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Toronto. 650. 700. Boston. Chicago. 200. 600. New York. Shortest Path Problems. We can assign weights to the edges of graphs, for example to represent the distance between cities in a railway network:. Shortest Path Problems.

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Shortest path problems

Toronto

650

700

Boston

Chicago

200

600

New York

Shortest Path Problems

  • We can assign weights to the edges of graphs, for example to represent the distance between cities in a railway network:

Applied Discrete Mathematics Week 14: Trees


Shortest path problems1
Shortest Path Problems

  • Such weighted graphs can also be used to model computer networks with response times or costs as weights.

  • One of the most interesting questions that we can investigate with such graphs is:

  • What is the shortest path between two vertices in the graph, that is, the path with the minimal sum of weights along the way?

  • This corresponds to the shortest train connection or the fastest connection in a computer network.

Applied Discrete Mathematics Week 14: Trees


Dijkstra s algorithm
Dijkstra’s Algorithm

  • Dijkstra’s algorithm is an iterative procedure that finds the shortest path between to vertices a and z in a weighted graph.

  • It proceeds by finding the length of the shortest path from a to successive vertices and adding these vertices to a distinguished set of vertices S.

  • The algorithm terminates once it reaches the vertex z.

Applied Discrete Mathematics Week 14: Trees


Dijkstra s algorithm1
Dijkstra’s Algorithm

  • procedure Dijkstra(G: weighted connected simple graph with vertices a = v0, v1, …, vn = z and positive weights w(vi, vj), where w(vi, vj) =  if {vi, vj} is not an edge in G)

  • for i := 1 to n

  • L(vi) := 

  • L(a) := 0

  • S := 

  • {the labels are now initialized so that the label of a is zero and all other labels are , and the distinguished set of vertices S is empty}

Applied Discrete Mathematics Week 14: Trees


Dijkstra s algorithm2
Dijkstra’s Algorithm

  • while zS

  • begin

  • u := the vertex not in S with minimal L(u)

  • S := S{u}

  • for all vertices v not in S

  • if L(u) + w(u, v) < L(v) then L(v) := L(u) + w(u, v)

  • {this adds a vertex to S with minimal label and updates the labels of vertices not in S}

  • end{L(z) = length of shortest path from a to z}

Applied Discrete Mathematics Week 14: Trees


Dijkstra s algorithm3

b

d

a

z

c

e

Dijkstra’s Algorithm

  • Example:

5

6

4

8

1

2

0

3

2

10

Step 0

Applied Discrete Mathematics Week 14: Trees


Dijkstra s algorithm4

b

d

5

6

4

8

a

1

z

2

0

3

2

10

c

e

Dijkstra’s Algorithm

4 (a)

  • Example:

2 (a)

Step 1

Applied Discrete Mathematics Week 14: Trees


Dijkstra s algorithm5

b

d

5

6

4

8

a

1

z

2

0

3

2

10

c

e

Dijkstra’s Algorithm

3 (a, c)

4 (a)

10 (a, c)

  • Example:

2 (a)

12 (a, c)

Step 2

Applied Discrete Mathematics Week 14: Trees


Dijkstra s algorithm6

b

d

5

6

4

8

a

1

z

2

0

3

2

10

c

e

Dijkstra’s Algorithm

3 (a, c)

4 (a)

10 (a, c)

8 (a, c, b)

  • Example:

2 (a)

12 (a, c)

Step 3

Applied Discrete Mathematics Week 14: Trees


Dijkstra s algorithm7

b

d

5

6

4

8

a

1

z

2

0

3

2

10

c

e

Dijkstra’s Algorithm

3 (a, c)

4 (a)

10 (a, c)

8 (a, c, b)

  • Example:

14 (a, c, b, d)

2 (a)

12 (a, c)

10 (a, c, b, d)

Step 4

Applied Discrete Mathematics Week 14: Trees


Dijkstra s algorithm8

b

d

5

6

4

8

a

1

z

2

0

3

2

10

c

e

Dijkstra’s Algorithm

4 (a)

3 (a, c)

8 (a, c, b)

10 (a, c)

  • Example:

14 (a, c, b, d)

13 (a, c, b, d, e)

2 (a)

12 (a, c)

10 (a, c, b, d)

Step 5

Applied Discrete Mathematics Week 14: Trees


Dijkstra s algorithm9

b

d

5

6

4

8

a

1

z

2

0

3

2

10

c

e

Dijkstra’s Algorithm

4 (a)

3 (a, c)

8 (a, c, b)

10 (a, c)

  • Example:

14 (a, c, b, d)

13 (a, c, b, d, e)

2 (a)

12 (a, c)

10 (a, c, b, d)

Step 6

Applied Discrete Mathematics Week 14: Trees


The traveling salesman problem
The Traveling Salesman Problem

  • The traveling salesman problem is one of the classical problems in computer science.

  • A traveling salesman wants to visit a number of cities and then return to his starting point. Of course he wants to save time and energy, so he wants to determine the shortest path for his trip.

  • We can represent the cities and the distances between them by a weighted, complete, undirected graph.

  • The problem then is to find the circuit of minimum total weight that visits each vertex exactly once.

Applied Discrete Mathematics Week 14: Trees


The traveling salesman problem1

Toronto

650

550

700

Boston

700

Chicago

200

600

New York

The Traveling Salesman Problem

  • Example: What path would the traveling salesman take to visit the following cities?

Solution: The shortest path is Boston, New York, Chicago, Toronto, Boston (2,000 miles).

Applied Discrete Mathematics Week 14: Trees


The traveling salesman problem2
The Traveling Salesman Problem

  • Question: Given n vertices, how many different cycles Cn can we form by connecting these vertices with edges?

  • Solution: We first choose a starting point. Then we have (n – 1) choices for the second vertex in the cycle, (n – 2) for the third one, and so on, so there are (n – 1)! choices for the whole cycle.

  • However, this number includes identical cycles that were constructed in opposite directions. Therefore, the actual number of different cycles Cn is (n – 1)!/2.

Applied Discrete Mathematics Week 14: Trees


The traveling salesman problem3
The Traveling Salesman Problem

  • Unfortunately, no algorithm solving the traveling salesman problem with polynomial worst-case time complexity has been devised yet.

  • This means that for large numbers of vertices, solving the traveling salesman problem is impractical.

  • In these cases, we can use approximation algorithms that determine a path whose length may be slightly larger than the traveling salesman’s path, but can be computed with polynomial time complexity.

  • For example, artificial neural networks can do such an efficient approximation task.

Applied Discrete Mathematics Week 14: Trees


Let us talk about
Let us talk about…

  • Trees

Applied Discrete Mathematics Week 14: Trees


Trees
Trees

  • Definition: A tree is a connected undirected graph with no simple circuits.

  • Since a tree cannot have a simple circuit, a tree cannot contain multiple edges or loops.

  • Therefore, any tree must be a simple graph.

  • Theorem: An undirected graph is a tree if and only if there is a unique simple path between any of its vertices.

Applied Discrete Mathematics Week 14: Trees


Trees1
Trees

  • Example: Are the following graphs trees?

Yes.

No.

No.

Yes.

Applied Discrete Mathematics Week 14: Trees


Trees2
Trees

  • Definition: An undirected graph that does not contain simple circuits and is not necessarily connected is called a forest.

  • In general, we use trees to represent hierarchical structures.

  • We often designate a particular vertex of a tree as the root. Since there is a unique path from the root to each vertex of the graph, we direct each edge away from the root.

  • Thus, a tree together with its root produces a directed graph called a rooted tree.

Applied Discrete Mathematics Week 14: Trees


Tree terminology
Tree Terminology

  • If v is a vertex in a rooted tree other than the root, the parent of v is the unique vertex u such that there is a directed edge from u to v.

  • When u is the parent of v, v is called the child of u.

  • Vertices with the same parent are called siblings.

  • The ancestors of a vertex other than the root are the vertices in the path from the root to this vertex, excluding the vertex itself and including the root.

Applied Discrete Mathematics Week 14: Trees


Tree terminology1
Tree Terminology

  • The descendants of a vertex v are those vertices that have v as an ancestor.

  • A vertex of a tree is called a leaf if it has no children.

  • Vertices that have children are called internal vertices.

  • If a is a vertex in a tree, then the subtree with a as its root is the subgraph of the tree consisting of a and its descendants and all edges incident to these descendants.

Applied Discrete Mathematics Week 14: Trees


Tree terminology2
Tree Terminology

  • The level of a vertex v in a rooted tree is the length of the unique path from the root to this vertex.

  • The level of the root is defined to be zero.

  • The height of a rooted tree is the maximum of the levels of vertices.

Applied Discrete Mathematics Week 14: Trees


Trees3
Trees

James

  • Example I: Family tree

Christine

Bob

Frank

Joyce

Petra

Applied Discrete Mathematics Week 14: Trees


Trees4
Trees

/

  • Example II: File system

usr

bin

temp

bin

spool

ls

Applied Discrete Mathematics Week 14: Trees


Trees5
Trees

  • Example III: Arithmetic expressions

+

-

y

z

x

y

This tree represents the expression (y + z)(x - y).

Applied Discrete Mathematics Week 14: Trees


Trees6
Trees

  • Definition: A rooted tree is called an m-ary tree if every internal vertex has no more than m children.

  • The tree is called a full m-ary tree if every internal vertex has exactly m children.

  • An m-ary tree with m = 2 is called a binary tree.

  • Theorem: A tree with n vertices has (n – 1) edges.

  • Theorem: A full m-ary tree with i internal vertices contains n = mi + 1 vertices.

Applied Discrete Mathematics Week 14: Trees