COMS 4701 Midterm Practice Exam Spring 2024 1. Consider the directed search graph shown below. S is the start state and G is the goal state. Transition costs are shown along the graph edges. Note that the transition between C and D is bidirectional. (a) For each of the search algorithms below, indicate the largest number of nodes that may possibly be expanded, not counting the goal state. Assume all algorithms conduct the goal test upon popping a node from the frontier. (Your answer may be ∞ .) Depth-first search with no reached table Depth-first search with reached table Breadth-first search with reached table (b) Suppose we run uniform-cost search on this search graph. List the order in which nodes are expanded (do not count the goal state) and give the final solution returned. Expanded nodes Returned solution (c) Suppose we currently have a heuristic function h(n) = 0 for all nodes n. Propose a change to the heuristic of a single node (indicate both node and heuristic value), such that h remains admissible and A* may expand fewer nodes than UCS. Also write out this shorter sequence of expanded nodes. New heuristic h(n) = x Expanded nodes 2. The graph below represents a particular constraint satisfaction problem. Nodes represent variables and directed edges indicate the presence of at least one unidirectional binary constraint between the two adjacent variables. There are no higher-order constraints. (a) For this part only, suppose the domains of the variables X and Y are all real numbers R. There are two implicit constraints between them: X2 – Y = 0 and 3X – Y = 2. Rewrite these two implicit constraints as one explicit constraint. (b) After making sure that the CSP is fully arc-consistent, we select A as the first variable to assign and then perform arc consistency again. At minimum which arcs must be checked? Which variables’ domains, if any, may be modified if no other arcs are checked? (c) Suppose again that we are starting with an arc-consistent CSP and that we assign A as our first variable. When we perform. arc consistency again, what are all the arcs that may be checked in the worst or maximum case? Which variables’ domains, if any, may be modified in this situation? 3. In this problem you will investigate the minimax tree shown below. The root node is MIN, it has three MAX children nodes, and each leads to four possible terminal nodes with utilities shown in the boxes. (a) Suppose we perform alpha-beta search, processing all nodes from left to right. Place an X in the boxes of all pruned branches (leave them all blank if no branches are pruned). Fill in the MIN and MAX nodes with their final values when the search concludes. (b) What are the values of α and β at the root node when the search concludes? Assume that all updates to these two parameters occur before a node returns its value. Briefly explain whether these values depend on the results of pruning during search. […]
