L02
Review: Sequences,
Sums, and Induction
Review
Sequences
L01
CIS 2910
Dr Charlie Obimbo - University of Guelph
2
Sequences
Definition
Sequences are ordered lists of elements or
objects. They can be finite (we call them
strings), or infinite.
We will e
Graph Coloring
CIS 2910
Joe Sawada
University of Guelph
CIS 2910
Graph Coloring
1/7
Graph Coloring
Consider the problem of assigning frequencies to various radio transmitting stations. If
the transmission ranges overlap, then it is not desirable for them
Connectivity
CIS 2910
Joe Sawada
University of Guelph
CIS 2910
Connectivity
1/1
Connectivity
Definition
Let G = (V, E) be a simple graph. A path is a sequence of vertices v0 , v1 , . . . , vn where
there is an edge between each consecutive pair of vertice
Planar Graphs
CIS 2910
Joe Sawada
University of Guelph
CIS 2910
Planar Graphs
1/8
Planar Graphs
Circuit Design
When designing electronic circuits or computer chips, it is often desirable to implement
the designs without wires crossing.
crosssings
When lay
CIS2910 Discrete Structures II
Lab 1
Arithmetic, Sums & Sequences
Dr. Charlie Obimbo
Week 1
Name:
1
Divisibility
1. Give a Yes or No answer to the following Questions
1. Is 724378724378 divisible by 4?
2. Is 6352600 divisible by 5?
3. Is 6783458 divisible
University of Guelph
CIS 2910 F13 Midterm (Oct. 15)
Instructor: Joe Sawada
First Name:
Last Name:
Student Number:
Problem 1: (5 marks)
Problem 6: (6 marks)
Problem 2: (6 marks)
Problem 7: (6 marks)
Problem 3: (5 marks)
Problem 8: (6 marks)
Problem 4: (5 m
CIS 2910 (Fall14)
Due: September 30th , 2014
Assignment 2
Solution
Dr. Charlie Obimbo
1. Mathematical Induction
Use mathematical induction to prove that n! > n3 for n > a.
[4 marks]
First find a.
n
3
n!
n3
4
6
27
5
120
125
24
64
6
720
216
Thus a = 5
Theor
Recurrence Relations
CIS 2910
Joe Sawada
University of Guelph
CIS 2910
Recurrence Relations
1 / 15
Recurrence Relations
Recall how sequences can be defined recursively
I
specify one or more initial terms a0 (a1 ) - the basis
I
give a rule for determining
NP-Completeness
CIS 3150
Joe Sawada
University of Guelph
CIS 3150
NP-Completeness
1 / 24
Clique
Clique
A clique is a subset V 0 of V such that every pair of vertices in V 0 are adjacent (a
complete subgraph).
Decision problem: Does G contain a clique of s
Terminology and Special Types of Graphs
CIS 2910
Joe Sawada
University of Guelph
CIS 2910
Terminology and Special Types of Graphs
1/1
Terminology for Undirected Graphs
Definition
Two vertices u and v are said to be adjacent or neighbors if there is an edg
Generation Algorithms
CIS 2910
Joe Sawada
University of Guelph
CIS 2910
Generation Algorithms
1 / 12
Generating algorithms
So far we have only looked at the counting problem: How many items are there?
Now, we want to handle the question: What are all the
Euler and Hamilton Paths
CIS 2910
Joe Sawada
University of Guelph
CIS 2910
Euler and Hamilton Paths
1/7
Euler Paths and Cycles
The Seven Bridges of K
onigsberg
The town of K
onigsberg has 4 land regions separated by a river with 7 bridges joining
them. A
Representing Graphs and Isomorphism
CIS 2910
Joe Sawada
University of Guelph
CIS 2910
Representing Graphs and Isomorphism
1/8
Representing Graphs
For large graphs, we want to apply computers to solve tasks such as:
I Finding the shortest path between two
Introduction to Graphs
CIS 2910
Joe Sawada
University of Guelph
CIS 2910
Introduction to Graphs
1 / 10
Problems Solved Using Graphs
Graphs are frequently used to model real world problems. Many of these
problems can be solved by applying well known graph
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CIS 2910 Fall 2008 Homework #3
Instructor: Joe Sawada
Make sure you include a completed and signed cover page as your rst page. In addition to
correctness, your answers must be clear, concise, and readable.
1. [6 marks] Implement the algorithm to generate
CIS 2910 Fall 2008 Homework #1 SOLUTIONS
Instructor: Joe Sawada
Make sure you include a completed and signed cover page as your rst page. In addition to
correctness, your answers must be clear, concise, and readable.
1. [3 marks] Ch 5.1 #24 How many strin