# Directed Graph

## Introduction

An integral part of storing, manipulating, and retrieving numerical data are *data structures* or as they are called in Dart: collections.
Arguably the most common data structure is the *list*. It enables efficient storage and retrieval of sequential data that can be associated with an index.

A more general (non-linear) data structure where an element may be connected to one, several, or none of the other elements is called a **graph**.

Graphs are useful when keeping track of elements that are linked to or are dependent on other elements. Examples include: network connections, links in a document pointing to other paragraphs or documents, foreign keys in a relational database, file dependencies in a build system, etc.

The package `directed_graph`

contains an implementation of a Dart graph that follows the
recommendations found in graphs-examples and is compatible with the algorithms provided by `graphs`

.
It is simple to use and includes methods that enable:

- adding/removing vertices and edges,
- the sorting of vertices.

The library provides access to algorithms for finding:

- the shortest path between vertices,
- the path with the lowest/highest weight (for weighted directed graphs),
- all paths connecting two vertices,
- cycles,
- a topological ordering of the graph vertices.

The class `GraphCrawler`

can be used to retrieve *paths* or *walks* connecting two vertices.

## Terminology

Elements of a graph are called **vertices** (or nodes) and neighbouring vertices are connected by **edges**.
The figure below shows a **directed graph** with unidirectional edges depicted as arrows.
Graph edges are emanating from a vertex and ending at a vertex. In a **weighted directed graph** each
edge is assigned a weight.

**In-degree**of a vertex: Number of edges ending at this vertex. For example, vertex H has in-degree 3.**Out-degree**of a vertex: Number of edges starting at this vertex. For example, vertex F has out-degree 1.**Source**: A vertex with in-degree zero is called (local) source. Vertices A and D in the graph above are local sources.**Directed Edge**: An ordered pair of connected vertices (v_{i}, v_{j}). For example, the edge (A, C) starts at vertex A and ends at vertex C.**Path**: A path [v_{i}, ..., v_{n}] is an ordered list of at least two connected vertices where each**inner**vertex is**distinct**. The path [A, E, G] starts at vertex A and ends at vertex G.**Cycle**: A cycle is an ordered*list*of connected vertices where each inner vertex is distinct and the first and last vertices are identical. The sequence [F, I, K, F] completes a cycle.**Walk**: A walk is an ordered*list*of at least two connected vertices. [D, F, I, K, F] is a walk but not a path since the vertex F is listed twice.**DAG**: An acronym for**Directed Acyclic Graph**, a directed graph without cycles.**Topological ordering**: An ordered*set*of all vertices in a graph such that v_{i}occurs before v_{j}if there is a directed edge (v_{i}, v_{j}). A topological ordering of the graph above is: {A, D, B, C, E, K, F, G, H, I, L}. Hereby, dashed edges were disregarded since a cyclic graph does not have a topological ordering.

**Note**: In the context of this package the definition of *edge* might be more lax compared to a
rigorous mathematical definition.
For example, self-loops, that is edges connecting a vertex to itself are explicitly allowed.

## Usage

To use this library include `directed_graph`

as a dependency in your pubspec.yaml file. The
example below shows how to construct an object of type `DirectedGraph`

.
The constructor takes an optional comparator function
as parameter. If a comparator is specified, vertices are sorted accordingly. For more information see comparator.

```
import 'package:directed_graph/directed_graph.dart';
// To run this program navigate to
// the folder 'directed_graph/example'
// in your terminal and type:
//
// # dart bin/directed_graph_example.dart
//
// followed by enter.
void main() {
int comparator(String s1, String s2) => s1.compareTo(s2);
int inverseComparator(String s1, String s2) => -comparator(s1, s2);
// Constructing a graph from vertices.
final graph = DirectedGraph<String>(
{
'a': {'b', 'h', 'c', 'e'},
'b': {'h'},
'c': {'h', 'g'},
'd': {'e', 'f'},
'e': {'g'},
'f': {'i'},
'i': {'l'},
'k': {'g', 'f'}
},
comparator: comparator,
);
print('Example Directed Graph...');
print('graph.toString():');
print(graph);
print('\nIs Acylic:');
print(graph.isAcyclic);
print('\nStrongly connected components:');
print(graph.stronglyConnectedComponents);
print('\nShortestPath(d, l):');
//print(graph.shortestPath('d', 'l');
print('\nInDegree(C):');
print(graph.inDegree('c'));
print('\nOutDegree(C)');
print(graph.outDegree('c'));
print('\nVertices sorted in lexicographical order:');
print(graph.sortedVertices);
print('\nVertices sorted in inverse lexicographical order:');
graph.comparator = inverseComparator;
print(graph.sortedVertices);
graph.comparator = comparator;
print('\nInDegreeMap:');
print(graph.inDegreeMap);
print('\nSorted Topological Ordering:');
print(graph.sortedTopologicalOrdering);
print('\nTopological Ordering:');
print(graph.topologicalOrdering);
print('\nLocal Sources:');
print(graph.localSources);
// Add edge to render the graph cyclic
graph.addEdges('i', {'k'});
graph.addEdges('l', {'l'});
graph.addEdges('i', {'d'});
print('\nCycle:');
print(graph.cycle);
print('\nShortest Paths:');
print(graph.shortestPaths('a'));
}
```

## Click to show the console output.

```
$ dart example/bin/directed_graph_example.dart
Example Directed Graph...
graph.toString():
{
'a': {'b', 'h', 'c', 'e'},
'b': {'h'},
'c': {'h', 'g'},
'd': {'e', 'f'},
'e': {'g'},
'f': {'i'},
'g': {},
'h': {},
'i': {'l'},
'k': {'g', 'f'},
'l': {},
}
Is Acylic:
true
Strongly connected components:
[[h], [b], [g], [c], [e], [a], [l], [i], [f], [d], [k]]
ShortestPath(d, l):
InDegree(C):
1
OutDegree(C)
2
Vertices sorted in lexicographical order:
[a, b, c, d, e, f, g, h, i, k, l]
Vertices sorted in inverse lexicographical order:
[l, k, i, h, g, f, e, d, c, b, a]
InDegreeMap:
{a: 0, b: 1, h: 3, c: 1, e: 2, g: 3, d: 0, f: 2, i: 1, l: 1, k: 0}
Sorted Topological Ordering:
{a, b, c, d, e, h, k, f, g, i, l}
Topological Ordering:
{a, b, c, d, e, h, k, f, i, g, l}
Local Sources:
[[a, d, k], [b, c, e, f], [g, h, i], [l]]
Cycle:
[l, l]
Shortest Paths:
{e: (e), c: (c), h: (h), a: (), g: (c, g), b: (b)}
```

## Weighted Directed Graphs

The example below shows how to construct an object of type `WeightedDirectedGraph`

.
Initial graph edges are specified in the form of map of type `Map<T, Map<T, W>>`

. The vertex type `T`

extends
`Object`

and therefore must be a non-nullable. The type associated with the edge weight `W`

extends `Comparable`

to enable sorting of vertices by their edge weight.

The constructor takes an optional comparator function as parameter. If a comparator is specified, vertices are sorted accordingly. For more information see comparator.

```
import 'package:directed_graph/directed_graph.dart';
void main(List<String> args) {
int comparator(
String s1,
String s2,
) {
return s1.compareTo(s2);
}
final a = 'a';
final b = 'b';
final c = 'c';
final d = 'd';
final e = 'e';
final f = 'f';
final g = 'g';
final h = 'h';
final i = 'i';
final k = 'k';
final l = 'l';
int sum(int left, int right) => left + right;
var graph = WeightedDirectedGraph<String, int>(
{
a: {b: 1, h: 7, c: 2, e: 40, g:7},
b: {h: 6},
c: {h: 5, g: 4},
d: {e: 1, f: 2},
e: {g: 2},
f: {i: 3},
i: {l: 3, k: 2},
k: {g: 4, f: 5},
l: {l: 0}
},
summation: sum,
zero: 0,
comparator: comparator,
);
print('Weighted Graph:');
print(graph);
print('\nNeighbouring vertices sorted by weight:');
print(graph..sortEdgesByWeight());
final lightestPath = graph.lightestPath(a, g);
print('\nLightest path a -> g');
print('$lightestPath weight: ${graph.weightAlong(lightestPath)}');
final heaviestPath = graph.heaviestPath(a, g);
print('\nHeaviest path a -> g');
print('$heaviestPath weigth: ${graph.weightAlong(heaviestPath)}');
final shortestPath = graph.shortestPath(a, g);
print('\nShortest path a -> g');
print('$shortestPath weight: ${graph.weightAlong(shortestPath)}');
}
```

## Click to show the console output.

```
$ dart example/bin/weighted_graph_example.dart
Weighted Graph:
{
'a': {'b': 1, 'h': 7, 'c': 2, 'e': 40, 'g': 7},
'b': {'h': 6},
'c': {'h': 5, 'g': 4},
'd': {'e': 1, 'f': 2},
'e': {'g': 2},
'f': {'i': 3},
'g': {},
'h': {},
'i': {'l': 3, 'k': 2},
'k': {'g': 4, 'f': 5},
'l': {'l': 0},
}
Neighbouring vertices sorted by weight
{
'a': {'b': 1, 'c': 2, 'h': 7, 'g': 7, 'e': 40},
'b': {'h': 6},
'c': {'g': 4, 'h': 5},
'd': {'e': 1, 'f': 2},
'e': {'g': 2},
'f': {'i': 3},
'g': {},
'h': {},
'i': {'k': 2, 'l': 3},
'k': {'g': 4, 'f': 5},
'l': {'l': 0},
}
Lightest path a -> g
[a, c, g] weight: 6
Heaviest path a -> g
[a, e, g] weigth: 42
Shortest path a -> g
[a, g] weight: 7
```

## Examples

For further information on how to generate a topological sorting of vertices see example.

## Features and bugs

Please file feature requests and bugs at the issue tracker.

## Libraries

- directed_graph
- Dart implementation of directed graphs. Provides methods to [...]