Graphs
Telerik Academy Alpha
DSA
Table of contents
What is a graph?
What is a graph?
- Set of nodes with many-to-many relationship between them is called graph
- Each node has multiple predecessors
- Each node has multiple successors
Where we use graphs?
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Few important real life applications of graph data structures are:
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Facebook: Each user is represented as a vertex(node) and two people are friends when there is an edge between two vertices. Similarly friend suggestion also uses graph theory concept.
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Google Maps: Various locations are represented as vertices(nodes) and the roads are represented as edges and graph theory is used to find shortest path between two nodes.
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Where we use graphs?
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Friends
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Different Connections
Where we use graphs?
-
Few important real life applications of graph data structures are:
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Recommendations on e-commerce websites: The “Recommendations for you” section on various e-commerce websites uses graph theory to recommend items of similar type to user’s choice.
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Graph theory is also used to study molecules in chemistry and physics.
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Graph definitions
Node, edge
- Node(vertex)
- Element of graph
- Can have name or value
- Keeps a list of adjacent nodes
- Edge
- Connection between two nodes
- Can be directed / undirected
- Can be weighted / unweighted
- Can have name / value
Directed graph
- Directed graph
- The edges have directions
Directed graph applications
| graph | vertex | edge |
|---|---|---|
| financial stock | currency | transaction |
| transportation | street intersection, airport | highway, airway route |
| telephone | person | placed call |
One-way streets
Undirected graph
- Undirected graph
- The edges have NO directions
Undirected graph applications
| graph | vertex | edge |
|---|---|---|
| communication | telephones, computers | fiber optic cables |
| mechanical | joints | rods, beams, springs |
| financia | stocks, currency | transactions |
Friend connections
Weighted graph
- Weight (cost) is associated with each edge
Paths
- Path (in undirected graph)
- Sequence of nodes \( n_1, n_2, ..., n_k \)
- Edge exists between each pair of nodes \( ni, ni+1 \)
- Examples:
- A,B,C is a path
- H,K,C is not a path
Paths
- Path (in directed graph)
- Sequence of nodes \( n_1, n_2, ..., n_k \)
- Direct edge exists between each pair of nodes \( ni, ni+1 \)
- Examples:
- A,B,C is a path
- A,G,K is not a path
Cycle
- Cycle
- Path that ends back at the starting node
- Example:
- A,B,C,G,A
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Simple path
- No cycles in path
- Acyclic graph
- Graph with no cycles
- Acyclic undirected graphs are trees
Connections
- Two nodes are reachable if
- Path exists between them
- Connected graph
- Every node is reachable from any other node
Connections
- Connected graph
- Unconnected graph
Representing graphs
Representing graphs
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Adjacency list
- Each node holds a list of its neighbors
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Adjacency matrix
- Each cell keeps whether and how two nodes are connected
- Set of edges
C# Representation - simple
using System.Collections.Generic;
public class Program
{
public static void Main()
{
Graph g = new Graph(new List<int>[] {
new List<int> {3, 6}, // successors of vertice 0
new List<int> {2, 3, 4, 5, 6},// successors of vertice 1
new List<int> {1, 4, 5}, // successors of vertice 2
new List<int> {0, 1, 5}, // successors of vertice 3
new List<int> {1, 2, 6}, // successors of vertice 4
new List<int> {1, 2, 3}, // successors of vertice 5
new List<int> {0, 1, 4} // successors of vertice 6
});
}
}
public class Graph
{
List<int>[] childNodes;
public Graph(List<int>[] nodes)
{
this.childNodes = nodes;
}
}
C# Representation - advanced
- Write your own using OOP (as an exercise):
- Class Node
- Class Connection (Edge)
- Class Graph
- Optional classes
- Using external library (redommended):
- QuickGraph