One example of a greedy algorithm is the Dijkstra algorithm for finding the lowest cost path through a weighted graph. The diagram below shows two weighted graphs that a student wants to investigate using Dijkstra's algorithm. In each case the task it to find the lowest cost of reaching every node from v1. Each graph has a single negative weight in it. V1 10 12 V2 V3 10 -9 V4 V1 15 8 V2 V3 30 -7 V1 Graph (a) Graph (b) One of the graphs will yield a correct analysis of the lowest cost for all vertices, and the other will produce an incorrect analysis. Which of the two graphs will produce the incorrect analysis, and explain why the greedy nature of Dijkstra's algorithm is responsible for the incorrect analysis. Your answer should include the key concept of an invariant.

consider alphabetical order if there are more than option thank you.

Given an undirected, weighted graph G(V, E) with n vertices and m edges, design an (O(m + n)) algorithm to compute a graph G1 (V, E1 ) on the same set of vertices, where E1 subset of E and E1 contains the k edges with the smallest edge weights , where k < m.

4-Clique Problem The clique problem is to find cliques in a graph. A clique is a set of vertices that are all adjacent - connected - to each other. A 4-clique is a set of 4 vertices that are all connected to each other. So in this example of the 4-Clique Problem, we have a 7-vertex graph. A brute-force algorithm has searched every possible combination of 4 vertices and found a set that forms a clique: https://en.wikipedia.org/wiki/Clique_problem You should read the Wikipedia page for the Clique Problem (and then read wider if need be) if you need to understand more about it. Note that the Clique Problem is NP-Complete and therefore when the graph size is large a deterministic search is impractical. That makes it an ideal candidate for an evolutionary search. For this assignment you must suppose that you have been tasked to implement the 4-clique problem as an evolutionary algorithm for any graph with any number of vertices (an n-vertex graph). The algorithm succeeds if it finds a...

Luby’s Algorithm: 1. Input: G = (V, E)2. Output: MIS I of G3. I ← ∅4. V ← V5. while V = ∅6. assign a random number r(v) to each vertex v ∈ V7. for all v ∈ V in parallel8. if r(v) is minimum amongst all neighbors9. I ← I ∪ {v}10. V ← V \ {v ∪ N(v)}Make Implementation of this algorithm in sequential form in Python. dont copy ans from bartleby its wrong

Consider a graph with five nodes labeled A, B, C, D, and E. Let's say we have the following edges with their weights: A to B with weight 3 A to C with weight 1 B to C with weight 2 B to D with weight 1 C to E with weight 4 D to E with weight 2 a. Find the shortest path from A to E using Dijkstra's algorithm. (Draw the finished shortest path) b. Use Prim to find the MST (Draw the finished MST) c. Use Kruskal to find the MST (Draw the finished MST) d. What's the difference between Prim and Kruskal algorithms? Do they always have the same result? Why or why not.

This is a graph-based Algorithms Question

Consider a graph with five nodes labeled A, B, C, D, and E. Let's say we have the following edges with their weights: A to B with weight 3 A to C with weight 1 B to C with weight 3 B to D with weight 1 C to E with weight 4 D to E with weight 2 a. Find the shortest path from A to E using Dijkstra's algorithm (Would anything change if B to C weight was changed from 3 to 4? To 1? What about 5?)

Use of Matching for Edge ColouringEdges in a matching can share the same colour since matching in a graph is defined as the collection of non-adjacent edges. As a result, we can create an algorithm based on this idea using the below stages. 1. Input: G = (V, E)2. Output: Minimal edge coloring of G3. color ← 04. while G = ∅5ドル. find maximal matching M of G6. color all M vertices with color7. G ← G − M8. color ← color + 1implementation of Python

Luby’s Algorithm: 1. Input: G = (V, E)2. Output: MIS I of G3. I ← ∅4. V ← V5. while V = ∅6. assign a random number r(v) to each vertex v ∈ V7. for all v ∈ V in parallel8. if r(v) is minimum amongst all neighbors9. I ← I ∪ {v}10. V ← V \ {v ∪ N(v)}Make Implementation of this algorithm in sequential form in Python

An undirected graph G = (V,E) is said to be k-colorable if all the vertices of G can be colored using k different colors such that no two adjacent vertices have the same color. Design an algorithm that runs in time O(n + e) to color a graph with two colors or determine that the graph is not 2-colorable.

Mark Zuckerberg, the CEO of Facebook, has hired you to lead the Facebook Algorithms Group. He has asked you to use various graph algorithms to analyze the world's largest social network. The Facebook Graph has 2.8 billion vertices, with each vertex being a Facebook user. Two vertices are connected provided those two users are "friends". The first decision you need to make is how you want to model the Facebook graph. Determine whether you should use an adjacency-list representation or an adjacency-matrix representation.

There are many applications of Shortest Path Algorithm. Consider the problem of solving a jumbled Rubik's Cube in the fewest number of moves. I claim that this problem can be solved using a Shortest Path Algorithm. Determine whether this statement is TRUE or FALSE. NOTE: if you want to check if this statement is TRUE, think about how the Rubik's Cube Problem can be represented as a graph. What are the vertices? Which pairs of vertices are connected with edges? What is your source vertex and what is your destination vertex? How would Dijkstra's Algorithm enable you to find the optimal sequence of moves to solve a jumbled cube in the fewest number of moves?