Question

There are eight small islands in a lake, and the state wants to build seven= bridges to connect them so that each island can be reached from any other one via one or more bridges. The cost of constructing a bridge is propor- tional to its length. The distances between pairs of islands are given in the following table. $$\begin{aligned} &\begin{array}{ccccccccc} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 1 & \- & 240 & 210 & 340 & 280 & 200 & 345 & 120 \\ 2 & \- & \- & 265 & 175 & 215 & 180 & 185 & 155 \\ 3 & \- & \- & \- & 260 & 115 & 350 & 435 & 195 \\ 4 & \- & \- & \- & \- & 160 & 330 & 295 & 230 \\ 5 & \- & \- & \- & \- & \- & 360 & 400 & 170 \\ 6 & \- & \- & \- & \- & \- & \- & 175 & 205 \\ 7 & \- & \- & \- & \- & \- & \- & \- & 305 \\ 8 & \- & \- & \- & \- & \- & \- & \- & - \end{array}\\\ &\text { A } \end{aligned}$$ Find which bridges to build to minimize the total construction cost.

Short answer

Construct bridges (1-8), (3-5), (2-4), (6-7), (2-8), (1-6), (3-8) with a total distance of 1135.

Step-by-step solution

- Understand the problem

The goal is to connect all eight islands with seven bridges such that each island can be reached from any other island and the total cost (distance) is minimized. This is a minimum spanning tree problem.

- Create the Graph

List all the islands as vertices and each possible bridge (with its distance) as edges. The given table shows the distances between each pair of islands.

- Select an Algorithm

Prim's or Kruskal's algorithm can be used to find the minimum spanning tree. For simplicity, the following steps will use Kruskal's algorithm.

- Sort All Edges

Sort all possible bridges by their distances in ascending order. This helps in building the minimum spanning tree by always picking the smallest edge that does not form a cycle.

- Initialize Disjoint Sets

Each island starts as its own set. Use the union-find structure to manage merging these sets as edges are added.

- Add Smallest Edges

Iterate through the sorted edges and add the smallest edge that does not form a cycle until there are exactly seven edges. This involves checking if two vertices are in the same set and merging them if they are not.

- List the Chosen Edges

The selected edges (bridges) are: (1, 8, 120), (3, 5, 115), (2, 4, 175), (6, 7, 175), (2, 8, 155), (1, 6, 200), (3, 8, 195). The corresponding distances should be summed up.

- Calculate Total Cost

Add the distances of the chosen edges: 120 + 115 + 175 + 175 + 155 + 200 + 195 = 1135.

Key concepts

Graph Theory

Graph theory is a branch of mathematics that studies the properties and applications of graphs. A graph consists of vertices (nodes) connected by edges (lines). It is a powerful tool for representing and solving problems involving network structures, such as computer networks, road maps, and social networks. In the context of our problem, the islands are the vertices and the bridges are the edges. The goal is to use graph theory to find the minimum way to connect all the islands with the least total distance.

Kruskal's Algorithm

Kruskal's Algorithm is a popular method for finding the Minimum Spanning Tree (MST) of a graph, which is a subset of the edges connecting all vertices with the minimum total edge weight and no cycles. Here are the steps involved in Kruskal's Algorithm:

• **Sort All Edges**: Begin by sorting all the edges in the graph according to their weight in ascending order.
• **Initialize Disjoint Sets**: Each vertex is initially its own set. We use a union-find data structure to keep track of the sets.
• **Add Edges**: Iterate through the sorted edges and include the smallest edge in the MST if it connects two different sets. Merge sets if an edge is added.

By following these steps, we ensure that the total weight of the edges in the MST is minimized. In our problem, the MST connects all eight islands with seven bridges, resulting in minimal total construction costs.

Data Structures and Algorithms

Data structures and algorithms are fundamental for efficiently solving computational problems. In our exercise, we utilize specific data structures and algorithms to find the MST.

• **Union-Find Data Structure**: This is employed to manage the sets of edges. It supports two main operations: find (determines the set a particular element is in) and union (merges two sets).
• **Greedy Algorithms**: Kruskal's Algorithm is an example of a greedy algorithm, which makes locally optimal choices at each step with the hope of finding a global optimum.

Understanding these concepts is crucial for solving MST problems effectively. The efficient management of edge selections and set operations is what enables Kruskal's Algorithm to find the minimum spanning tree with minimal computational cost.