Question

A university has eight buildings that need to be connected so that each building's computer network is accessible to the networks in the other buildings. The distance between buildings is given in units of 1000 yards. These distances between buildings are given in the table that follows. The distance from building \(i\) to building \(j\) is the same as the distance from building \(j\) to building \(i .\) $$\begin{array}{c|c|c|c|c|c|c|c|c|} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\\hline 1 & \- & 1.6 & 1.4 & 0.5 & 1.2 & 1.5 & 1.8 & 2.3 \\\2 & & \- & 0.9 & 1.8 & 1.2 & 2.6 & 2.3 & 1.1 \\\3 & & & \- & 2.6 & 1.7 & 2.5 & 1.9 & 1.0 \\\4 & & & & \- & 0.7 & 1.6 & 1.5 & 0.9 \\\5 & & & & & \- & 0.9 & 1.1 & 0.8 \\\6 & & & & & & \- & 0.6 & 1.0 \\\7 & & & & & & & \- & 0.5 \\\8 & & & & & & & & \- \\\\\hline\end{array}$$ Which pairs of buildings should be directly connected to connect all the buildings with a minimum total network length? What is the length of a minimum network? What are the different possible minimum networks?

Short answer

Connect buildings using edges: (1,4), (7,8), (5,8), (6,7), (3,8), (4,5), and (2,3). Minimum total network length is 5.0 units.

Step-by-step solution

Understanding the Problem

We have 8 buildings and the goal is to connect them with minimal total cable length, making sure each building can access the network of every other building. This is a classic problem that requires finding a Minimum Spanning Tree (MST) in a graph where nodes represent buildings and edges are the distances.

Represent the Problem as a Graph

Each building represents a node, and the distances given are the weights of edges between these nodes. Our task is to choose the edges such that they're connected with the least possible total weight.

Choosing an MST Algorithm

To find the MST, we can use either Prim’s or Kruskal’s algorithm. Both are effective ways to accomplish this; however, Kruskal's is often preferred when dealing with lists of edges, as in this problem.

Applying Kruskal's Algorithm - Sorting Edges

List all the edges in the graph along with their distances and sort them in ascending order. The edges are: (1,4,0.5), (7,8,0.5), (5,8,0.8), (3,8,1.0), (6,7,0.6), (4,5,0.7), (5,6,0.9), (4,8,0.9), etc.

Selecting Edges to Form MST

Start adding edges one by one from the sorted list to form the MST, while ensuring no cycle is formed. Using a Union-Find data structure can help efficiently check for cycles. Continue this process until you have included 7 edges (for 8 buildings), ensuring all nodes are connected minimally.

Calculate Minimum Network Length

Add together the weights of the edges included in the MST. These include (1,4,0.5), (7,8,0.5), (5,8,0.8), (6,7,0.6), (3,8,1.0), (4,5,0.7), and (2,3,0.9). The total length is the sum of these weights.

Key concepts

Graph Theory

Graph Theory is a field within mathematics and computer science focused on the study of graphs. A graph comprises nodes (or vertices) connected by edges. These edges can have weights, representing the cost, distance, or capacity between nodes. In many real-world applications, such as road networks or communication setups, we can model problems using graphs.

In the context of our problem at the university, each building is a node, and the distances between them are the weights of the edges connecting these nodes. The goal here is to connect every building, forming a network with the minimal total weight. This specific task is related to finding the Minimum Spanning Tree (MST) of a graph. An MST is a subset of the edges that connects all nodes with the least possible total weight and without any cycles.

Understanding these graph theory principles is crucial in solving numerous practical problems efficiently, making it a valuable toolset for anyone involved in computational tasks.

Kruskal's Algorithm

Kruskal's Algorithm is one of the most popular methods for finding a Minimum Spanning Tree (MST) in a graph. It is especially efficient when the graph is presented as an edge list, which is the case for the university buildings problem.

The algorithm works by following these steps:

• First, sort all edges of the graph in ascending order of their weights.
• Start with an empty tree and add edges one by one from the sorted list, as long as they do not form a cycle.
• The process continues until the tree spans all nodes and includes exactly (n-1) edges, where n is the number of nodes.

The strength of Kruskal's Algorithm is in its simplicity and use of the Union-Find data structure, which helps it efficiently manage and track connected components. This makes it a go-to choice when dealing with graph problems involving edge lists.

Union-Find Data Structure

The Union-Find Data Structure, also known as Disjoint Set Union (DSU), is a powerful tool used for tracking and merging sets. In the context of Kruskal's Algorithm, it aids in efficiently checking whether two nodes are in the same connected component, which is essential to avoid forming cycles.

This data structure supports two essential operations:

• Find: Determines the root of the set in which a specific element is located. Path compression can be used to optimize this operation, flattening the structure whenever Find is called.
• Union: Connects two sets into one, using techniques like union by rank or size to keep the structure balanced.

By applying Union-Find within Kruskal's Algorithm, we ensure the minimum spanning tree forms without cycles, facilitating efficient construction of the optimal network setup for the university's buildings. This efficiency proves fundamental in managing larger data sets, showcasing the excellent synergy between Kruskal's method and the Union-Find Data Structure.