Question 1: Apply A* method for finding the shortest route from S0 to We consider the problem described by the Graph below. The initial state is S0 and the goal state is S6. The heuristic of the states are presented in the table below SO 3 4 State (Si) Heuristic h(Si) si 1 S3 SO 3 S2 3 S2 S3 7 S4 1 S5 4 S5 S6 S4 S6

Using Dijkstra algorithm, visualize the steps to get shortest path from node A to all other nodes.

Use Best-first algorithm to find the path solution to the following graph and consider the values on edges are as h(n), where S is the start state and G is the goal state. A F B G C E -5-

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3. Kleinberg, Jon. Algorithm Design (p. 519, q. 28) Consider this version of the Independent Set Problem. You are given an undirected graph G and an integer k. We will call a set of nodes I "strongly independent" if, for any two nodes v, u € I, the edge (v, u) is not present in G, and neither is there a path of two edges from u to v. That is, there is no node w such that both (v, w) and (u, w) are present. The Strongly Independent Set problem is to decide whether G has a strongly independent set of size at least k.   Show that the Strongly Independent Set Problem is NP-Complete.

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 b. Use Prim to find the MST c. Use Kruskal to find the MST d. What's the difference between Prim and Kruskal algorithms? Do they always have the same result? Why or why not.

10. Apply Djikstra's Algorithm to find a shortest path from A to Z. Be sure to show each step of the algoriothm on a separate picture as discussed in class. Work carefully and pay attention to details. You need to clearly show how you are applying the algorithm. Also be sure to clearly name the verticies of a final shortest path B

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.

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I need to use Dijkstra’s algorithm to find the shortest path from a to z for the graph. Please help