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Algorithm & Data Structures
gy,
What is a Graph?What is a Graph? A graph G = (V,E) is composed of:V: set of vertices^ E: set of edges connecting the vertices in V An edge e = (u,v) is a pair of vertices E lExample:^ a^
b^ V= {a,b,c,d,e}^ E= {(a,b),(a,c),(a,d),(b^
) (^ d) (^ ) c^
(b,e),(c,d),(c,e),(d,e)} d^
e
ApplicationsApplications
CS
electronic circuits
CS
networks (roads, flights,communications)
karachi Islamabad^ LAX
Abbott Lahore
Maree Lahore
Muza-rabad
Terminology:Adjacent and IncidentAdjacent and Incident If (v^0 v^1 ) is an edge in an undirected graphIf (v^0 , v^1 ) is an edge in an undirected graph,^ v^0 and v^1 are adjacent^ The edge (v
0 v^1 ) is incident on vertices v
0 and v^1
The edge (v^0
, v^1 ) is incident on vertices v
0 and v^1
If <v^0 , v^1 > is an edge in a directed graphi^ dj^
t t^ d^
i^ dj^ t f v^0 is adjacent to v
1 , and v^1 is adjacent from v
0
The edge <v^0
, v^1 > is incident on v
0 and v^1 docsity.com
Terminology:Degree of a VertexDegree of a Vertex^ The degree of a vertex is the number of edgesincident to that vertexdi^
d^ h For directed graph,^ the in-degree of a vertex
v^ is the number of edges that have^ v^ as the headthat have^ v^ as the head the out-degree of a vertex
v^ is the number of edges that have^ v^ as the tail
Examples
3
2
p
G^1
1 1 1 1 G^2
3 3
G^2
directed graph 30 in:1, out: 1 in-degreeout-degree^
in: 1, out: 2 1 in: 1, out: 0 (^2) G 3
Terminology:PathPath^ h^
f path:^ sequence ofvertices v^ ,v^1
,.. .vsuch k^ that consecutive
3
2
that consecutive vertices v^ and vi^
arei+ adjacent.
3
j
3
3 a^
a^ b
b c ed
c ed 7 d^
e^ d^
e a b e d c^
b e d c
Graphs: Examplep^
p 1 6
2 3
6 7 4 5 Legend^
8
Directional path
Bi-directional paths
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Oriented (Directed) Graph
p
A graph where edges are directedA graph where edges are directed
Directed vs. Undirected Graph
p
An^ undirected graph
is one in which the pair An^ undirected graph
is one in which the pair of vertices in a edge is unordered, (v
0 , v^1 ) =
(v^1 v^0 )(v^1 ,v^0 ) A directed graph is one in which each edge isa directed pair of vertices <v
v > != a directed pair of vertices, <v
0 , v^1 > != <v
1 ,v^0 > t il^
h^ d tail^
head
Graph RepresentationsGraph Representations^ Adjacency Matrix^ Adjacency Lists
Adjacency MatrixAdjacency Matrix^ Let G=(V,E) be a graph with n vertices.h^ dj^
i^ f^ i^
di^ i^
l
The adjacency matrix of G is a two-dimensionaln by n array, say adj_mat If the edge (v
i^ , vj^ ) is in E(G), adj_mat[i][j]= If there is no such edge in E(G), adj
mat[i][j]=
t e e s^ o suc
edge^
(G), adj_^
at[ ][j]^0
The adjacency matrix for an undirected graph issymmetric; the adjacency matrix for a digraphsymmetric; the adjacency matrix for a digraphneed not be symmetric
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Examples for Adjacency Matrixp^
f^ j^
y (^0 01 23 10111) ⎡⎤ ⎢⎥ (^1011) ⎢⎥ ⎢⎥
(^010) ⎡⎤ ⎢⎥ (^101) ⎢⎥ ⎢⎥ (^3 12) ⎢⎥ (^1101) ⎢ ⎥^1110 ⎣⎦
2 ⎢^ ⎥^0 0 0 ⎣⎦^ G^2
G^1
G^2
(^40512)
1 2 3
3
(^67)
7
2 G 1
0 2 G^3
6 G^4
G^3
G^4
An undirected graph with n vertices and e edges ==> n head nodes and 2e list nodes
Graph Traversalp Problem:^ Search for a certain node or traverseProblem:^ Search for a certain node or traverse all nodes in the graph Depth First SearchDepth First Search^ Once a possible path is found, continue the searchuntil the end of the pathuntil the end of the path Breadth First SearchSt^ t^
l^ th^ t^ ti
d^ d^
i^ h
Start several paths at a time, and advance in eachone step at a time
Depth-First SearchDepth First Search
B^ A C^ D B^ A C^ D F^ E G^ H I^ J^
K^ L M^ N^
O^ P M^ N^
O^ P
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More BFSMore BFS 0 1 2
3 0
1 2
3
B^ A C^ D^
A^ B^
C^ D
E^ F^
G^ H^4
E^ F^
G^ H^4
J^ I K^ L^
I^ J^
K^ L
M^ N^
O^ P^
M^ N^
O^ P^5
BFS Pseudo-Code Algorithm BFS(s): Input: A vertex s in a graphOutput: A labeling of the edges as “discovery” edges and “cross edges”i iti li^ t i^ L
t^ t i^
t initialize container L
to contain vertex s 0 i^ ←^^0 while L^ is not empty doi^ create container L
to initially be emptyi+ for each vertex
v^ in Ldoi^ if edge^ e^ incident on v dog let^ w^ be the other endpoint of
e if vertex w is unexplored thenlabel^ e^ as a discovery edgelabel^ e^ as a discovery edgeinsert^ w^ into Li+1else label e as a cross edgei i
26
i^ ←^ i^ + 1
Applications: Finding a Pathpp^
g
Find path from source vertex s to destinationdvertex d Use graph search starting at s and terminating as soon as we reach d^ Need to remember edges traversed Use depth – first search? Use breath – first search?Use breath
first search?
DFS vs. BFS^ E
F^ B^
A^ start
DFS Process
G^ DC destinationDFS on CC DFS on BBB
Call DFS on DD Return to call on BB
DFS on A^ A A A
A C ll^ S^ G^
f^ d d^ ti^
ti^ d^! Call DFS on GG (^) D B
found destination - done!Path is implicitly stored in DFS recursionPath is: A, B, D, G B^ A
Path is: A, B, D, G
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DFS vs. BFS^ E
FB GD
A^ start
BFS Process^
G^ DC destinationfront rear^
frontrear
frontrear
frontrear
A Initial call to BFS on A
B Dequeue A frontrear
frontrear
frontrearCD (^) Dequeue B
frontrearD Dequeue C
Initial call to BFS on AAdd A to queue
Dequeue AAdd B
Dequeue BAdd C, D
Dequeue CNothing to add
frontrear Gfound destination - done!Dequeue DPath must be stored separatelyAdd G Add G
Referencef Rada MihalceaRada Mihalcea John Lewis and William Loftus Data structure through C in depthData structure through C in depth Data Structure through C
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