**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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Write a function that returns true (1) if there is an edge between given two vertices u and v, otherwise false (0), in an directed graph which is represented by adjacency lists.
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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…
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ou are given a directed graph G = (V, E) and two vertices s and t. Moreover, each edge of this graph is colored either blue or red. Your goal is to find whether there is at least one path from s to t such that all red edges in this path appear after all blue edges (the path may not contain any blue edges or any red edges, but if it has both types of edges, all red edges should appear after all blue edges). Design and analyze an algorithm for solving this problem in O(n + m) time.
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Relaxing the Edges |V|+1 times on a directed graph will compute whether the graph has negative cycle or not True or False
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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.