Problem 34 - Discrete Mathematics with Applications

Question

Using Dijkstra's algorithm, find a shortest path and its length from vertex \(a\) to the other vertices. $$\left.\begin{array}{c|cccccc} & a & b & c & d & e & f \\ a & \infty & 3 & 5 & \infty & \infty & \infty \\ b & \infty & \infty & \infty & 2 & \infty & \infty \\ c & \infty & \infty & \infty & 3 & \infty & \infty \\ d & 4 & \infty & \infty & \infty & 3 & 6 \\ e & \infty & 5 & \infty & \infty & \infty & \infty \\ f & \infty & \infty & 4 & \infty & \infty & \infty \end{array}\right]$$

Step-by-Step Solution

Answer

The shortest path and its length from vertex \(a\) to the other vertices using Dijkstra's algorithm are: - Vertex \(b\): Path = \(a \to b\), Length = 3 - Vertex \(c\): Path = \(a \to c\), Length = 5 - Vertex \(d\): Path = \(a \to b \to d\), Length = 5 - Vertex \(e\): Path = \(a \to b \to d \to e\), Length = 6 - Vertex \(f\): Path = \(a \to b \to d \to f\), Length = 9

Step 1: Initialize unvisited set and cost list

Create a set of unvisited vertices, \(V\), containing all vertices in the graph: \(V=\{a, b, c, d, e, f\}\). Assign the cost list for these vertices: \(C(a) = 0\), \(C(b) = \infty\), \(C(c) = \infty\), \(C(d) = \infty\), \(C(e) = \infty\), and \(C(f) = \infty\).

Step 2: Find lowest cost unvisited vertex

Find the vertex with the lowest cost in the unvisited set, which is vertex \(a\) with cost 0. Mark \(a\) as visited and remove it from the unvisited set: \(V = \{b, c, d, e, f\}\).

Step 3: Update costs to unvisited neighbors

Update the costs of the unvisited neighbors of vertex \(a\), which are vertices \(b\) and \(c\), as follows: - For vertex \(b\): The current cost is \(C(b) = \infty\). The alternative path through vertex \(a\) consists of \(C(a) + P(a, b) = 0 + 3 = 3\). Since \(3 < \infty\), update the new cost for vertex \(b\), \(C(b) = 3\). - For vertex \(c\): The current cost is \(C(c) = \infty\). The alternative path through vertex \(a\) consists of \(C(a) + P(a, c) = 0 + 5 = 5\). Since \(5 < \infty\), update the new cost for vertex \(c\), \(C(c) = 5\).

Step 4: Repeat steps 2 and 3

Repeat the process by selecting the unvisited vertex with the lowest cost, updating the costs of its unvisited neighbors, and marking it as visited until all vertices are visited. At the end of the process, the cost list will be: \(C(a) = 0, C(b) = 3, C(c) = 5, C(d) = 5, C(e) = 6, C(f) = 9\).

Step 5: Conclusion

The shortest path and its length from the starting vertex \(a\) to the other vertices are as follows: - Vertex \(b\): Path = \(a \to b\), Length = 3 - Vertex \(c\): Path = \(a \to c\), Length = 5 - Vertex \(d\): Path = \(a \to b \to d\), Length = 5 - Vertex \(e\): Path = \(a \to b \to d \to e\), Length = 6 - Vertex \(f\): Path = \(a \to b \to d \to f\), Length = 9

Key Concepts

Shortest Path Problem

The Shortest Path Problem is a fundamental concept in computer science and graph theory. It focuses on finding the minimum path between two points in a graph. Imagine you are in a maze. You want to get from your starting point to your destination quickly, with the least effort.
This problem is not just theoretical but has real-world applications too. It can be seen in navigation systems, logistics, and network routing. For instance, when you use GPS navigation, it's solving a Shortest Path Problem to direct you from your current location to your destination.

In the context of Dijkstra's algorithm, you're provided with a beginning vertex and tasked with discovering the short path to other vertices in the graph. Each edge between the vertices has a 'weight', representing the effort or distance needed to move from one vertex to another. The goal is to accumulate the lowest possible total cost from the start vertex to any of the vertices you're interested in.
It's essential to understand that this problem assumes all the edge weights are non-negative, which fits perfectly with Dijkstra's approach.

Graph Theory

Graph Theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph consists of vertices (also called nodes) and edges (connections between nodes). Think of vertices as places you can be, and edges as routes you can take to get from one vertex to another.

Graphs can be directed or undirected, meaning edges have a direction in which they can be traversed or not. They can also be weighted or unweighted, indicating whether edges have a numerical value associated with them. Weighted graphs are commonly used in Shortest Path Problems since the weights can represent distances or costs.
In the exercise, we're dealing with a directed, weighted graph, where each path between vertices has a direction and a specific value. Understanding these terms is essential to leverage algorithms like Dijkstra's effectively. By applying Graph Theory, you can model and solve complex problems in fields like computer networking, social networks, and transportation systems.

Algorithmic Complexity

Algorithmic Complexity refers to the amount of resources, such as time and memory, that an algorithm requires to execute. An efficient algorithm is one that uses resources sparingly while producing the correct result. In the case of Dijkstra's Algorithm, understanding its complexity provides insight into its performance and suitability.

When analyzing Dijkstra's algorithm, we typically consider its time complexity. Its running time is \(O(V^2)\) for a simple implementation using an adjacency matrix. Here, \(V\) is the number of vertices in the graph. However, with more efficient data structures like priority queues, the time complexity can be reduced to \(O(V \log V + E)\), where \(E\) is the number of edges.
These complexities indicate how the algorithm might scale with the size of the input. For graphs with a large number of vertices and edges, using an optimized method to store and manage data is crucial. Recognizing which algorithm to apply based on its complexity ensures that we can address the Shortest Path Problem efficiently, even in expansive networks.