Directed Chinese Postman Problem in Urban Routing
In a city, a municipal service is modeled as a directed network where intersections are represented as vertices and one-way streets as directed edges. Due to one-way restrictions, some intersections may not have an equal number of incoming and outgoing roads. (a) Explain how you would verify whether this directed network has an Eulerian circuit. […]
Part b
Assuming the network does not have an Eulerian circuit, outline the method to transform it into an Eulerian circuit using the CPP approach. Specifically, describe how to select additional routes (or duplicate existing ones) by pairing vertices with surplus and deficit degrees using shortest path algorithms.
Matching Odd Vertices in the Chinese Postman Problem
In the Chinese Postman Problem, a non-Eulerian graph contains vertices with an odd degree. Explain the process of matching these odd vertices to duplicate edges to create an Eulerian circuit. Detail how to determine the optimal pairings of odd vertices to minimize the total added distance to the route.
Graph Optimization in Transportation Networks
Consider an urban area modeled as a weighted graph where intersections are nodes and streets are edges. Each edge is assigned two distinct weights: one for the construction cost and another for travel time. Your task is to formulate a comprehensive network model and analyze it from different optimization perspectives. Refer to parts (a) and […]
Modeling Transportation Networks Using Graph Theory
Consider a city transportation network represented as a directed weighted graph. The network must meet specific regulatory and operational constraints for safety and efficiency.\n\n(a) Formulate the constraints that ensure no intersection (node) has more than 4 incoming roads and at least 2 outgoing roads, while also ensuring the network does not contain cycles that may […]
Optimizing Network Connections with MST and Shortest Path
A telecommunications company is designing a network to connect six cities (A, B, C, D, E, F) with communication cables. The distances between cities (in km) are given as weighted edges in the following network graph:\n\n- City A to City B: 4 km\n- City A to City C: 3 km\n- City B to City C: […]
Part b
Determine the shortest path from city A to city F in the original network graph. Then, identify if any minimal extension is needed to the MST so that this shortest path is included within the MST. If an extension is needed, specify the edge(s) to be added.
Determining the Shortest Path Using BFS
Given a description of an unweighted graph with vertices labeled A through F, explain in detail how the Breadth-First Search (BFS) algorithm can be used to find the shortest path from vertex A to vertex F. In your answer, outline the initialization of the algorithm, the process of visiting nodes level by level using a […]
Adjacency Matrix Analysis in Parametrized Networks
Consider a network of \(n\) nodes in which the relationship between nodes is modeled by an adjacency matrix with parametric edge weights. In this network, the weight of an edge from node \(i\) to node \(j\) is given by \(a_{ij} = p \cdot k_{ij}\), where \(p\) is a parameter and \(k_{ij}\) is a known constant […]
Optimizing Paths on Voronoi Edges
Consider a scenario where a city is divided into regions using a Voronoi diagram, with the edges of the diagram representing natural barriers. Explain how you would combine standard graph-based shortest path algorithms with the unique structure of Voronoi edges to optimize pathfinding between two points. Your explanation should discuss how the region boundaries can […]