Intro to Graphs
Muhammad Magdi
Agenda
- Motivation
- Graph Terminology
- Graph Classification
- Graph Representation
- Graph Traversal
- Common Problems
Hundreds of interesting computational problems are
couched in terms of graphs.--CLRS
Graphs have applications in a host of different domains,
including mapping, transportation, electrical engineering, and computer networks.--Data Structures and Algorithms in C++
Pairwise connections between items play a critical role in a vast array of computational applications.
--Algorithms Book
Motivation
Motivation
- What's the shortest route from Cairo to Alex?
- What's the fastest way to get from Cairo to Alex?
- What's the min cost to ship some orders from a group of warehouses to the customres?
- Is there a short-circuit in a given circuit?
- Can a given circuit be put onto a chip without crossing wires?
- How to match some people with their limited interest list?
- How to send this peace of info through internet with minimum latency?
- How can a compiler build a modular system?
- How to suggest friends in a social network?
- and more...
Graph Terminology
Nodes/Vertices (n)
Usually used to represent object in a given problem
Edges/Arcs (m)
Usually used to represent relationships between those objects (nodes)
Graph
A way of representing relationships between pairs of objects. Visualized as circles connected by lines representing nodes and edges respectively
Nodes a, b are called adjacents (Neighbors), iff there's a direct edge between them.
Adjacency Relation
A sequence of nodes connected by edges.
Path
A Path whose first and last nodes are the same.
Cycle
A graph having multiple edges between the same pair of nodes.
Multigraph
An edge connecting a node to itself.
Self-loop
A graph that's neither multigraph nor having self-loops.
Simple Graph
Graph Classification
Directed
A graph whose all edges are having directions.
Undirected
A graph whose all edges are NOT having directions.
Graph Classification
Weighted
A graph having a weight (number) associated with each edge.
Unweighted
A graph whose all edges are considered equivalent in length (weight).
Graph Classification
Connected
A graph in which there's a path between every pair of nodes.
Disconnected
A graph consisted of a set of connected components (subgraphs).
Graph Classification
Cyclic
Acyclic
A graph with no cycles.
A graph that is/contains a cycle.
Graph Classification
Dense
A graph having a relatively few of the possible edges missing
Sparse
A graph having a relatively few of the possible edges present
m∝n
m \propto n
m∝n2
m \propto n^2
Graph Classification
Explicit
Implicit
A graph whose nodes or edges are not explicitly represented as objects in computer memory, but are determined algorithmically in runtime from some input.
--Wikipedia
Graph Classification
Special Types of Graphs
A graph in which each pair of nodes is directly connected by an edge
Complete Graph
Special Types of Graphs
m=2n∗(n−1)
m = \frac{n * (n-1)}{2}
The most dense graph
A graph that can be divided into 2 sets such that there's no edge between nodes within the same set.
Bipartite Graph
Special Types of Graphs
- Connected Acyclic Graph
- Undirected Graph in which there's exactly one unique path between each pair of nodes
that means:
1. One Connected Component
2. Edges = nodes - 1
3. Acyclic
Tree Graph
Special Types of Graphs
Directed Acyclic Graph
DAG
Special Types of Graphs
Graph Representation
Intro to Graphs Muhammad Magdi
Intro to Graphs
By Muhammad Magdi
Intro to Graphs
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