Find the least number of cables required to connect 100...

Name: Problem 41, Chapter 6: Discrete Mathematics and its Applications
Uploaded: 2019-11-04T10:52:41-08:00
Duration: 1 min 8 s
Channel: Clarissa Noh
Description: Find the least number of cables required to connect 100 computers to 20 printers to guarantee that every subset of 20 computers can directly access 20 different printers. (Here, the assumptions about cables and computers are the same as in Example 9.) Justify your answer.

Find the least number of cables required to connect 100 computers to 20 printers to guarantee that every subset of 20 computers can directly access 20 different printers. (Here, the assumptions about cables and computers are the same as in Example 9.) Justify your answer.

Key Concepts

Bipartite Graph

This concept involves representing the problem as a graph where the two disjoint sets (in this case, computers and printers) form the two sides. Edges (or cables) connect vertices from one set to the other. Modeling the situation in this way facilitates the use of combinatorial and graph theoretical tools to analyze connections and matchings between the two sets.

Hall's Marriage Theorem

A fundamental result in matching theory for bipartite graphs, Hall's Marriage Theorem provides a necessary and sufficient condition for the existence of a perfect matching between two sets. In this context, it ensures that for every subset of computers the set of connected printers is sufficiently large (i.e., has at least as many printers as computers in the subset) so that a complete matching is possible for any group of 20 computers.

Degree Conditions and Edge Minimization

This concept involves determining the least number of edges needed while satisfying the connection requirements. It includes analyzing the minimum number of connections (or cables) each printer and computer must have to guarantee a matching for any subset of computers. Such analysis is integral to problems of combinatorial optimization that seek to minimize resource usage while ensuring complete connectivity or coverage conditions.

Extremal Combinatorics

This branch of combinatorics deals with determining the extreme (maximum or minimum) number of elements, such as cables, within a structured set given certain conditions. In this problem, extremal combinatorial techniques are used to establish a lower bound on the number of cables needed to ensure that every group of 20 computers can access 20 distinct printers.

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