1. Graphs: As a mathematics branch
Map of Königsberg 1 2 3 4
Leonhard Paul Euler
Can we cross every bridge exactly once and return to the same place?
(There is no other way but through the bridges to cross the rivers)
Euler’s solution is considered as the first Graph theorem in history.
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2. Eulerian Graph 1
Map of Königsberg 2 3 4
Starting from a vertex, can we travel every edges exactly once and return to the same vertex?
Leonhard Paul Euler
No!
If we can, such a path is called an Eulerian Circuit
Theorem [Euler 1737]: A graph has an Eulerian Circuit iff the degree of every vertex is even.
The first Graph theorem in history.
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3. Graphs: Algorithmic approach
Many problems are unsolvable (usually are associated to infinite graphs) ;
Many problems are intractable (NPC problems),
And yet, many applicative problems are manageable;
(in terms of |V| and |E|)
G = (V, E)
V: a set of vertices
E: a set of edges, E  V  V 1
Example,
V: = {1,2,3,4,5,6}
E: = {(1,2), (1,5), (2,3), (2,5), (3,4), (4,5), (4,6)} 2 5 6 3 4
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4. Undirected Graphs
G= (V,E) is an undirected graph, if (v, w)  E  (w, v)  E
Example,
V: = {1,2,3,4,5,6}
E: = {(1,2), (1,5), (2,3), (2,5), (3,4), (4,5), (4,6)
(2,1), (5,1), (3,2), (5,2), (4,3), (5,4), (6,4) } 1
V: = {1,2,3,4,5,6}
E: = { {1,2}, {1,5}, {2,3}, {2,5},
{3,4}, {4,5}, {4,6} } 2 5 6 3 4
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5. Weighted Graphs Every edge in E has a value associated to it. Example,1
Example,
2.3 3
V: = {1,2,3,4,5,6}
E: = {(1,2,3), (1,5,2.3), (2,3,4.5), (2,5,2/3),
(3,4,7.1), (4,5,10), (4,6,12)} 2
2/3 5 6
4.5 10 12 3
7.1 4
Weighted graph: G= (V,E), where E  V  V  Q
rational numbers
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6. A walk on a graph
(v1, v2,.... vn) Vn , such that (vi , vi+1)  E, for 1  i < n, is called a walk of G= (V,E)
Example:
(1,2,4,7,9,8,5,4,7,6,5,2,3)
a walk 2 4 7 9 1 3 5 8 6
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7. More Terminologies
(v1, v2,.... vn) Vn , such that (vi , vi+1)  E, for 1  i < n, is called a walk of G= (V,E)
A path is a walk such that no vertex in the walk is repeated.
A trail is a walk such that no edge in the walk is repeated. (Any path is a trial)
A circuit is a closed trial, i.e, v1 = vn.
A cycle is a circuit such that v1 = vn is only repeated vertex.
A spanning is a walk such that every vertex is included
A Eulerian circuit is a circuit that contains every edge.
A Hamiltonian cycle is a spanning cycle.
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8. A Walk: 1,2,4,7,9,8,5,4,7,6,5,2,3 (not a trial) 2 4 7 9 1 3 5 8 6
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9. A Circuit: 1,2,4,7,9,8,5,4,7,6,5,2,3,1 (not a cycle)
A Trial: 1,2,4,7,9,8,5,4,0,7,6,5,2,3 (not a path)
A Circuit: 1,2,4,7,9,8,5,4,7,6,5,2,3,1 (not a cycle) 2 4 7 9 1 3 5 8 6
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10. A Path: 1,2,4,0,7,6,5,3
A Cycle: 1,2,4,0,7,6,5,3,1
There are many other cycles in this graph 2 4 7 9 1 3 5 8 6
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11. A graph without any cycle in it is called acyclic2 4 7 9 1 3 5 8 6
An acyclic Graph
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12. Representation of Graphs. (Data Structures)
Adjacency Matrix
Good for dense graph. 1 2 3 4 5 6 7 8 9 2 4 7 9 1 3 5 8 6
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13. Representation of Graphs.
Adjacency Matrix for a Weighted Graph 1 2 3 4 5 
2.3
4.5
2/3
7.1 10 12
2.3 3 1
2/3 4 5
4.5 10 12 2
7.1 3
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14. Representation of Graphs. (Data Structures)
Adjacency List
A commonly used representation 2 4 7 1 3 5 6 9 8 2 4 7 9 1 3 5 8 6
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15. Representation of Graphs.
Adjacency List for a Weighted Graph
(1, 3)
(4, 2.3) 1
(2, 4.5)
(4, 2/3) 2
(3, 7.1) 3
(4, 10)
(5, 12) 4 5
2.3 3 1
2/3 4 5
4.5 10 12 2
7.1 3
How to implement?
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16. Representation of Graphs. (Data Structures)
Adjacency List
Should be easy to insert/delete vertices and edges
Should be easy to retrieve.
Should not use to much memory. 2 4 7 1 2 3 4 5 6 7 8 9 1 3 4 5 1 7 3 4 6 1 6 9 5 9
array (dynamic)
linked lists
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17. STL: C++ Standard Template Library
Read: Weiss Ch. 4.8 of Weiss and Chapter 19 of Savitch
#include <vector>
#include <deque>
#include <stack>
#include <set>
#include <map>
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18. <deque> (Double Ended Queue)
student 2
student 1
#include <iostream>
#include <deque>
deque<student> IT279;
while (....) {
......
IT279.push_front(s);
IT279.push_back(s); }
for (int i=0;i<ITK.size();i++) {
cout << "\nName:" << ITK[i].name << " ";
cout << "SSN:" << ITK[i].ssn;
ITK
IT279[0]
IT279[1]
IT279[2]
IT279[3]
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19. pop back and pop front on a deque
student 1
student 2
student 3
student 4
student 5
student 1
student 2
student 3
student 4
student 5
IT279
IT279
IT279[0]
IT279[0]
IT279[1]
IT279[0]
IT279[2]
....
IT279.pop_front();
IT279.pop_back();
IT279[1]
IT279[3]
IT279[2]
IT279[4]
IT279[4]
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20. <set> Red-Black Tree 10 6 12 3 8 15 5 9
#include <iostream>
#include <set>
set<int> S;
while (....) {
int e; // an element of S
S.insert(e); }
set<int>::iterator itr;
itr = S.begin();
S.erase(itr);
for (itr= S.begin(); itr != S.end(); itr++)
cout << *itr; 6 12 3 8 15 5 9
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21. <map> Red-Black Tree key, associated value in, double
10, 1.2
Red-Black Tree
#include <iostream>
#include <map>
map<int, double> M;
int i;
double w;
while (....) {
// e is an element of M
pair e = make_pair(i,w);
M.insert(e); }
map<int, double>::iterator itr;
itr = M.begin();
M.erase(itr);
for (itr= M.begin(); itr != M.end(); itr++)
cout << “\n(“ << itr->first << “,” << itr->second << “)”;
6, 3.4
12, 1.2
3, 3.4
8, 5.6
15, 0.3
5, 2.8
9, 0.1
key, associated value
in, double
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22. Using STL to implement Graphs
typedef map<int, double> AdjacencyList;
typedef map<int, AdjacencyList> Graph; 3
2.3
Graph G;
0.6 3, 1 4 5
4.5
6.3 10 12 1, 4,
G[4] 2 3
7.1
G[5] 0, 2, 5,
(1, 3)
(4, 2.3) 1
(2, 4.5)
(4, 0.6) 2
(3, 7.1) 3
(1, 6.3)
(4, 10)
(5, 12) 4 5
G[0]
G[3]
G[1]
G[2]
4, 2.3
4, 10
2, 4.5
3, 7.1
1, 3.0
1, 6.3
5, 12
4, 0.6
(G[3])[5] = 12
(G[1])[4] = 0.6
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23. Create Random Graphs typedef map<int, double> AdjacencyList;
typedef map<int, AdjacencyList> Graph;
// Create a graph of n vertices, d is the maximum out degree
Graph create_graph(int n, int d, bool weighted=true) {
Graph g;
for (int i=0;i<n;i++) { // Prepare vertex i
int outdegree = rand()%d;
AdjacencyList aj; // Prepare the Adjacency List for vertex i
for (int j=0; j<outdegree; j++) {
int v = rand()%n;
if (i==v) continue;
aj[v]=(weighted ? (double)(rand()%100): 1.0; }
g[i]=aj;
return g;
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24. Some basic graph operations
class graph {
public:
static void add_v(Graph & g, int v, AdjacencyList aj);
static void remove_v(Graph & g, int v);
static void display(Graph & g);
..... };
void graph::add_v(Graph & g, int v, AdjacencyList aj) {
g[v]=aj; }
void graph::display(Graph & G) {
Graph::iterator p;
AdjacencyList::iterator q;
for (p=G.begin();p!=G.end();p++) {
cout << p->first << ":: ";
for (q=p->second.begin();q!=p->second.end();q++)
cout << p->first << "-->" << q->first
<< " [" << q->second << "], ";
cout << " *\n"; }
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25. Remove a vertex
void graph::remove_v(Graph & g, int v) {
g.erase(v);
Graph::iterator p;
for (p=g.begin();p!=g.end();p++) p->second.erase(v); } 2 4 7 1 2 3 4 5 6 7 8 9 1 3 4 5
remove 7 1 7 3 4 6 1 6 9 5 9
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26. Given a graph, determine is the graph is acyclic2 4 7 9 1 3 5 8
Not An acyclic Graph 6
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27. Given a graph, determine is the graph is acyclic2 4 7 9 1 3 5 8
Not An acyclic Graph 6
No vertex in this cycle has in-degree 0
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28. Procedure to determine if a given graph is acyclic
Repeatedly remove vertices with in-degree 0; 2 4 7 9 1 3 5 8
If we can’t remove all vertices, then the graph is not acyclic 6
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29. Procedure to determine if a given graph is acyclic
Repeatedly remove vertices with in-degree 0; 2 4 7 9 1 3 5 8
If all vertices can be removed, then the given graph is acyclic 6
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30. C++ program to determined if a given graph is acyclic
bool graph::acyclic(Graph & G) {
Graph tempG = G;
while (tempG.size() >= 2) {
int v = findVertexwithZeroIn(tempG);
if (v == -1) return false;
remove_v(tempG, v); }
return true;
// -1 mean no zero indegree vertex.
int graph::findVertexwithZeroIn(Graph & g){
for (Graph::iterator p=g.begin();p!=g.end();p++) {
if (!someTo(g, p->first)) return p->first; }
return -1;
bool graph::someTo(Graph & g, int v) {
for (Graph::iterator p=g.begin();p!=g.end();p++)
if (p->second.find(v) != p->second.end()) return true;
return false; }
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