Standard 2: Problem Solving

Transcription

1 Unit 2 Day 17

2 Standard 2: Problem Solving Objective 1: Students will be given opportunities to become "computational thinkers" by applying a variety of problem-solving techniques as they create solutions to problems that are situated in a variety of contexts. e. Create algorithms that meet specified objectives Objective 2: Students will be introduced to selected topics in discrete mathematics including Boolean logic, functions, graphs and the binary number system. b. Explore algorithms for fundamental practical problems, such as searching, sorting and finding paths in graphs.

3 Standard 6: Computing Systems Objective 5: Students will demonstrate a knowledge of networking concepts. a: Understand the Internet concepts of speed, bandwidth, wired, wireless, cell wired, security, etc. b: Understand different network types, features, and capabilities publicly switched networks, domain name server (DNS), addressing, LAN, WAN, VPN, etc. c: Show an understanding of common network problems, IP addressing, and how to solve simple network scenarios.

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7 JOURNAL: DEFINE NETWORK

8 NETWORKS: WHY DO THEY MATTER? Our society is linked by many networks: telephone networks, utility supply networks, computer networks, and road networks. For a particular network there is usually some choice about where the roads, cables, or radio links can be placed. We need to find ways of efficiently linking objects in a network.

9 Another Type Of Network: A Spanning Tree

10 Spanning Tree A spanning tree is a graph where all points are connected.

11 Minimal Spanning Tree A minimal spanning tree is a subgraph with minimal total connections. (A graph where you can get from one point to all other points with the least number of connections.)

12 Meet Our Problem

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14 Once upon a time there was a city that had no roads. Getting around the city was particularly difficult after rainstorms because the ground became very muddy cars got stuck in the mud and people got their boots dirty. The mayor of the city decided that some of the streets must be paved, but didn t want to spend more money than necessary because the city also wanted to build a swimming pool. The mayor therefore specified two conditions: Enough streets must be paved so that it is possible for everyone to travel from their house to anyone else s house only along paved roads, and The paving should cost as little as possible

15 Here is the layout of the city. The number of paving stones between each house represents the cost of paving that route. Find the best route that connects all the houses, but uses as few counters (paving stones) as possible.

16 Journal: How many roads do you have? How many stones did you use? What strategies did you use to come up with your solution(s)?

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18 STUDENT SOLUTIONS

19 Strategies: One good strategy to find the best solution is to start with an empty map, and gradually add counters until all of the houses are linked, adding the paths in increasing order of length, but not linking houses that are already linked. Different solutions are found if you change the order in which paths of the same length are added. Two possible solutions are shown below.

20 Strategies: Another strategy is to start with all of the paths paved, and then remove paths you don t need. This takes much more effort, however.

21 Why it Matters: Computer scientists call the representations of these networks graphs. Real networks can be represented by a graph to solve problems such as designing the best network of roads between local cities, or plane flights around the country. There are also many other algorithms that can be applied to graphs, such as finding the shortest distance between two points, or the shortest route that visits all the points.

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23 Create Your Own

24 Create Your Own Minimal Spanning Tree 1. Put some dots on your paper 2. Connect the dots with as many lines as you want 3. Put a number next to each line to give it a length 4. The houses are represented by circles, the muddy roads by lines, and the length of a road is given by the number beside the line 5. Give your map to a partner. They must create a MINIMAL Spanning Tree

25 Scratch To play at home: or

26 Journal What ALGORITHM can you come up with that describes how many roads are needed for a best solution? Is there a relationship between the number of houses and the number of roads? What are some other uses for minimal spanning trees?

27 The Algorithm Solution?

28 Credits Created by: Patti Smith and Utah ECS Group, June 2015 Modified by Dawnene Young, 2015