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Minimal Spanning Trees
Networks are everywhere in modern society: roads, wires, water and gas pipes all connect one place to another. Computers are built of networks at many levels, from the microscopic connections between transistors in a chip to the cables and satellites that link the internet around the world. People who build networks often need to work out the most efficient way to make connections, which can be a difficult problem.
This puzzle shows students the decisions involved in linking a network between houses in a muddy city. It can lead on to a discussion of minimal spanning tree algorithms for optimizing networks.
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Mordechai (Moti) Ben-Ari from the Weizmann Institute of Science, Israel has programmed The Muddy City Minimal Spanning Trees Unplugged activity in Scratch which can be downloaded in a zip file of the complete set of activities . Please read the ReadMe.txt for documentation.
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Blaine Booher from the University of Cincinnati has a variation on the Minimal Spanning Trees activity called Internet Routing where students learn a brief overview of the history of the internet and its impact on society. Students work to understand internet routing and how all of the computers on the internet are interconnected by using the analogy of roads in a town. This activity was developed for high school students.
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NCTM Illuminations has an activity called Road Trip! to find the shortest path to save expenses that investigates three different methods to determine the shortest route: the Nearest Neighbour method, the Cheapest Link method, and the Brute Force method.
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As another extension, try the online games from Wiliam Cook to demonstrate the Travelling Salesman Problem(TSP) below:
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Cre8ate Maths UK has a resource called Working for Efficiency that looks at 3 different network problems that are encountered in practical logistical planning. The resource helps in learning how the Prim's algorithm works in the real world.
Note: You will need to register (free) as a teacher at Cre8ate Maths UK to access this resource
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An older version of this activity can be downloaded in PDF format here. The content is similar to the current version, but there's some extra technical information.
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Wikipedia: Travelling Salesman Problem : Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once.
Wikipedia: Route Inspection Problem or The Chinese Postman Problem : to find a shortest closed path or circuit that visits every edge of a (connected) undirected graph. When the graph has an Eulerian circuit (a closed walk that covers every edge once), that circuit is an optimal solution.
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The Mathmaniacs web site has a similar activity (lesson 13)
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2 July Maths UK website for Mathematics has some useful resources in Graphs and Networks:
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TES Connect UK contributor numskull has the Minimum Connector Program . Instructions: Create and solve minimum connector problems interactively using the power of Excel. Use for whole-class work with a projector, or use the practice sheets for individual/small group work at a workstation. The answer is illustrated step-by-step on a matrix, which can also be input by the user if preferred (for the graphical version of the algorithm, simply draw the graph alongside or on a piece of paper). On the practice sheets your answers can also be checked by the program.
Note: Teachers will need to register on TES Connect UK in order to access resources. -
Mr Barton Maths UK has some short presentations for introducing some concepts below:
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nrich Maths has the following activities with notes and solutions provided:
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MathsNet has the following free interactive activities:
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The Student Room UK has the following revision sections:
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Christine Mumford has excellent explanation and demonstration of the Travelling Salesman Algorithm Heuristics: Nearest Neighbour, Multi-Fragment, Farthest Insertion developed by Howard Plummer
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Pasi Sahlberg has a set of Minimum Connector Problems using Prims and Kruskal's algorithms with solutions attached