**BFS and its Applications:**You are given a directed graph \( G = (V, E) \), and a special vertex \( v \). Provide an algorithm based on Breadth-First Search (BFS) that determines in \( O(V + E) \) time whether \( v \) is part of a cycle.

BFS means Breadth first Search, It traverse the graph level by level and For BFS traversal Queue…

Answered: BFS and its applications: You are given…

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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We are given an undirected connected graph G = (V, E) and vertices s and t.Initially, there is a robot at position s and we want to move this robot to position t by moving it along theedges of the graph; at any time step, we can move the robot to one of the neighboring vertices and the robotwill reach that vertex in the next time step.However, we have a problem: at every time step, a subset of vertices of this graph undergo maintenance andif the robot is on one of these vertices at this time step, it will be destroyed (!). Luckily, we are given theschedule of the maintenance for the next T time steps in an array M [1 : T ], where each M [i] is a linked-listof the vertices that undergo maintenance at time step i.Design an algorithm that finds a route for the robot to go from s to t in at most T seconds so that at notime i, the robot is on one of the maintained vertices, or output that this is not possible. The runtime ofyour algorithm should ideally be O((n + m) ·T ) but you will…
One can manually count path lengths in a graph using adjacency matrices. Using the simple example below, produces the following adjacency matrix: A B A 1 1 B 1 0 This matrix means that given two vertices A and B in the graph above, there is a connection from A back to itself, and a two-way connection from A to B. To count the number of paths of length one, or direct connections in the graph, all one must do is count the number of 1s in the graph, three in this case, represented in letter notation as AA, AB, and BA. AA means that the connection starts and ends at A, AB means it starts at A and ends at B, and so on. However, counting the number of two-hop paths is a little more involved. The possibilities are AAA, ABA, and BAB, AAB, and BAA, making a total of five 2-hop paths. The 3-hop paths starting from A would be AAAA, AAAB, AABA, ABAA, and ABAB. Starting from B, the 3-hop paths are BAAA, BAAB, and BABA. Altogether, that would be eight 3-hop paths within this graph. Write a program…
Consider an undirected graph with n nodes and m edges. The goal is to find a path between two specified nodes u and v that maximizes the minimum weight of any edge along the path. Assume that all edge weights are positive and distinct. Design an algorithm to solve this problem with a time complexity of O(m log n).
Relaxing the Edges |V|+1 times on a directed graph will compute whether the graph has negative cycle or not True or False
Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k.
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Algorithms
Note: Solve the following question and please don't repost previous answers and don't give a computer generated answer. Problem: Draw a simple, connected, directed, weighted graph with 9 vertices and 17 edges, each with unique edge weights. Identify one vertex as a “start” vertex and illustrate a running of Dijkstra’s algorithm on this graph.
Consider an undirected graph G with 100 nodes. The maximum number of edges to be included in G so that the graph is not connected is
Give a linear time algorithm via pseudo code that takes as input a directed acyclic graph G (V, E) and two vertices u and v, that returns the number of simple paths from u to v in G. Your algorithm needs only to count the simple paths, not list them. Explain why your code runs in linear time.
Write an algorithm using depth-first search that determines if a graph is bipartite. What is the complexity of your algorithm?
Consider the following directed graph. E N B C Which of the following are valid cycles? A, A, D A, D, C, A A, A, A, A, A C, E, B, C A E, D, C, B, E C, E, B

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BFS and its applications: You are given a directed graph G = (V, E), %3D and a special vertex v. Give an algorithm based on BFS that determines in O(V +E) time whether v is part of a cycle.