1. CMSC 341 Lecture 21 Graphs (Introduction)
Prof. Neary
Based on slides from previous iterations of this course (Dr. Katherine Gibson)

2. Basic Graph Definitions
A graph G = (V,E) consists of a finite set of vertices, V, and a finite set of edges, E.
Each edge is a pair (v, w) where v, w  V
V and E are sets, so each vertex v  V is unique, and each edge e  E is unique.
Edges are sometimes called arcs or lines
Vertices are sometimes called nodes or points
UMBC CMSC 341 Graphs (Introduction)

3. UMBC CMSC 341 Graphs (Introduction)
Graph Example
A vertex represents an airport and stores the three-letter airport code
An edge represents a flight route between two airports and stores the mileage of the route
849
PVD
1843
ORD
142
SFO
LGA
1743
802
337
1387
HNL
2555
1099
LAX
1233
DFW
1120
MIA
UMBC CMSC 341 Graphs (Introduction)

4. UMBC CMSC 341 Graphs (Introduction)
Graph Applications
Graphs can be used to model a wide range of applications including
Intersections and streets within a city
Roads/trains/airline routes connecting cities/countries
Computer networks
Electronic circuits
What else?
UMBC CMSC 341 Graphs (Introduction)

5. UMBC CMSC 341 Graphs (Introduction)
Edge Types
Directed edge
Ordered pair of vertices (u,v)
First vertex u is the origin
Second vertex v is the destination
(e.g., a flight)
Undirected edge
Unordered pair of vertices (u,v)
(e.g., a flight route)
flight
AA 1206
ORD
PVD
849
miles
ORD
PVD
UMBC CMSC 341 Graphs (Introduction)

6. UMBC CMSC 341 Graphs (Introduction)
Graph Types
Directed graph
All the edges are directed (edges are ordered pairs)
(e.g., route network)
Sometimes called a digraph
Undirected graph
All the edges are undirected (unordered pairs)
(e.g., flight network)
Sparse and dense graphs
“Few” and “many” edges, respectively
UMBC CMSC 341 Graphs (Introduction)

7. Example: Undirected Graph
All edges are two-way
Edges are unordered pairs
V = { 1, 2, 3, 4, 5 }
E = { (1,2), (2,3), (3,4), (2,4), (4,5), (5,1) } 2 3 1 5 4
UMBC CMSC 341 Graphs (Introduction)

8. Example: Directed Graph
All edges are “one-way” (as shown by arrows)
Edges are ordered pairs
V = { 1, 2, 3, 4, 5 }
E = { (1,2), (3,2), (4,3), (2,4), (4,5), (5,1), (5,4) } 2 3 1 5 4
UMBC CMSC 341 Graphs (Introduction)

9. Graph Terminology and Vocabulary

10. UMBC CMSC 341 Graphs (Introduction)
Terminology
End vertices (or endpoints) of an edge
U and V are the endpoints of a
Edges incident on a vertex
a, d, and b are incident on V
Adjacent vertices
U and V are adjacent
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
UMBC CMSC 341 Graphs (Introduction)

11. UMBC CMSC 341 Graphs (Introduction)
More Terminology
Path
Sequence of alternating vertices and edges
Begins with a vertex
Ends with a vertex
Each edge is preceded and followed by its endpoints
Simple path
Path such that all its vertices and edges are distinct
Examples
P1=(V,X,Z) is a simple path
P2=(U,W,X,Y,W,V) is a path that is not simple V a b P1 d U X Z P2 h c e W g f Y
UMBC CMSC 341 Graphs (Introduction)

12. UMBC CMSC 341 Graphs (Introduction)
More Terminology
Cycle
Circular sequence of alternating vertices and edges
Each edge is preceded and followed by its endpoints
Acyclic: without a cycle
Simple cycle
Cycle such that all its vertices and edges are distinct
Examples
C1=(V,X,Y,W,U,V) is simple
C2=(U,W,X,Y,W,V,U) is a cycle that is not simple V a b d U X Z C2 h e C1 c W g f Y
UMBC CMSC 341 Graphs (Introduction)

13. UMBC CMSC 341 Graphs (Introduction)
Vocabulary
Vertex w is adjacent to vertex v if and only if (v, w)  E.
For undirected graphs, with edge (v, w), and hence also (w, v), w is adjacent to v and v is adjacent to w.
An edge may also have:
Weight or cost -- an associated value
Label -- a unique name
The degree of a vertex, v, is the number of vertices adjacent to v. Degree is also called valence.
UMBC CMSC 341 Graphs (Introduction)

14. UMBC CMSC 341 Graphs (Introduction)
Degrees
For directed graphs vertex w is adjacent to vertex v if and only if (v, w)  E
In-degree of a vertex w is the number of edges (v,w)
X has in-degree 3
Out-degree of a vertex w is the number of edges(w,v)
X has out-degree 2 X U V W Z Y a c b e d f g h i j
UMBC CMSC 341 Graphs (Introduction)

15. UMBC CMSC 341 Graphs (Introduction)
Connectedness
An undirected graph is connected if there is a path from every vertex to every other vertex
A directed graph is strongly connected if there is a path from every vertex to every other vertex
A directed graph is weakly connected if there would be a path from every vertex to every other vertex, disregarding the direction of the edges
A complete graph is one in which there is an edge between every pair of vertices
A connected component of a graph is any maximal connected subgraph. Connected components are sometimes simply called components
UMBC CMSC 341 Graphs (Introduction)

16. UMBC CMSC 341 Graphs (Introduction)
Graph Algorithms
Graph Traversal
Breadth First Search (BFS)
Depth First Search (DFS)
Uniform Cost Search (UCS)
A/A* Search
Etc...
Shortest Paths
Bellman-Ford
Dijsktra’s
Floyd-Warshall
UMBC CMSC 341 Graphs (Introduction)

17. UMBC CMSC 341 Graphs (Introduction)
Graph Traversal
Depth First Search (DFS)
DFS(v):
Mark v as ”visited”
For each edge v -> u:
DFS(u)
(could use a stack instead of recursion)
UMBC CMSC 341 Graphs (Introduction)

18. UMBC CMSC 341 Graphs (Introduction)
Graph Traversal
Breadth First Search
BFS(v):
Initialize a queue Q
Mark v as visited, and enqueue v
While Q is not empty:
w = Q.dequeue()
For each edge w -> u:
If u is not visited, mark it as visited and enqueue it
UMBC CMSC 341 Graphs (Introduction)

19. A Graph ADT Has some data elements Has some operations
Vertices and Edges
Has some operations
getDegree( u ) -- Returns the degree of vertex u (outdegree of vertex u in directed graph)
getAdjacent( u ) -- Returns a list of the vertices adjacent to vertex u (list of vertices that u points to for a directed graph)
isAdjacentTo( u, v ) -- Returns TRUE if vertex v is adjacent to vertex u, FALSE otherwise.
Has some associated algorithms to be discussed.
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UMBC CMSC 341 Graphs

20. Adjacency Matrix Implementation
Uses array of size |V|  |V| where each entry (i ,j) is boolean
TRUE if there is an edge from vertex i to vertex j
FALSE otherwise
store weights when edges are weighted
Very simple, but large space requirement = O(|V|2)
Appropriate if the graph is dense.
Otherwise, most of the entries in the table are FALSE.
For example, if a graph is used to represent a street map like Manhattan in which most streets run E/W or N/S, each intersection is attached to only 4 streets and |E| < 4*|V|. If there are intersections, the table has 9,000,000 entries of which only 12,000 are TRUE.
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UMBC CMSC 341 Graphs

21. Undirected Graph / Adjacency Matrix2 1 3 4 5
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UMBC CMSC 341 Graphs

22. Directed Graph / Adjacency Matrix1 5 2 3 4
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UMBC CMSC 341 Graphs

23. Weighted, Directed Graph / Adjacency Matrix1 8 2 2 5 6 5 7 2
4 and 1 are adjacent to 5. (5,1) and (5,4) have weights of 8 and 5 respectively.
2 is not adjacent to 4, but 4 is adjacent to 2 and edge (2,4) has weight of 6. 3 4 3
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UMBC CMSC 341 Graphs

24. Adjacency Matrix Performance
Storage requirement: O(|V|2 )
Performance:
getDegree ( u )
O(|V|)
isAdjacentTo( u, v )
O(1)
getAdjacent( u )
Storage: O(|V|^2)
getDegree(u) O(|V|) -- Returns outdegree of u. Must look at (|V|-1) elements on the u-th row
getAdjacent(u) O(|V|) -- Returns list of vertices that are adjacent to u. must look down the u-th row
isAdjacentTo(u,v) O(1) -- random access to element[u,v]
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UMBC CMSC 341 Graphs

25. Adjacency List Implementation
If the graph is sparse, then keeping a list of adjacent vertices for each vertex saves space. Adjacency Lists are the commonly used representation. The lists may be stored in a data structure or in the Vertex object itself.
Vector of lists: A vector of lists of vertices. The i- th element of the vector is a list, Li, of the vertices adjacent to vi.
If the graph is sparse, then the space requirement is O( |E| + |V| ), “linear in the size of the graph”
If the graph is dense, then the space requirement is O( |V|2 )
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UMBC CMSC 341 Graphs

26. Vector of Lists 5 2 3 4 8 1 6 7 1 2 2 4 3 2 4 3 5 5 1 4
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UMBC CMSC 341 Graphs

27. Adjacency List Performance
Storage requirement: O(|V| + |E|)
Performance: O
getDegree( u )
isAdjacentTo( u, v )
getAdjacent( u )
Storage: O(|V| + |E|)
getDegree(u) -- O(D(u)) – Returns outdegree of u. Requires counting list length in each case except
when degree is kept in a separate vector which is then O(1)
getAdjacent(u) -- O(D(u)) –Returns list of vertices that are adjacent to u. Requires construction of a list of
each vertex on the adjacency list of vertex u
isAdjacentTo(u,v) - O(D(u)) -- Requires searching the adjacency list of vertex u
for the presence of vertex v to see if v is adjacent to u
storage -- O(|V|+|E|) for “Array of Lists”-- entry for each
vertex and list of entries for each adjacent vertex
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UMBC CMSC 341 Graphs