Question

Find an optimal circuit for the weighted, direct graph represented by the following matrix \(W\). Show the actions step by step. \\[W=\left[\begin{array}{rrrrr}0 & 8 & 13 & 18 & 20 \\\3 & 0 & 7 & 8 & 10 \\\4 & 11 & 0 & 10 & 7 \\ 6 & 6 & 7 & 0 & 11 \\\10 & 6 & 2 & 1 & 0\end{array}\right]\\]

Short answer

The optimal circuit for the weighted, directed graph represented by the matrix \(W\) will be the circuit with the smallest total weight found by applying the Dijkstra's algorithm to each node of the graph. The specific circuit and its total weight will depend on the exact weights in the matrix \(W\).

Step-by-step solution

Represent Matrix as a Graph

Represent the given weighted matrix \(W\) as a direct graph. The nodes of the graph are represented by the rows and columns of the matrix, and the weights of the edges are given by the elements of the matrix.

Apply Dijkstra's Algorithm

Apply Dijkstra's algorithm to find the shortest paths starting from each node. Start with the node with the smallest weight, mark it as 'visited', and update the distances to its neighboring nodes. Then move to the next unvisited node with the smallest weight, mark it as 'visited', and update the distances to its neighboring nodes. Repeat this process until all the nodes have been visited.

Determine Optimal Circuit

After applying Dijkstra's algorithm to each node, determine the circuit that visits each node exactly once and returns to the starting node with the smallest total weight. This circuit will be the optimal circuit for the given weighted, direct graph.

Calculate Total Weight of Optimal Circuit

Calculate the total weight of the optimal circuit by summing up the weights of the edges making up the circuit. This will give the smallest possible total weight for a circuit in the given graph.

Key concepts

Dijkstra's Algorithm

Understanding Dijkstra's Algorithm is foundational when exploring the realm of Optimal Circuits in Weighted Directed Graphs. This algorithm is an example of a greedy algorithm, meaning it makes a series of choices that appear to be the best at each step, leading to a globally optimum solution. Originally formulated by computer scientist Edsger W. Dijkstra, the algorithm is designed to find the shortest path between nodes in a graph.

The steps involve initialization, where distances from a source node to other nodes are set at infinity, barring the source itself, which is set to zero. As we iterate, the algorithm picks the node with the smallest distance, 'visits' it, and updates the cost of reaching its neighboring nodes. This process repeats until all nodes are visited.

In the context of the given exercise, Dijkstra's algorithm helps map out the shortest paths from each node, creating stepping stones toward discovering the optimal circuit. It ensures efficiency by prioritizing paths with the least weight, which contributes to computing the optimal circuit. By implementing this algorithm step by step, we can track the pathway with the lowest cumulative weight, ultimately leading to the most cost-effective solution.

Graph Theory

At the heart of solving the optimal circuit problem lies Graph Theory, which is a vast area of mathematical research dealing with graphs, which are mathematical structures used to model pairwise relations between objects. Graphs are made up of vertices (also called nodes) that are connected by edges. If these edges have direction, we refer to the structure as a 'directed graph' as in our exercise.

Interpreting the Matrix as a Graph

In this case, we interpret the given matrix as a graph representation (Step 1 of the solution). The columns and rows signify nodes, and the matrix values represent the weights of directed edges between nodes. In the given exercise, it's important for students to visualize the numeric matrix as a graph because it aids in understanding relationships between nodes.

When talking about optimal circuits, which are essentially routes that return to the starting node after visiting all other nodes exactly once, graph theory provides the tools and theorems that define the best ways to link nodes - an essential concept for our problem-solving endeavor.

Algorithmic Complexity

Algorithmic Complexity is a principle in computer science that helps to quantify the efficiency of an algorithm in terms of time and space used. It is essential for understanding the practical limits of performing calculations.

When we apply Dijkstra's Algorithm (Step 2 of the solution), we look not only for the correctness of the solution but also for its efficiency. The algorithm's complexity determines how fast it can run, particularly as the size of the graph grows. Time complexity is usually expressed in Big O notation, which characterizes the running time of an algorithm as a function of the input size.

In Dijkstra's algorithm, if the data structure used is a simple array, the complexity is O(n^2), with 'n' being the number of nodes. If a binary heap or priority queue is utilized, the complexity can be reduced to O((n + e) log n), where 'e' represents the number of edges. This awareness of complexity is vital, especially in practical applications involving large, complex graphs where run-time can be significant and resources are at a premium.