**Title: Algorithm for Finding the Shortest Cycle in a Directed Weighted Graph Containing a Specific Edge****Introduction:**Given a directed weighted graph \( G \) with \( n \) vertices and \( m \) edges, where all edges have positive weights, this page will guide you through deriving an algorithm to identify the shortest cycle containing a specific edge \( (u, v) \).**Definition:**- **Graph \( G \):** A structure consisting of vertices and edges.- **Directed Graph:** Each edge has a direction, going from one vertex to another.- **Weighted Graph:** Each edge has a weight (a positive number in this case).- **Cycle:** A path starting and ending at the same vertex without traversing any vertex more than once except for the starting/ending vertex.- **Shortest Cycle containing \( (u, v) \):** For a specified edge \( (u, v) \), it's the cycle that has the smallest total weight among all cycles that include the edge \( (u, v) \).**Problem Statement:**- **Objective:** To find the shortest cycle in \( G \) that includes the edge \( (u, v) \) or determine that no such cycle exists.- **Constraint:** The algorithm must run in \( O(m \log n) \) time complexity.**Algorithm Explanation:**1. **Initialize Data Structures:** - Use appropriate data structures to store the graph information, such as adjacency lists to represent the graph and a priority queue for the shortest path calculations.2. **Weight Calculation:** - Apply Dijkstra's algorithm or another shortest path algorithm that can handle positive weights efficiently to find the shortest path from \( v \) to \( u \) (excluding the edge \( (u, v) \)).3. **Cycle Identification:** - Calculate the total weight of the cycle by adding the weight of edge \( (u, v) \) to the shortest path found from \( v \) to \( u \). - Check if a cycle exists by verifying if the shortest path from \( v \) to \( u \) is non-existent or has infinite weight (indicating no viable path).4. **Output Result:** - If a valid cycle is identified, report the cycle and its total weight. - If no such cycle exists, report the absence of such a cycle.**Mathematical Representation:**

So, the Dijkstra Algorithm may be used to do this. The vertex name, I presume, may be changed to…

Let G be a directed weighted graph with n vertices and m edges such that the edges in G have positive weights. Let (u, v) be an edge in G. By a shortest cycle containing edge (u, v) we mean a cycle containing (u, v) that is of minimum weight, among all cycles containing (u, v) (if such a cycle exists). Give an O(mlgn)-time algorithm that computes a shortest cycle in G containing the edge (u, or reports that no cycle containing edge (u, v) exists.

Let G be a directed weighted graph with n vertices and m edges such that the edges in G have positive weights. Let (u, v) be an edge in G. By a shortest cycle containing edge (u, v) we mean a cycle containing (u, v) that is of minimum weight, among all cycles containing (u, v) (if such a cycle exists). Give an O(mlgn)-time algorithm that computes a shortest cycle in G containing the edge (u, or reports that no cycle containing edge (u, v) exists.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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