Graph terminology and algorithm and tree.pptx

Design & Analysis of Algorithms
Lecture#09
Graph

Lecture Contents
Graph
Graph Terminologies
Graph Representations (Adjacency Matrix, Adjacency Link List)
Graph Traversals (Depth First, Breadth First)
Spanning Tree
Minimum Spanning Tree
 Kruskalls’ Algorithm
 Prim’s Algorithm
Shortest Path
 Dijkstra’s Algorithm
 Bellman Ford Algorithm
 A* Algorithm

Graph
A graph G (V, X) consists of two sets V & X. V is a set of
vertices & X is a set of Edges
Every edge e ϵ X joins two vertices in V
Graph G = (V, X)
where V = Set of vertices
AND X = set of edges
AND X  V x V
Example:
G = (V, X)
V = {A, B, C, D, E, F}
X = {(A,B), (A,C), (B,D), (D,E), (D,F), (C,E), (E,F), (F,C)}

Directed Graph (Digraph)
A directed graph is a graph in which all edges are directed. Directed
graph is also called a digraph.
G = (V, E)
V = {A, B, C, D, F, G}
E = {(A,B), (A,C), (B,D), (D,E), (D,F), (C,E), (E,F), (F,C)}

Undirected Graph
A undirected graph is a graph in which all edges are bidirectional.
In an undirected graph G(V,E),
∀e ϵ E
e1 (u, v) = e2 (u, v)

Connected Graph
A graph is connected if there is path between every two nodes of
that graph.

Complete Graph
A graph is complete if there is edge between every two nodes of that
graph.
Total number of edges in complete graph are given as:
𝑇𝑜𝑡𝑎𝑙 𝐸𝑑𝑔𝑒𝑠 =
𝑛(𝑛 − 1)
2

Degree, In Degree, Out Degree
Degree:
Number of edges connecting a vertex is called the
degree of node.
Degree of vertex A is 6
In Degree:
Number of edges pointing into a vertex is called its in
degree
In Degree of vertex A is 2
Out Degree:
Number of edges going out of a vertex is called its out
degree
Out Degree of vertex A is 4

Adjacency
Two nodes are said to be adjacent if they are connected through an edge.
In the example below nodes B&C, C&D, D&E are adjacent to each other.
Adjacent Nodes

Weighted Graph
A graph in which every edge has some weight (numeric
value) associated with it.
It refers to the effort while going from one vertex to other
using this edge Weighted Graph

Unweighted Graph
A graph having no weight value associated with
any of its edge is called an unweighted graph
Unweighted Graph

Reachable Node
A node is reachable from another node if there exists a path between them.
Path exists between A & E

Multigraph, Pseudo Graph
Multigraph
A graph is said to be multigraph if it allows more than
one paths between two nodes
Pseudo Graph
A graph is pseudo graph if it allows if it allows self
loops
Multigraph
Pseudo Graph

Reachable Node
A node is reachable from another node if
there exists a path between them.
For given graph paths exist between:
A & B
A & D
A & F
B & F
C & F
Path exists between A & E

Path
A path of length k is sequence of v1, v2, v3, …., vk of vertices such that (vi, vi+1) for i
= 0, 1, 2, …, k-1 is an edge of graph.
1. ABCD is a path
2. ABD is not a path

Cycle
A cycle is a path starting and ending at same vertex
ABCA makes a cycle in this graph

Hamiltonian Cycle
A Hamiltonian Cycle is a cycle in an undirected graph which visits every vertex
of graph and also returns to starting vertex.
Hamiltonian Cycle:
A, B, C, D, E, F, G, H, A

Eulerian Cycle
An Eulerian Cycle of a connected directed / undirected graph is a cycle that
traverses each vertex of a graph exactly once. There may be one vertex that is
visited more than once.
Eulerian Cycle:
A, B, C, D, E, F, C, G back to A

Subgraph
A subgraph H of graph G is a graph that has its vertices and edges as
subsets of vertices and edges of G respectively.
Graph G
V = {A, B, C, D, E, F, G}
Edges={(A,B), (A,G), (B,C), (B, G), (C,D), (C,F), (C,G), (D,E),
(D,F), (E,F)}
Subgraph H
V = {A, B, C, G}
Edges={(A,B), (A,G), (B,C), (B, G), (C,G)}

Tree
A connected graph without any cycles is call a tree graph or simply a tree.
For a tree graph, there is a unique path between every two nodes of the graph
Tree

Graph Representation
Graphs are represented through two different ways:
1. Adjacency Matrix
2. Adjacency List

Graph Adjacency Matrix Representation
Say a graph has V={1,2,3,…,n} vertices then graph is represented by a
NXN matrix A such that:
A[i, j] = 1 if there is edge between nodes i & j
= 0 otherwise
Graph G
Graph G 1 2 3 4
1 0 0 1 0
2 0 0 1 1
3 1 1 0 1
4 0 1 1 0
Space Complexity: ϴ|V2|

Graph Adjacency Matrix Representation
Adjacency matrix is dense representation i.e. too much space required for large
number of vertices
Can be very efficient for small graphs

Graph Adjacency List Representation
For each vertex, a list of adjacent vertices (neighbors) is
maintained
For the given graph:
Adj[1] = {3}
Adj[2]={3, 4}
Adj[3]={1, 2, 4}
Adj[4]={2, 3}
A possible variation can be to store the list of incoming edges
instead of outgoing edges
Graph G

Graph Adjacency List Representation
If a graph is directed then sum of lengths of all adjacency lists is equal to the
number of edges in graph
If the graph is undirected then this number is doubled
Graph G
Adjacency List for G

Graph Traversals
Graphs are traversed using two different methods:
1. Breadth First Traversal
2. Depth First Traversal

A Typical Graph Traversal
For a connected graph, pick one node as root node and visit all nodes of the
graph in some order
Starting from root we iterate for each node of the graph
During every iteration we visit the node and capture the neighboring nodes of
node in hand
Keep repeating until all nodes are visited
For graphs with disconnected nodes, we may need to visit subgraphs
containing all nodes disconnected with root node separately

Spanning Tree (ST)
“A spanning tree is a subset of Graph G, which has all the vertices covered
with minimum possible number of edges”
A spanning tree can contain no cycle
A spanning tree cannot be disconnected
A graph can have multiple spanning trees
Spanning tree T1
Graph G
Spanning tree T2

Spanning Tree (ST)
“A spanning tree is a subset of Graph G, which has all the vertices
covered with minimum possible number of edges”
Number of vertices for all spanning trees of a graph are same
Number of edges for all spanning trees of a graph are same
All spanning trees form a minimally connected graph i.e. one
takes away one edge from ST and it is no more a spanning tree
Adding one edge to ST will introduce a cycle i.e. ST is maximally
acyclic
Graph G
Spanning tree T1

Spanning Tree (ST) … Applications
Network Planning & Design
Network Routing Protocols
Cluster Analysis
Approximation Algorithms for NP-hard Problems
Image Registration & Segmentation Algorithms
Minimax Process Control
etc

Minimum Spanning Tree (MST)
“Minimum spanning tree is subset of edges of a connected,
weighted undirected graph that connects all vertices
together and bears the minimum possible total edge
weight”
Minimum Spanning Tree is also called Minimum Weight
Spanning Tree
MST is a tree so it does not contain any cycle
Graph K
Spanning Tree T1
Spanning Tree T2

Graph terminology and algorithm and tree.pptx

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