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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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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ACM-ICPC

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
vector
deque
stack
set
map 1 6 9 5 9
array (dynamic)
linked lists
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ACM-ICPC

17. <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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ACM-ICPC

18. <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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ACM-ICPC

19. 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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ACM-ICPC

20. Some basic graph operations
Remove a vertex 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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ACM-ICPC

21. 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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ACM-ICPC

22. 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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ACM-ICPC

23. 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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ACM-ICPC

24. Topological Sort (topological order)
Let G = (V, E) be a directed graph
with |V| = n.
A Topological Sort is a sequence v1, v2,.... vn, such that
{ v1, v2,.... vn } = V, and
for every i and j with 1  i < j  n, (vj , vi)  E. 7 1 6 5 3 2 4 7 5
There are more than one such order in this graph!! 2 4 3 1 6
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ACM-ICPC

25. If the graph is not acyclic, then there is no topological sort for the graph.
Not an acyclic Graph 2 4 7 9 1 3 5 8
No way to arrange vertices in this cycle without pointing backwards 6
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ACM-ICPC

26. Algorithm for finding Topological Sort
Principle: vertex with in-degree 0 should be listed first. 2 1 7 2 7
Algorithm: 1
Input G=(V,E)
Repeat the following steps until no more vertex left:
Find a vertex v with in-degree 0;
If can’t find such v and V is not empty
then stop the algorithm; this graph is not acyclic
2. Remove v from V and update E
3. Process v
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27. Topological Sort (topological order) of an acyclic graph7 1 6 5 3 2 4 7 5 2 4 3 1 6
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28. Fail to find a topological sort (topological order)1 2 8 2 4 7 9 1 3 5 8
the graph is not acyclic 6
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ACM-ICPC

29. Topological Sort (topological order) of an acyclic graph2 8 5 4 3 7 9 6 1 2 4 7 9 1 3 5 8 6
This graph is acyclic
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ACM-ICPC

30. Common Algorithm Techniques
Back Tracking....
Greedy Algorithms: Algorithms that make decisions based on the current information; and once a decision is made, the decision will not be revised
Divide and Conquer....
Dynamic Programming.....
Problem: Minimized the number of coins for change.
In real life, we can use a greedy algorithm to obtain the
minimum number of coins. But in general, we need dynamic
programming to do the job
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ACM-ICPC

31. Greedy Algorithms  Dynamic Programming
Problem: Minimized the number of coins for 66¢
{ 25¢, 12¢, 5¢, 1¢ }
{ 25¢, 10¢, 5¢, 1¢ }
25¢ 2
50¢
10¢ 1 5¢ 1¢ 5
66¢
25¢ 2
50¢
12¢ 1 5¢ 0¢ 1¢ 4 4¢ 7
66¢
25¢ 2
50¢
12¢ 0¢ 5¢ 3
15¢ 1¢ 1 6
66¢
16¢
16¢ 6¢ 4¢ 1¢ 4¢ 0¢ 0¢
Greedy method
Greedy method
Dynamic method
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ACM-ICPC

32. Finding the shortest path between two vertices
Starting Vertex
Starting Vertex
Unweighted Graphs 
Weighted Graphs 2 5 1 2 2 3 3 5 1 1 1 5 3 3 7 7 7 2 2 1 9 10 12 8 8 2
Both are using the
greedy algorithm. w w
Final Vertex
Final Vertex
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ACM-ICPC

33. Construct a shortest path Map
Starting Vertex
Starting Vertex a
0,-1 b c
1,a
1,a d e
2,a
2,a f
2,a x
w-1,y n
w,x
Final Vertex
Final Vertex
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ACM-ICPC

34. Construct a shortest path Map for weighted graph
Starting Vertex
Starting Vertex 
0,-1 2 5   1 1 2
2,0
3,1
5,0 1 5   3 4 7
3,1
7,1 1 9  5
10,2
8,4
w-1,y
: decided n
w,x
Final Vertex
Final Vertex
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ACM-ICPC