarity()` |
O(n m) |
✔️ | ✔️ | 0.0.10 |

`jaroSimilarityOf()` |
O(n m) |
✔️ | ✔️ | 0.0.10 |

`jaroWinklerSimilarity()` |
O(n m) |
✔️ | ✔️ | 0.0.10 |

`jaroWinklerSimilarityOf()` |
O(n m) |
✔️ | ✔️ | 0.0.10 |

Here,

`n`

and`m`

are the length of first and second string respectively.

## Links

## Libraries

- algorithmic
- A collection of all available algorithms.
- algorithmic.searching
- A collection of index search algorithms.
- algorithmic.sorting
- A collection of list sorting algorithms.
- algorithmic.string
- A collection of algorithms for strings or similar data structures.
- algorithmic.string.metrics
- A collection of algorithms for strings or similar data structures.