True or False (If your answer to the question is "False", explain why, and providecorrection when possible).(a) Let h(n) be the heuristics for the node n, h(m) be the heuristics for the node m,d(m,n) be the actual minimal cost from node m to n in a graph. A* satisfies themonotone restriction iff d(m,n) <= |h(n)-h(m)|.(b) If an A* heuristics is admissible then it satisfies the monotone restriction.(c) Best-first search guarantees optimality in its returned solution.(d) Least-cost-first search guarantees optimality in its returned solution.(e) If all edges are with unit cost, then Breadth-first search guarantees optimality inits returned solution.

In the field of Artificial Intelligence, search algorithms play a crucial role in finding optimal…

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Artificial Inteligence
a) write the algorithm b) Briefly explain why the algorithm solves the problem, and the correctness c ) State and briefly explain the running time of your algorithm
Question 8. A connected graph G has 4 vertices and 4 edges of costs 1, 2, 3 and respectively 4. (a) Show that G is not a tree. (b) Show that G has a single minimum spanning tree. (Hint: For example, think how Prim's algorithm constructs the minimum spanning tree). (c) Draw a connected graph G with 4 nodes and 4 edges of costs 1, 2, 3, 4, respectively, so that the minimum spanning tree contains the edge of cost 4. The 4 nodes are below, you need to draw the 4 edges and indicate their cost. a O b d
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.
3. 4. Given a directed acyclic graph G = (V, E) and two vertices s, te V, design an efficient algorithm that computes the number of different directed paths from s to t. Define the incidence matrix B of a directed graph with no self-loop to be an nxm matrix with rows indexed by vertices, column indexed by edges such that Bij = -1 1 0 if edge j leaves vertex i, if edge jenters vertex i, otherwise. Let BT be the transpose of matrix B. Find out what the entries of the n x n matrix BBT stand for.
5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…
Can you help me solve this exercise? Please note that the greedy approach described in the advice paragraph does not work.
5. Consider the following modification of the independent set problem. As input, you receive an undirected graph G = (V, E). As output, you must decide if G contains an independent set UCV such that U contains at least 99% of the vertices in V (that is, JU| > 0.99|V|). Prove that this modification of the independent set problem is NP-Complete.
Question 1: In graph theory, a graph X is a "complement" of a graph F if which of the following is true? Select one: a. If X is isomorph to F, then X is a complement of F. b. If X has half of the vertices of F (or if F has half of the vertices of X) then X is a complement of F. c. If X has the same vertex set as F, and as its edges ONLY all possible edges NOT contained in F, then X is a complement of F. d. If X is NOT isomorph to F, then X is a complement of F. Question 2: Which statement is NOT true about Merge Sort Algorithm: Select one: a. Merge Sort time complexity for worst case scenarios is: O(n log n) b. Merge Sort is a quadratic sorting algorithm c. Merge Sort key disadvantage is space overhead as compared to Bubble Sort, Selection Sort and Insertion Sort. d. Merge Sort adopts recursive approach
Assume you are to write a program to analyze the social connection between students in MTSU. Each student is a node in a undirected graph. An edge is added between two nodes if these two students have close social connections, i.e., in the same club, or in the same department. Which would be a better representation for this graph? Question 45 options: adjacency matrix representation adjacency list representation