Graph Theory and Approximation Algorithms (a) Find the maximum flow possible from vertex s to vertex t in the given graph. Show all your steps Given an example to show why reverse flow edges should be included to get the optimal flow 12 v1 V4 20 16 8. v2 V3 15 (b) Given below are the distances between A, B, C, D, E, F. You have to start from A, visit all the vertices and return back to the A. Find the optimal path using the approximation algorithm discussed in class. Show all your steps. (*) A B C DE F A|0 4 4 3 5 5 4 0 2 4 3 3 C 4 2 0 5 6 4 D3 4 E5 3 6 4 B 5 0 4 5 0 5 F 4 5 5 0

„ Run the Dijkstra's algorithm on the following graph beginning at node S. 5 12 6 3 5 16 Problem 3-(1). After each iteration, show the D values of all nodes (initial D values are shown above each node in red). Problem 3-(2). Show the shortest path from S to every other node generated by the algorithm 8 e.

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).

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Given N cities represented as vertices V₁, V2, un on an undirected graph (i.e., each edge can be traversed in both directions). The graph is fully-connected where the edge eij connecting any two vertices vį and vj is the straight-line distance between these two cities. We want to search for the shortest path from v₁ (the source) to VN (the destination). ... Assume that all edges have different values, and €₁,7 has the largest value among the edges. That is, the source and destination have the largest straight-line distance. Compare the lists of explored vertices when we run the uniform-cost search and the A* search for this problem. Hint: The straight-line distance is the shortest path between any two cities. If you do not know how to start, try to run the algorithms by hand on some small cases first; but remember to make sure your graphs satisfy the conditions in the question.

Computer science question help

I need the algorithm, proof of correctness and runtime analysis for the problem. No code necessary ONLY algorithm. And runtime should be O(n + m) as stated in the question.

Problem 2. The two problems below can be solved using graph coloring. For each problem, represent the situation with a graph, say whether you should be coloring vertices or edges and why, and use the coloring to solve the problem. a. Your Quidditch league has 5 teams. You will play a tournament next week in which every team will play every other team once. Each team can play at most one match each day, but there is plenty of time in the day for multiple matches. What is the fewest number of days over which the tournament can take place? b. Ten members of Math Club are driving to a math conference in a neighboring state. However, some of these students have dated in the past, and things are still a little awkward. Each student lists which other students they refuse to share a car with; these conflicts are recorded in the table below. What is the fewest number of cars the club needs to make the trip? Do not worry about running out of seats, just avoid the conflicts. Student: A B C D E F…