Graph Introduction.ppt
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A graph is a pair (V, E) where V is a set of vertices and E is a set of edges connecting the vertices. Graphs can be represented using an adjacency matrix, where a matrix element A[i,j] indicates if there is an edge from vertex i to j, or using adjacency lists, where each vertex stores a list of its neighboring vertices. Graphs find applications in modeling networks, databases, and more. Common graph operations include finding paths and connectivity between vertices.
![Graphs 9
Adjacency Matrix Representation
Assume V = {1, 2, …, n}
An adjacency matrix represents the graph as a n x n
matrix A:
A[i, j] = 1 if edge (i, j) E (or weight of edge)
= 0 if edge (i, j) E
1
2 4
3
a
d
b c
A 1 2 3 4
1 0 1 1 0
2 0 0 1 0
3 0 0 0 0
4 0 0 1 0](https://image.slidesharecdn.com/graph1-240201115017-1579e605/85/Graph-Introduction-ppt-9-320.jpg)
![Graphs 11
Adjacency Matrix Representation
Pros:
Simple to implement
Easy and fast to tell if a pair (i, j) is an edge:
simply check if A[i, j] is 1 or 0
Can be very efficient for small graphs
Cons:
No matter how few edges the graph has, the
matrix takes O(n2) in memory](https://image.slidesharecdn.com/graph1-240201115017-1579e605/85/Graph-Introduction-ppt-11-320.jpg)
![Graphs 12
Adjacency Lists Representation
A graph is represented by a one-dimensional array L of
linked lists, where
L[i] is the linked list containing all the nodes adjacent
to node i.
The nodes in the list L[i] are in no particular order
Example:
Adj[1] = {2,3}
Adj[2] = {3}
Adj[3] = {}
Adj[4] = {3}
1
2 4
3](https://image.slidesharecdn.com/graph1-240201115017-1579e605/85/Graph-Introduction-ppt-12-320.jpg)
![Graphs 14
Adjacency Lists Representation
Pros:
Saves on space (memory): the representation
takes O(|V|+|E|) memory.
Good for large, sparse graphs (e.g., planar maps)
Cons:
It can take up to O(n) time to determine if a pair
of nodes (i, j) is an edge: one would have to
search the linked list L[i], which takes time
proportional to the length of L[i].](https://image.slidesharecdn.com/graph1-240201115017-1579e605/85/Graph-Introduction-ppt-14-320.jpg)
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Graph Introduction.ppt
- 1. Graphs 1 Graphs hierarchical (1 to many) graph (many to many) first ith last sequence/linear (1 to 1)
- 2. Graphs 2 What is a Graph? A graph is a pair (V, E), where V is a set of nodes, called vertices E is a collection of pairs of vertices, called edges V(G) and E(G) represent the sets of vertices and edges of G, respectively Example: A tree is a special type of graph! a b c d e V= {a, b, c, d, e} E= {(a,b), (a,c), (a,d), (b,e), (c,d), (c,e), (d,e)}
- 3. Graphs 3 John David Paul brown.edu cox.net cs.brown.edu att.net qwest.net math.brown.edu cslab1b cslab1a Applications Electronic circuits Printed circuit board Integrated circuit Transportation networks Highway network Flight network Computer networks Local area network Internet Web Databases Entity-relationship diagram
- 4. Graphs 4 What can we do with graphs? Find a path from one place to another Find the shortest path from one place to another Determine connectivity Find the “weakest link” (min cut) • check amount of redundancy in case of failures Find the amount of flow that will go through them
- 5. Graphs 5 Edge and Graph Types Directed edge ordered pair of vertices (u,v) first vertex u is the origin second vertex v is the destination Undirected edge unordered pair of vertices (u,v) Directed graph (Digraph) all the edges are directed e.g., route network Undirected graph all the edges are undirected e.g., flight network Mixed graph some edges are undirected and some edges are directed e.g., a graph modeling a city map a b a b a b c d e Directed edge Undirected edge Mixed graph
- 6. Graphs 6 Terminology End vertices (or endpoints) of an edge u and v are the endpoints of a Edges incident to a vertex a, d, and b are incident to v Adjacent vertices u and v are adjacent Degree of a vertex x has degree 5 Parallel edges h and i are parallel edges Self-loop j is a self-loop x u v w z y a c b e d f g h i j
- 7. Graphs 7 Terminology (cont.) Outgoing edges of a vertex (a, b) and (a, c) are outgoing edges of vertex a Incoming edges of a vertex (b, c), (d, c) and (a, c) are incoming edges of vertex c In-degree of a vertex c has in-degree 3 b has in-degree 1 Out-degree of a vertex a has out-degree 2 b has out-degree 1 a b d c
- 8. Graphs 8 Graph Representation For graphs to be computationally useful, they have to be conveniently represented in programs There are two computer representations of graphs: Adjacency matrix representation Adjacency lists representation
- 9. Graphs 9 Adjacency Matrix Representation Assume V = {1, 2, …, n} An adjacency matrix represents the graph as a n x n matrix A: A[i, j] = 1 if edge (i, j) E (or weight of edge) = 0 if edge (i, j) E 1 2 4 3 a d b c A 1 2 3 4 1 0 1 1 0 2 0 0 1 0 3 0 0 0 0 4 0 0 1 0
- 10. Graphs 10 Adjacency Matrix Representation Undirected Graph Directed Graph The adjacency matrix for an undirected graph is symmetric; the adjacency matrix for a digraph need not be symmetric
- 11. Graphs 11 Adjacency Matrix Representation Pros: Simple to implement Easy and fast to tell if a pair (i, j) is an edge: simply check if A[i, j] is 1 or 0 Can be very efficient for small graphs Cons: No matter how few edges the graph has, the matrix takes O(n2) in memory
- 12. Graphs 12 Adjacency Lists Representation A graph is represented by a one-dimensional array L of linked lists, where L[i] is the linked list containing all the nodes adjacent to node i. The nodes in the list L[i] are in no particular order Example: Adj[1] = {2,3} Adj[2] = {3} Adj[3] = {} Adj[4] = {3} 1 2 4 3
- 13. Graphs 13 Adjacency Lists Representation Undirected Graph Directed Graph
- 14. Graphs 14 Adjacency Lists Representation Pros: Saves on space (memory): the representation takes O(|V|+|E|) memory. Good for large, sparse graphs (e.g., planar maps) Cons: It can take up to O(n) time to determine if a pair of nodes (i, j) is an edge: one would have to search the linked list L[i], which takes time proportional to the length of L[i].
Editor's Notes
- #2: 2/1/2024 11:50 AM