Lemma 8.7 Let A denote the set output by the algorithm. Then, profit (4) 2 (1) OPT.
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- P(M|D)P(D) given P(D|M) = solve the following p(M) а. Draw the probability tree for the situation. b. Draw the reverse tree for the situation.Given the following Truth Table: Inputs ABC Outputs Y 000 001 010 011 100 101 110 111 Y= 1. Construct a K-Map on Scrap Paper 2. Perform Grouping on Scrap Paper 3. Give the SIMPLEST Boolean Expression for x in the answer box below. 1 X 1 0 X 1 1 1 4. Give sum of minterms and sum of don't cares. Em( )+ d( Should you have an empty set please answer with N/A )Suppose that a manufacturing company builds n different types of robots, sayrobots 1, 2, . . . , n. These robots are made from a common set of m types of materials, saymaterials 1, 2, . . . , m. The company has only a limited supply of materials for each year,the amount of materials 1, 2, . . . , m are limited by the numbers b1, b2, . . . , bm, respectively.Building robot i requires an aij amount from material j. For example, building robot 1requires a11 from material 1, a12 from material 2, etc. Suppose the profit made by sellingrobot i is pi. Write an integer linear program for maximizing the annual profit for thecompany
- 1. Let S = {1,2...n} be a set activities, where activity i € S has a start time si, a finish time f; and a profit p, asSociated with it. Design an dynamic programming algorithm for computing a set of compatible activities of maximum profit. Argue that the time complexity of you algorithm is O(n log(n)). (You must start with the correct definitions, and then write a recurrence relation.)Minimize the function given below using imap F(ABC)=€(0,1,3,4,6,8,9,11,12,14)There are three financial aid counselors. If a student’s last initial is from A – H, let them know that their counselor is Jon Stewart. If a student’s last initial is from I – Q, let them know that their counselor is Chelsea Handler. If a student’s last initial is from R – Z, let them know that their counselor is Brian Williams.Ask student for the FICO score, if it’s less than 660 tell that this student cannot have a loan.
- 2. In a two-class problem, the log odds is defined as P(C₁|x) log p(C₂\x) Write the discriminant function in terms of the log odds.no handwritten Alice and Bob are playing a match to see who is the first to win n games, for some fixed n > 0. Suppose Alice and Bob are equally competent, that is, each of them wins a game with probability 1/2. Further, suppose that they have already played i + j games, of which Alice won i and Bob won j. Give an efficient algorithm to compute the probability that Alice will go on to win the match. For example, if i = n − 1 and j = n − 3, then the probability that Alice will win the match is 7/8, since she must win any of the next three games.Given the following Truth Table: Inputs AB Outputs 100 01 10 11 1. Construct a K-Map on Scrap Paper 2. Perform Grouping on Scrap Paper 3. Give the SIMPLEST Boolean Expression for x in the answer box below. Y= X 1 1 0 4. Give sum of minterms and sum of don't cares. Σm( )+ d( Should you have an empty set please answer with N/A )
- There are n men and women, each of which has ranked all n members of the other sex in order of preference. Consider the following algorithm for matching them up: Start with an arbitrary matching and as long as there is a man and woman who prefer each other to whoever they are currently matched with, switch them. Will this algorithm always result in a stable matching? O true O falseHat-check problem. Use indicator random variables to solve the following problem, which is known as the hat-check problem. Each of n customers gives a hat to a hat-check person at a restaurant. The hat-check person gives the hats back to the customers in a random order. What is the expected number of customers who get back their own hat?3. Suppose that we want to know which agents are connected by a walk of length two in a given network. That is, want to know the set of ordered pairs (i, j) such that there exists a walk of length 2 between any i and j. Define n as the number of nodes and m as the number of links. (a) One way to do this is to input the adjacency matrix A, calculate A², then output all pairs (i, j) for which A(i) > 0. Defining scalar multiplication and addition as "basic operations," up to a constant c, how many "basic operations" are required to calculate A², in terms of n and m? (Hint: First think about how many operations are required to calculate each entry of A², then multiply by the number of entries.) (b) Suppose that we want to know the number of agents who are connected by a walk of length k in a given network, where k≥ 1. Up to a constant, how many basic operations are required to calculate Ak, in terms of n and m? (c) (Harder) We showed in class (Lecture 4) that a simple adaptation of…