📗 [4 points] Which order of goal check is possible with , without specifying the order of successors when putting them in the queue (i.e. you can rearrange the order of the branches)? Enter the correct choices as a list, comma separated, without parentheses, for example, "1, 2, 4".

📗 Choices:

📗 Answer (comma separated vector):

📗 [4 points] Which order of goal check is possible with , without specifying the order of successors when putting them in the queue (i.e. you can rearrange the order of the branches)? That is, if the 2 nodes in the choices can be visited in any order without violating BFS, mark this choice as "possible".

📗 Note: some of the choices may be repeated, if you think the choice is correct, please select all repeated ones as well.

📗 Choices:

📗 [2 points] Consider \(n + 1\) = + \(1\) states. The initial state is \(1\), the goal state is \(n\). State \(0\) is a dead-end state with no successors. For each non-\(0\) state \(i\), it has two successors: \(i + 1\) and \(0\). We may expand the same states many times, because we do not keep track of which states are checked previously. How many states (including repeated ones) will be expanded by ? Break ties by expanding the state with the index first.

📗 Note: the tie-breaking rule may be different from the convention used during the lectures, please read the question carefully.

📗 Answer: . Calculate

📗 [2 points] Consider n + 1 = + 1 states. The initial state is 1, the goal state is n. State 0 is a dead-end state with no successors. For each non-0 state i, it has two successors: i + 1 and 0. There is no cycle check nor CLOSED list (this means we may expand (or goal-check) the same nodes many times, because we do not keep track of which nodes are checked previously). How many goal-checks will be performed by Breadth First Search? Break ties by expanding the node with the smaller index first.

📗 Answer: . Calculate

📗 [3 points] Suppose the states are integers between \(1\) and \(x\). The initial state is \(1\), and the goal state is . The successors of a state \(i\) are \(2 i\) and \(2 i + 1\), if exist. What is the smallest value of \(x\) so that the worst case space complexity (number of states stored in the list (queue or stack)) of DFS (Depth First Search) is larger than or equal to BFS (Breadth First Search)?

📗 Note: the worst case space complexity for BFS is \(b^{d}\) and for DFS is \(\left(b - 1\right) D + 1\). "Worst case" means you can re-order the successors and search in the order that maximizes the space requirement.

📗 Answer: . Calculate

📗 [3 points] In a by grid, Tom is located at (, ) and Jerry is located at (, ). Tom uses to find Jerry and the successors of a state (one cell in the grid) are the four neighboring states on the grid (the cells above, below, to the left and to the right). What is the imum number of states that need to be expanded to find (and expand) the goal state? The order in which the successors are added can be arbitrary. Include both the initial and the goal states.

📗 Highlight one example state expansion path (order does not matter).

📗 Answer: 

📗 [4 points] In a by grid, Tom is located at (, ) and Jerry is located at (, ). Tom uses to find Jerry and the successors of a state (one cell in the grid) are the four neighboring states on the grid (the cells above, below, to the left and to the right). What is the imum number of states that need to be expanded to find (and expand) the goal state? The order in which the successors are added can be arbitrary. Do not count repeated expansion of the same state. Include both the initial and the goal states.

📗 Answer: . Calculate

📗 [4 points] Given there are states in a search tree with levels (max depth, the root is at depth 0 or level 0 but counts as one level) and no goal states, what is the imum possible number of states expanded during an IDS (Iterative Deepening Search)? The same state may be expanded multiple times during different iterations (with different depth limits) of the search.

📗 Note: the number of levels is fixed in this question, a tree with more or fewer levels is not valid. The answer obtained from not counting the root as one of the levels will be accepted too.

📗 Answer: . Calculate

📗 [3 points] Suppose the states are integers between \(1\) and . The initial state is \(1\), and the goal state is . The successors of a state \(i\) are \(2 i\) and \(2 i + 1\), if exist. If it is allowed to put the successors into the stack (list) in any order, what are the minimum and maximum numbers of states that will be expanded during a DFS (Depth First Search)? Enter two numbers.

📗 Answer (comma separated vector): . Calculate

📗 [4 points] Suppose the states are integers between \(1\) and . The initial state is \(1\), and there are two goal states, the optimal one: , and another one with (strictly) higher cost (length of path from initial state to goal state): \(x\). The successors of a state \(i\) are \(2 i\) and \(2 i + 1\), if exist. What is the smallest value of \(x\) such that DFS (Depth First Search) does not find the optimal goal in the worst case? 

📗 Answer: . Calculate

📗 [4 points] Suppose the state space has \(n\) = states that form a tree with root state \(0\). What is the shape of the tree that makes iterative deepening realize that a goal does not exist as quickly as possible (i.e. one that minimizes the number of expanded nodes)? Draw one example of such a tree.

📗 Answer:

📗 [4 points] Suppose the state space has \(n\) = states that form a tree with root state \(0\). What is the shape of the tree that makes iterative deepening realize that a goal does not exist as quickly as possible (i.e. one that minimizes the number of expanded nodes)? Enter the number of nodes searched in this case.

📗 Answer: . Calculate

📗 [4 points] Imagine a world where each person has friends. Alice and Bob are \(d\) = "friendship links" away (i.e. if \(d\) = 1, Alice and Bob are friends; if \(d\) = 2, there is a third person X such that Alice and X are friends, and Bob and X are friends; and so on). Imagine a algorithm that has access to the friendship links. The algorithm starts at Alice and the goal is to find Bob. In the best (luckiest) case, how many people the algorithm needs to visit (including Alice and Bob)?

📗 Answer: . Calculate

📗 [2 points] Consider a search graph which is a tree, and each internal node has children. The only goal node is at depth (root is depth 0). How many total goal-checks will be performed by in the luckiest case (i.e. the smallest number of goal-checks)? If a node is checked multiple times you should count that multiple times.

📗 Answer: .  Calculate

📗 [3 points] Consider on a tree, where the nodes are denoted by numbers. The search visited the following sequence of nodes (expansion path): . Reconstruct one possible tree. \(0\) is the initial state and is the goal state. You must include all nodes from \(0\) to .

📗 Answer: 

📗 Note: to erase an edge, draw the same edge again.

📗 [2 points] Recall in uniform-cost search, each node has a path-cost from the initial node (sum of edge costs along the path), and the search expands the least path-cost node first. Consider a search graph with \(n\) = nodes: \(1, 2, ..., n\). For all \(1 \leq i < j \leq n\), there is a directed edge from \(i\) to \(j\)  with an edge cost . The initial node is 1, and the goal node is \(n\). How many (unique) goal-checks (the same nodes expanded twice is counted only once) with uniform-cost search perform? Break ties by expanding the node with the smaller index first.

📗 Answer: .  Calculate

📗 [2 points] Consider a search tree where the root is at depth 0, each internal node has children, and all leaves are at depth . There is a single goal state at depth . How much (in number of states including the root and the goal) is sufficient so always succeeds? Select all that applies.

📗 Choices:

📗 Calculator: . Calculate

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