Displaying Data with Graphs. Chapter 6 Mathematics of Data Management (Nelson) MDM 4U

Transcription

1 Displaying Data with Graphs Chapter 6 Mathematics of Data Management (Nelson) MDM 4U

2 Cause and Effect Diagrams Developed by Dr. Kaoru Ishikawa in 1943 (Japan) Picture composed of lines and symbols designed to represent a meaningful relationship between an effect and its causes Effect (characteristics that need improvement) on the right and causes on the left 2

3 Cause and Effect Diagrams People Materials Work Methods C a u s e s Primary Cause Secondary Cause Quality Characteristic E f f e c t Environment Equipment Measurement Cause-and-Effect Diagram Ishikawa Diagram Also called a Fishbone Diagram 3

4 Constructing Cause and Effect Diagrams 4

5 Constructing Cause and Effect Diagrams 5

6 Graphs You will learn basic principles of graphs as well as how they can be applied to problem solving.

7 The Seven Bridges of Königsberg Running through the city was the River Pregel. It separated the city into two mainland areas and two large islands. There were 7 bridges connecting the various areas of land.

8 The Seven Bridges of Königsberg The residents of Königsberg wondered whether they could wander around the city, crossing each of the seven bridges once and only once. Can you find a way?

9 Reformulating the problem With this observation, we can re-draw the bridges of Königsberg as follows: A B C D

10 Conditions for a solution Euler s Eureka! moment was realising that whenever you cross into a bit of land, you also have to cross back out of it. Therefore, for a bridge tour to be possible, there must be an even number of bridges coming out of every bit of land. (Except for the starting and finishing points.)

11 An impossible problem! If we look again at the map of Königsberg, we see that there are an odd number of bridges coming out of every bit of land, so such a walk around the city is impossible. By solving the problem the way he did, Euler invented the subject of graph theory. A graph is a collection of nodes and edges. A B C D

12 What Is A Graph A collection of vertices and edges (sets and relations) Vertex Edge Vertex As shown in the last section, visualizing the set of relations can be easier when shown in graphical rather than textual form. Types of graphs Directed Undirected

13 Directed Graphs Edges are one-way A B Direction of travel on streets Peppermint Patty Charlie Brown Little Red Linus Sally Relationships ( likes ) V = {Charlie Brown, Peppermint Patty, Little Red, Linus, Sally} E = {(Charlie Brown, Little Red), (Peppermint Patty, Charlie Brown), (Sally, Linus)}

14 Directed Graphs (2) Organizational structure ( reports to ) CEO VP Information Technology VP Finance Webmaster Database Administrator Payroll Accounting

15 Undirected Graphs Edges are two-way Alberta towns, cities and the highways that connect them. Edmonton Red Deer Banff Canmore Calgary V = {Edmonton, Red Deer, Calgary, Canmore, Banff} E = {(Banff, Canmore), (Calgary, Canmore), (Calgary, Red Deer), (Edmonton, Red Deer)}

16 Graphs Can Be Labeled The annotations can provide additional information. Speed limits Calgary 110 km Red Deer Debts owed $100 Bill Mary $25

17 Adjacent Vertices In a directed graph vertices are adjacent if they are connected by an edge. In an undirected graph: v1 is adjacent to v2 if there is a direct edge from v1 to v2 Edmonton Red Deer Banff Canmore Calgary Adjacent: Banff-Canmore, Canmore-Calgary etc. Not adjacent: Red Deer and Canmore etc.

18 Paths There is a path between vertices if they are directly connected by an edge or indirectly connected. Edmonton Red Deer Banff Canmore Calgary Medicine Hat Lethbridge Red Deer and Lethbridge are not adjacent but there is at least one path from Red Deer to Lethbridge.

19 Paths (Directed Graphs) There is a path from v 1 to v n (v 1, v 2, v 3..v n-1, v n ) if v 1 is adjacent to v 2, v 2 is adjacent to v 3...v n-1 is adjacent to v n. The Love graph You Her Him Somebody else

20 Cyclical Paths A path that starts and ends at the same vertex. Calgary Lethbridge Medicine Hat One cycle = (Calgary, Medicine Hat, Lethbridge, Calgary)

21 Degree Of A Vertex It s the number of edges that are connected to it. Red Deer Canmore Calgary Medicine Hat Lethbridge Degree of Calgary = 4 Degree of Medicine Hat, Lethbridge = 2 Degree of Canmore and Red Deer = 1

22 Multigraphs Most graphs don t allow for multiple edges between a pair of vertices. Multigraphs allow multiple edges to connect a pair of vertices. Example: path finding when alternatives are possible. A B

23 Applications Of Graphs Logistics and supply (multigraph) m A Air: 4 hours, $700 Truck: 2days, B

24 Application Of Graphs (2) Visualizing social networking connections e.g., 6 degrees of separation A B

25 Application Of Graphs (3) Disease transmission: examining which people had intimate contact in order to determine who may have become infected. Original

26 Gantt Chart for Software Launch Project