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- Review the 8-puzzle problem. Consider that the initial state is 1 3 48 27 6 5and the goal state is1 2 38 47 6 5Apply the breath-first-search method to obtain the path from the initial state to the goal state. Youneed to show the corresponding search tree. As soon as you find the goal state, you can stop the searchprocess.What is the best case, worst case time complexity for determining if a given directed edge exists in a graph. Assume the outgoing edges from each Vertex are stored in an adjacency list for that vertex. V is the total number of vertices in the graph, and E is the total number of outgoing edges in the graph, and you can find the vertex itself in constant time from a HashTable or HashMap ? Best Case: [ Select ] Worst Case: [ Select]Is it true that A* tree search always returns an optimal solution?
- Kruskal's algorithm is a minimum-spanning-tree algorithm which finds an edge of the least possible weight that connects any two trees in the forest. This algorithm is a Greedy Algorithm. The steps to find a MST using this algorithm are as follows: Sort all the edges in non-decreasing order of their weight. Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If cycle is not formed, include this edge. Else, discard it. Repeat step2 until there are (V-1) edges in the spanning tree. The graph must have 10 vertices and 20 weighted edges.An independent set in a graph G is a set S of vertices of G (so S is a subset of the vertex set V(G) of G) such that no two vertices from S are adjacent. A caterpillar is a tree T such that the graph obtained from T by removing all leaves is a path. Problem: Instance: a caterpillar T Problem: find the number of independent sets in T Design a recursive algorithm based on Divide and Conquer strategy for solving this problem. Prove its correctness. Estimate the time complexity of the algorithm.Consider eight points on the Cartesian two-dimensional x-y plane. a g C For each pair of vertices u and v, the weight of edge uv is the Euclidean (Pythagorean) distance between those two points. For example, dist(a, h) = V4? + 1² = v17 and dist(a, b) = v22 + 0² = 2. Using the algorithm of your choice, determine one possible minimum-weight spanning tree and compute its total distance, rounding your answer to one decimal place. Clearly show your steps.
- The below adjacency matrix M represents a directed graph. 0 1 1 M=|1 0 1 1 0 Select one: O True O FalsePlease implement the A* search algorithm in Python for the problem scenario below. Please explain your code and your approach. PROBLEM SCENARIO On holiday, a flight currently wants to travel to Bucharest from Arad. But there isno direct way to Bucharest from Arad. However, the cities are connected witheach other like a graph. The distance between the connected cities are given. Theflight wants to travel through the most optimal way. To find the optimal path totravel, another information is provided: the straight line distance between any cityand the final destination (Bucharest). Now apply A* search to determine the most optimal value for the route Arad toBucharest and help the flight. You have to use the straight line distance as theheuristic value for the cities. City Heuristic value City Heuristic valueArad 366 Mehadia 241Bucharest 0 Neamt 234Craiova 160 Oradea 380Eforie…Let T be a tree. Let a be the minimum weight and b be the maximum weight over all the edge weights in T. Then the strength of T is (a2+b2). Input: An undirected weighted graph G. Task: Give an efficient algorithm for computing a spanning tree of G that maximizes the strength of the tree. Give an efficient algorithm (to the best of your knowledge) and a formal proof of correctness for your algorithm. An answer without any formal proof will be considered incomplete. Analyze the running time of the algorithm. You must write down your algorithm as pseudocode
- You are given the problem below. Describe a simple brute force or backtracking algorithm that solves it. State the runtime of your algorithm. Problem: You are given a tree T. Each node in T has a weight (which can be positive or negative). Find the path in T with the largest total weight.The clique problem is finding cliques in a diagram. A clique is a set of vertices that are adjacent to each other. The 4-clique is a set of four knots all connected. So, in this example of the 4-clique problem, we have a graph with 7 vertices. A brute force algorithm searched all possible combinations of four vertices and found a set that formed a clique. If you want to understand more about it, the problem (and if possible read on). Note that the clique problem is NP-complete, so deterministic search is not practical for large graph sizes. This makes it an ideal candidate for evolutionary exploration. In this problem, we have to assume that we are given the problem of implementing the 4-clique problem as an evolutionary algorithm for an arbitrary graph with an arbitrary number of vertices (an n-vertex graph). If 4 cliques are found, the algorithm succeeds. 1. Provide an algebraic expression, in terms of n, for the size of the phenotypic search space (the number of possible…2. The greedy algorithm for the vertex cover problem picks a highest degree vertex v, includes v in the cover, deletes all edges incident to v, and repeats until there are no edges left. Show that this algorithm is not a c-approximation algorithm for any c. Hint: consider a bipartite graph G = (A, B, E) with |A| = n and B ULBi, where |B;| = [n/i]. Vertices in B; have degree i, and the neigh- borhoods of different vertices are disjoint.