📗 [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

📗 [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] The Nash equilibrium of the following simultaneous move zero-sum game is (U, L): the entry marked by \(x\). What is the smallest and largest possible integer values of \(x\)? Enter two numbers. (U, L) can be one of possibly many Nash equilibria.

📗 Note: if there is only one possible value, enter the same value twice; and if no values are possible, enter \(0, 0\).

📗 Answer (comma separated vector): . Calculate

📗 [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] What is the projected variance of and onto the principal component ? Use the MLE (Maximum Likelihood Estimate) formula for the variance: \(\sigma^{2} = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}\) with \(\mu = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} x_{i}\).

📗 Answer: . Calculate

📗 [3 points] You have a dataset with unique data points which you want to use k-means clustering on. You setup the experiment as follows: you apply k-means with different k's: \(k\) = . Which \(k\) value will minimize the total distortion? Enter -1 if the answer depends on the data points.

📗 Answer: .

📗 [4 points] When using the Genetic Algorithm, suppose the states are \(\begin{bmatrix} x_{1} & x_{2} & ... & x_{T} \end{bmatrix}\) = , , , . Let \(T\) = , the fitness function (not the cost) is \(\mathop{\mathrm{argmax}}_{t \in \left\{0, ..., T\right\}} x_{t} = 1\) with \(x_{0} = 1\) (i.e. the index of the last feature that is 1). What is the reproduction probability of the state with the highest reproduction probability?

📗 Answer: . Calculate

📗 [2 points] Consider a game board consisting of bits initially at . Each player can simultaneously flip any number of bits in a move, but needs to pay the other player one dollar for each bit flipped. The player who achieves wins and collects dollars from the other player. What is the game theoretic value (in dollars) of this game for the first player?

📗 Note: "game theoretic value" is what we called "value of the game" in the lectures.

📗 Answer: . Calculate

📗 [4 points] In GoogSoft, software engineers A and B form a two-person team. Their year-end bonus depends on their relative performance. The bonus outcomes are summarized in the following table. The value of slacking to each person is \(s\) = . The total payoff to each person is the sum of the bonus and the value from slacking. What is the smallest value of \(x\) such that both players will work hard in a Nash equilibrium?

📗 Answer: . Calculate

📗 [4 points] Suppose K-Means with \(K = 2\) is used to cluster the data set and initial cluster centers are \(c_{1}\) = and \(c_{2}\) = \(x\). What is the largest value of \(x\) if cluster 1 has \(n\) = points initially (before updating the cluster centers). Break ties by assigning the point to cluster 2.

📗 Answer: . Calculate

📗 [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):

📗 [3 points] Let \(h_{1}\) be an admissible heuristic from a state to the optimal goal, A* search with which ones of the following \(h\) will be admissible? Enter the correct choices as a list, comma separated, without parentheses, for example, "1, 2, 4".

📗 Choices:

📗 Answer (comma separated vector): .

📗 [4 points] Enter the largest integer value of \(A\) such that \(B\) will be alpha-beta pruned? Min player moves first. In the case alpha = beta, prune the node. Enter 100 if you think the answer is infinity.

📗 Answer: .

📗 [4 points] You will receive 4 points for this question and you can choose to donate x points (a number between 0 and 4). Your final grade for this question is the points you keep plus twice the average donation (sum of the donations from everyone in your section divided by the number of people in your section, combining both versions). Enter the points you want to donate (an integer between 0 and 4).

📗 Answer: (The grade for this question will be updated later).

📗 [1 points] Please enter any comments including possible mistakes and bugs with the questions or your answers. If you have no comments, please enter "None": do not leave it blank.

📗 Answer: .