Design a greedy algorithm for this problem and find how long it will take to cross the bridge by using this algorithm. Find an instance with the smallest number of people for which your greedy algorithm does not vield a minimum crossing time.
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- Problem 3: A day at the beach. A group of n people are lying on the beach. The beach is represented by the real line R and the location of the i-th person is some integer r; e Z. Your task is to prevent people from getting sunburned by covering them with umbrellas. Each umbrella corresponds to a closed interval I = [a, a + L] of length L e N, and the i-th person is covered by that umbrella if æ¡ € I. Design a greedy algorithm for covering all people with the minimum number of umbrellas. The input consists of the integers x1,..., En, and L. The output of your algorithm should be the positions of umbrellas. For example, if the input is r1 = 1, x2 = 3, r3 = 5, and L = 2, then an optimum solution is the set of two umbrellas placed at positions 2 and 5, covering intervals [1,3] and [4,6]. 1 2 3 4 5 6 Prove that your algorithm is correct and that it runs in time polynomial in n.Consider nonnegative integer solutions of the equation x1+x2+x3+x4+x5+x6=30. How many different solutions are there? How many solutions also satisfy: for every i∈{1,2,3,4,5,6}, xi is positive and even?There are n ≥ 2 married couples who need to cross a river. They have a boat that can hold no more than two people at a time. To complicate matters, all the husbands are jealous and will not agree on any crossing procedure that would put a wife on the same bank of the river with another woman's husband without the wife's husband being there too, even if there are other people on the same bank. Can they cross the river under such constraints? Solve the problem for n = 2.
- Problem 4: Let S = {s1, s2, . . . , sn} and T = {t1, t2, . . . , tm}, n ≤ m, be two sets of integers. (i) Describe a deterministic algorithm that checks whether S is a subset of T. What is the running time of your algorithm?Four individuals wish to traverse an unstable bridge; they all start on the same side. You have 17 minutes to get everyone to the opposite side. Night has fallen, and they have only one torch. Two individuals may cross the bridge at once at most. Any crossing party, whether one or two individuals, must carry a torch. The torch must be carried back and forth; it cannot, for instance, be hurled. Person 1 crosses the bridge in one minute, person 2 in two minutes, person 3 in five minutes, and person 4 in ten minutes. A dyad must trek together at the slower individual's cadence. Create the algorithm specification that solves the problem.For f(a, b) = (a | b) | b (a) Simplify f(a, b). (b) Find DNF for f(a, b). (c) Is f(a, b) satisfiable?
- Question: Suppose there are n people in a group, each aware of a scandal no one else in the group knows about. These people communicate by telephone; when two people in the group talk, they share information about all scandals each knows about. For example, on the first call, two people share information, so by the end of the call, each of these people knows about two scandals. The gossip problem asks for G(n), the minimum number of telephone calls that are needed for all n people to learn about all the scandals. a). Find G(1), G(2), G(3), and G(4). b). Use mathematical induction to prove that G(n) 4. [Hint: In the inductive step, have a new person call a particular person at the start and at the end.] c). Prove that G(n)= 2n – 4 for n> 4.Correct answer will be upvoted else downvoted. Annie's nursery is a limitless 2D plane. She has n potatoes to plant, and the I-th potato should be planted at (xi,yi). Beginning at the point (0,0), Annie starts strolling, in one stage she can travel one unit right or up (expanding her x or y facilitate by 1 individually). Anytime (X,Y) during her walk she can establish a few potatoes at discretionary focuses utilizing her potato weapon, burning-through max(|X−x|,|Y−y|) units of energy to establish a potato at (x,y). Track down the base absolute energy needed to establish each potato. Note that Annie might establish quite a few potatoes from any point. Input :The primary line contains the integer n (1≤n≤800000). The following n lines contain two integers xi and yi (0≤xi,yi≤109), addressing the area of the I-th potato. It is conceivable that a few potatoes ought to be planted in a similar area. Output :Print the base absolute energy to establish all potatoes.For f (a, b) = (a | b) | b(a) Simplify f (a, b).(b) Find DNF for f (a, b).(c) Is f (a, b) satisfiable?
- Let x and y be integers such that x = 3 (mod 10) and y = 5 (mod 10). Find the integer z such that 97x + 3y³ z (mod 10) and 0 ≤ z ≤9.Consider the following bridge crossing problem where n people with speeds s1, ··· , sn wish to cross the bridge as quickly as possible. The rules remain: • It is nighttime and you only have one flashlight. • A maximum of two people can cross at any one time • Any party who crosses, either 1 or 2 people must have the flashlight with them. • The flashlight must be walked back and forth, it cannot be thrown, etc. • A pair must walk together at the rate of the slower person’s pace. Give an efficient algorithm to find the fastest way to get a group of people across the bridge. You must have a proof of correctness for your method.You are an investor who receives daily price quotes for a stock. The span of a stock's price on a given day is the number of consecutive days, from the given day going backwards, on which its price was less than or equal to its price on the day we are considering. Thus, the Stock Span Problem is as follows: Given a series of daily price quotes for a stock, find the span of the stock on each day of the series. Assume you are given seven daily stock quotes: 3, 10, 4, 7, 9, 6, and 8. Assume further that these stock quotes are stored in the array quotes. Show a step-by-step, manual desk-check execution of the algorithm below showing the values of all variables and arrays for each step in each cycle of each loop, as demonstrated in clase Algorithm: A Simple Stock Span Algorithm SimpleStockSpan (quotes) spans Input: quotes, an array with n stock price quotes Output: spans, an array with n stock price spans 1 spans CreateArray (n) 2 for i-0 to n do k+1 span_endFALSE while i-k 20 and not…