Lecture 14 data structures and algorithms

Data Structure and
Algorithm (CS-102)
Ashok K Turuk

Graph
• A graph is a collection of nodes (or
vertices, singular is vertex) and edges (or
arcs)
– Each node contains an element
– Each edge connects two nodes together (or
possibly the same node to itself) and may
contain an edge attribute

• A directed graph is one in which the edges
have a direction
• An undirected graph is one in which the
edges do not have a direction
2

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
• Example:

a

b

E= {(a,b),(a,c),(a,d),
(b,e),(c,d),(c,e), (d,e)}

c
d

V= {a,b,c,d,e}

e

some problems that can be
represented by a graph

computer networks
airline flights
road map
course prerequisite structure
tasks for completing a job
flow of control through a
program
• many more
•
•
•
•
•
•

a digraph
A
E

B

C

D

V = [A, B, C, D, E]
E = [<A,B>, <B,C>, <C,B>, <A,C>, <A,E>, <C,D>]

Graph terminology
• The size of a graph is the number of nodes in it
• The empty graph has size zero (no nodes)
• If two nodes are connected by an edge, they are
neighbors (and the nodes are adjacent to each other)
• The degree of a node is the number of edges it has
• For directed graphs,
– If a directed edge goes from node S to node D, we call S the
source and D the destination of the edge
• The edge is an out-edge of S and an in-edge of D
• S is a predecessor of D, and D is a successor of S

– The in-degree of a node is the number of in-edges it has
– The out-degree of a node is the number of out-edges it has

6

Terminology:
Path
• path: sequence of
vertices v1,v2,. . .vk
such that consecutive
vertices vi and vi+1 are
adjacent.

3

2
3

3

3

a

b

a

c

b
c

e d

d
abedc

e
bedc
7

More Terminology
• simple path: no repeated vertices
a

b
bec

c
e

d

• cycle: simple path, except that the last vertex is the same
as the first vertex
a

b
acda

c
d

e

Even More Terminology
•connected graph: any two vertices are connected
by some path

connected

not connected

• subgraph: subset of vertices and edges forming a graph
• connected component: maximal connected subgraph. E.g.,
the graph below has 3 connected components.

Subgraphs Examples
0
1

0

0

2
3
G1
0

1

(i)

3

0
1

2
3

0

0

1

1

2

(i)

0
1

1

G3

2

2

(ii)
(iii)
(iv)
(a) Some of the subgraph of G1
0

2

1

2

(ii)

(iii)
(b) Some of the subgraph of G3

(iv)

More…
• tree - connected graph without cycles
• forest - collection of trees
tree

tree

forest
tree
tree

Connectivity
• Let n = #vertices, and m = #edges
• A complete graph: one in which all pairs of vertices
are adjacent
• How many total edges in a complete graph?
– Each of the n vertices is incident to n-1 edges, however, we
would have counted each edge twice! Therefore,
intuitively, m = n(n -1)/2.

• Therefore, if a graph is not complete, m < n(n -1)/2

n 5
m (5

More Connectivity
n = #vertices
m = #edges
• For a tree m = n - 1

n 5
m 4

If m < n - 1, G is not
connected

n 5
m 3

Oriented (Directed) Graph
• A graph where edges are directed

Directed vs. Undirected Graph
• An undirected graph is one in which the
pair of vertices in a edge is unordered,
(v0, v1) = (v1,v0)
• A directed graph is one in which each
edge is a directed pair of vertices, <v0,
tail
head
v1> != <v1,v0>

Graph terminology
• An undirected graph is connected if there is a
path from every node to every other node
• A directed graph is strongly connected if there is
a path from every node to every other node
• A directed graph is weakly connected if the
underlying undirected graph is connected
• Node X is reachable from node Y if there is a path
from Y to X
• A subset of the nodes of the graph is a connected
component (or just a component) if there is a path
from every node in the subset to every other node
in the subset
16

graph data structures
• storing the vertices

– each vertex has a unique identifier and,
maybe, other information
– for efficiency, associate each vertex with a
number that can be used as an index

• storing the edges

– adjacency matrix – represent all possible
edges
– adjacency lists – represent only the
existing edges

storing the vertices
• when a vertex is added to the graph,
assign it a number

– vertices are numbered between 0 and n-1

• graph operations start by looking up
the number associated with a vertex
• many data structures to use
– any of the associative data structures
– for small graphs a vector can be used
• search will be O(n)

the vertex vector
0

A
4

1

E

B

C
2

D
3

0
1
2
3
4

A
B
C
D
E

adjacency matrix
0

A
4

1

E

B

0
1
2
3
4

A
B
C
D
E
0 1 2 3 4

C
2

D
3

0
1
2
3
4

0
0
0
0
0

1
0
1
0
0

1
1
0
0
0

0
0
1
0
0

1
0
0
0
0

0
1

Examples for Adjacency Matrix
0
2

1

3

2

G1

G2

0

1

1

1

0

1

0

1
1

0
1

1
0

1
1

1

0

1

0

0

0

1

1

1

0

symmetric

Asymmetric

maximum # edges?

a V2 matrix is needed for
a graph with V vertices

many graphs are “sparse”
• degree of “sparseness” key factor in
choosing a data structure for edges

– adjacency matrix requires space for all
possible edges
– adjacency list requires space for existing
edges only

• affects amount of memory space
needed
• affects efficiency of graph operations

adjacency lists
0

0
1
2
3
4

A
4

1

E

B

C
2

D
3

0
1
2
3
4

1
2
1

A
B
C
D
E

2
3

4

0
1

2
3

0
1
2
3

1
0
0
0

2
2
1
1

3
3
3
2

0

G1
0
1
2

1
0

1

2
G3

2

Least-Cost Algorithms
Dijkstra’s Algorithm

Find the least cost path from a given node to all
other nodes in the network
N = Set of nodes in the network
s = Source node
M = Set of nodes so far incorporated by the
algorithm
dij = link cost from node i to node j, dii = 0
dij = ∞ if two nodes are not directly connected
dij >= 0 of two nodes are directly connected
Dn = cost of the least-cost path from node s to
node n that is currently known to the algorithm

1. Initialize
M = {s} [ set of nodes so far
incorporated consists of only the
source node]
Dn = dsn for n != s (initial path costs to
the neighbor nodes are simply the link
cost)
2. Find the neighbor node not in M that
has the least-cost path from node s
and incorporate that node into M.

Find w !Є M such that Dw = min { Dj } for j
!Є M
3. Update the least-cost path:
Dn = min[ Dn , Dw + dwn ] for all n !Є M

If the latter term is minimum, path from
s to n is now the path from s to w,
concatenated with the link from w to n

8
5
2
1

2

3
1 2

7

4

6
3

1

8

3

1 5
2

3
5

6

4

It = Iteration
It

M

D2 Path D3 Path D4 Path D5 Path D6 Path

1 {1}

2

1 -2

5

2 {1, 4}

2

1-2

3 {1,2,4}

2

1-2

1-3

1

1-4

∞

4 1-4-3

1

1-4

4 1-4-3

1

1-4

-

∞

-

2 1-4-5

∞

-

2 1-4-5

∞

-

Lecture 14 data structures and algorithms

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Lecture 14 data structures and algorithms