Graph

1
Graph

• C and Data Structures
• Baojian Hua
• bjhua_at_ustc.edu.cn

2
Whats a Graph?

• Graph a group of vertices connected by edges

3
Why Study Graphs?

• Interesting broadly used abstraction
• not only in computer science
• challenge branch in discrete math
• Ex the 4-color problem
• hundreds of known algorithms
• with more to study
• numerous applications

4
Broad Applications
Graph Vertices Edges
communication telephone cables
software functions calls
internet web page hyper-links
social relationship people friendship
transportation cities roads

5
Graph Terminology

1
2
3
4
6
5
A sample graph taken from chapter 22 of
Introduction to Algorithms.
6
Graph Terminology

7
Graph ADT

• A graph is a tuple (V, E)
• V is a set of vertices v1, v2, , vn
• E is a set of edges ltvi, vjgt (1lti, jltn)
• Typical operations
• graph creation
• search a vertex (or an edge)
• traverse all vertexes

8
Example
G (V, E) V 1, 2, 3, 4, 5, 6 E (1,
2), (2, 5), (3, 5), (3,
6), (4, 1), (4, 2), (5,
4), (6, 6)

9
Representation

• Two popular strategies
• array-based (adjacency matrix)
• Keep an extensible two-dimensional array M
  internally
• Mij holds the edge ltvi, vjgt, if there exists
  one
• linear list-based (adjacency list)
• for every vertex vi, maintain a linear list
  listltvigt
• listltvigt stores vis out-going edges

10
Adjacency Matrix
0
5
1
2
3
4






0
1
2
Note the mapping function map (n) n-1
3
4
5
11
Adjacency List

1
1-gt2
2
2-gt5
3
3-gt5
3-gt6
4
4-gt1
4-gt2
5
5-gt4
6
6-gt6
12
graph ADT Interface

• // This slides assume all graphs directed.
• // Undirected ones are similar and easier.
• // In file graph.h
• ifndef GRAPH_H
• define GRAPH_H
• define T Graph_t
• typedef struct T T
• T Graph_new ()
• void Graph_insertVertex (T g, poly data)
• void Graph_insertEdge (T g, poly from, poly to)
• // more to come later
• undef T
• endif

13
Client Code
graph g newGraph () insertVertex (g,
1) insertVertex (g, 2) insertVertex (g,
6) insertEdge (g, 1, 2) insertEdge (g, 2,
5) insertEdge (g, 6, 6)

14
Graph Implementation 1 Adjacency Matrix

• // adjacency matrix-based implementation
• include matrix.h
• include dict.h
• include graph.h
• struct Graph_t
• Matrix_t m
• // remember the index
• Dict_t d

0
2
3
1




0
1
2
3
15
Matrix Interface

• // file matrix.h
• ifndef MATRIX_H
• define MATRIX_H
• define T Matrix_t
• typedef struct T T
• T Matrix_new ()
• void Matrix_assign (matrix m, int i, int j)
• int Matrix_getNextIndex (matrix m)
• endif
• // Implementation may make use of a two-
• // dimensional extensible array, leave to you.

16
Adjacency Matrix-based Graph Creation

• T Graph_new ()
• T g malloc (sizeof (g))
• g-gtm Matrix_new () // an empty matrix
• g-gtd Dict_new () // an empty dictionary
• return g

17
Adjacency Matrix-based Inserting Vertices

• void Graph_insertVertex (T g, poly data)
• int i Matrix_getNextIndex (g-gtmatrix)
• Dict_insert (g-gtdict, data, i)
• return

0
1
2
3
4





0
2
3
0
1




0
1
1
2
2
3
3
4
18
Graph Implementation 1 Inserting Edges

• void Graph_insertEdge (T g, poly from, poly to)
• int f Dict_Lookup (g-gtdict, from)
• int t Dict_Lookup (g-gtdict, to)
• Matrix_assign (g-gtmatrix, f, t)
• return

0
1
2
3
4





0
2
3
0
1




0
1
1
2
2
3
3
4
19
Summary

• Pros
• Relatively easy to implement
• But need another level of indirection from data
  to array indexes
• Searching vertices or edges can be fast
• May be too space-consuming
• many empty slots in matrix it the graph has many
  vertices but few edges

20
Graph Representation 2 Adjacency List
struct T List_Tv vertices struct Tv
poly data List_Te edges struct Te Tv
from Tv to

• include linked-list.h
• include graph.h
• define T Graph_t
• define Tv Vertex_t
• define Te Edge_t
• define List_Tv \ LinkedList_t
• define List_Te \
• LinkedList_t
• typedef struct Tv Tv
• typedef struct Te Te

21
Graph Representation 2 Adjacency List

• struct T
• List_Tv vertices
• struct Tv
• poly data
• List_Te edges
• struct Te
• Tv from
• Tv to

22
Adjacency List-based Graph Creation

• // Convention for colors
• // graph, linkedList, data, vertex, edge
• T Graph_new ()
• T g malloc (sizeof (g))
• g-gtvertices LinkedList_new ()
• return g

23
Adjacency List-basedCreating New Vertex

• // Convention for colors
• // graph, linkedList, data, vertex, edge
• Tv Vertex_new (poly data)
• Tv v malloc (sizeof (v))
• v-gtdata data
• v-gtedges LinkedList_new ()
• return v

24
Adjacency List-basedCreating New Edge

• // Convention for colors
• // graph, linkedList, data, vertex, edge
• Te Edge_new (Tv from, Tv to)
• Te e malloc (sizeof (e))
• e-gtfrom from
• e-gtto to
• return e

25
Adjacency List-basedInserting New Vertex

• void Graph_insertVertex (T g, poly data)
• Tv v Vertex_new (data)
• LinkedList_insertTail (g-gtvertices, v)
• return

26
Adjacency List-basedInserting New Edge

• void Graph_insertEdge (T g, poly from, poly to)
• Tv vf lookupVertex (g, from)
• Tv vt lookupVertex (g, to)
• Te e Edge_new (vf, vt)
• LinkedList_insertTail (vf-gtedges, e)
• return
• // insert 0-gt4

27
Adjacency List-basedInserting New Edge

• void Graph_insertEdge (T g, poly from, poly to)
• Tv vf lookupVertex (g, from)
• Tv vt lookupVertex (g, to)
• Te e Edge_new (vf, vt)
• LinkedList_insertTail (vf-gtedges, e)
• return

28
Adjacency List-basedInserting New Edge

• void Graph_insertEdge (T g, poly from, poly to)
• Tv vf lookupVertex (g, from)
• Tv vt lookupVertex (g, to)
• Te e Edge_new (vf, vt)
• LinkedList_insertTail (vf-gtedges, e)
• return

29
Adjacency List-basedInserting New Edge

• void insertEdge (graph g, poly from, poly to)
• Tv vf lookupVertex (g, from)
• Tv vt lookupVertex (g, to)
• Te e Edge_new (vf, vt)
• LinkedList_insertTail (vf-gtedges, e)
• return

0
0-gt1
0-gt2
0-gt3
0-gt4
1
2
3
4
30
Adjacency List-basedInserting New Edge

• void insertEdge (T g, poly from, poly to)
• Tv vf lookupVertex (g, from)
• Tv vt lookupVertex (g, to)
• Te e Edge_new (vf, vt)
• LinkedList_insertTail (vf-gtedges, e)
• return

0
0-gt1
0-gt2
0-gt3
0-gt4
1
2
3
4
31
Example

32
Client Code for This ExampleStep 1 Cook Data

• T g Graph_new ()
• Int_t n1 Int_new (1)
• Int_t n2 Int_new (2)
• Int_t n3 Int_new (3)
• Int_t n4 Int_new (4)
• Int_t n5 Int_new (5)
• Int_t n6 Int_new (6)

2
3
1
4
6
5
33
Client Code ContinuedStep 2 Insert Vertices

• Graph_insertVertex (g, n1)
• Graph_insertVertex (g, n2)
• Graph_insertVertex (g, n3)
• Graph_insertVertex (g, n4)
• Graph_insertVertex (g, n5)
• Graph_insertVertex (g, n6)

2
3
1
1
2
3
4
6
5
4
6
5
34
Client Code ContinuedStep 3 Insert Edges

• Graph_insertEdge (g, n1, n2)
• Graph_insertEdge (g, n2, n5)
• Graph_insertEdge (g, n3, n5)
• Graph_insertEdge (g, n3, n6)
• Graph_insertEdge (g, n4, n1)
• Graph_insertEdge (g, n4, n2)
• Graph_insertEdge (g, n5, n4)
• Graph_insertEdge (g, n6, n6)
• // Done! -)

1
2
3
4
6
5
35
All in PictureAn Empty Graph

• // Ill make use of this convention for colors
• // graph, linkedList, data, vertex, edge

g
next
data
36
All in PictureAfter Inserting all Vertices

• // Ill make use of this convention for colors
• // graph, linkedList, data, vertex, edge

g
next
/\
data
1
6
5
2
3
4
data
next
/\
/\
/\
/\
/\
37
All in PictureAfter Inserting all Edges (Part)

• // Ill make use of this convention for colors
• // graph, linkedList, data, vertex, edge

g
next
/\
data
1
6
5
2
3
4
data
next
/\
/\
/\
/\
from
from
to
to
/\
/\