Algorithms
Breadth First Search (BFS)
Depth First Search (DFS)
Dijkstra’s Algorithm

Famous and Productive Problem of Graph
Theory
Four Color Problem: Is it true that any map drawn in the plane may have its regions colored with four colors in such a way that any two regions having a common border have different colors?

Is there a way to color this map using four different colors and still ensure that bordering states all have different colors?

What Is Graph Theory?
The Study of Graphs
DEFN: Structures used to model relationships between objects
Often used to describe a collection of nodes
(vertices) and edges

Different Types of Graphs
Graphs come in two distinct forms
Undirected vs. Directed Graphs
Although they are grouped under the umbrella term of graph, these graph have specific characteristics that distinguish them from each other

Undirected Graph
DEFN: Set of nodes connected by edges but there is no distinction between vertices associated with each edge
EXPLANATION: In the eyes of the graph, there is no difference between the edges that connect nodes. Basically the edge that connects vertices (6,4) is the very same edge that connects vertices
(5,2)

Undirected Graphs

Directed Graph
DEFN: Set of nodes connected by edges, where the edges have a direction associated with them
MAJOR INDICATION: THE
ARROWS!!
EXPLANATION: Major difference between edges.
The edge that connects vertices (7,8) is DIFFERENT from the edge that connects vertices (3,10)

Directed Graphs

Applications
Used to model many types of relationships
Have use in many aspects of physical, biological, and social system
Many problems of practical interest can be represented by graphs

Computer Science Applications
Can be used to represent network of communication, data organization, flow of computation, and more
Practical Example
The links on a website can represented by a directed graph. The vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B.

Chemistry
Other Applications
Math
Sociology

Algorithms
What is an ALGORITHM?
DEFN: Set of step by step procedure for calculation
Input Output
Set of Directions
“Algorithm”

Types of Algorithms
Simplistic Algorithms Complex Algorithms
Main purpose: Based on given problem, can you create step by step instructions to produce an appropriate solution?

Breadth First Search Algorithm
Algorithm used for searching a graph
Starts at the root (top of the graph) and does a layer by layer search
Moves left to right during its search

BFS
Does a layer by layer search

Depth First Search Algorithm
Another algorithm used for searching a graph
Starts at the root (top of the graph)
Moves left to right during its search
Major Difference: Starts at the root and explores as far as possible along each branch before backtracking

DFS
Does a depth search before backtracking

Dijkstra’s Algorithm
Discovered by Dutch computer scientist Edsger
Dijkstra in 1956
DEFN: Graph search algorithm that solves the shortest path problem with edge path cost to produce a shortest path tree
SIMPLY: How can we get from the start position to the end position in the least amount of “steps”

Weights
The major thing to notice in this graph is that the edges have specific numbers given to them called “weights”

Undirected graph implementation of Dijkstra’s algorithm
Directed graph implementation of Dijkstra’s algorithm
The addition of the value of weights to the edges is the major difference between the normal implementation of undirected and directed graphs and Dijkstra implementation of those graphs

How To Tackle Dijkstra Problems
The idea is to get from the start position to the end position
You want to take the road that will give you the least value in the end
End
Start

Example
The best way to leave how to do this is to run through an example
Dijkstra's Algorithm for Shortest Route
Problems

Famous and Productive Problem of Graph
Theory
Four Color Problem: Is it true that any map drawn in the plane may have its regions colored with four colors in such a way that any two regions having a common border have different colors?

Solution to The Four Color
Problem

Four Color Problem