4. The procedure find-good-value returns, with probability 1-1/n³, a value y that is sandwiched between f(i1) and f(i2), where i1, i2 E I are 8-good. This requires the existence of at least a fraction of 8 28-good integers in I. The running time of find-good-value is O(log² n), for fixed 6.
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- 7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).Prove that I(X; Y |Z) ≥ I(X; Y ) . Note: X, Y, and Z are random variables. X and Z are independent.Correct answer will be upvoted else downvoted. Computer science. way from block u to obstruct v is a grouping u=x0→x1→x2→⋯→xk=v, where there is a street from block xi−1 to hinder xi for each 1≤i≤k. The length of a way is the amount of lengths over all streets in the way. Two ways x0→x1→⋯→xk and y0→y1→⋯→yl are unique, if k≠l or xi≠yi for some 0≤i≤min{k,l}. Subsequent to moving to another city, Homer just recalls the two exceptional numbers L and R yet fails to remember the numbers n and m of squares and streets, separately, and how squares are associated by streets. Be that as it may, he accepts the number of squares ought to be no bigger than 32 (in light of the fact that the city was little). As the dearest companion of Homer, if it's not too much trouble, let him know whether it is feasible to see as a (L,R)- constant city or not. Input The single line contains two integers L and R (1≤L≤R≤106). Output In case it is difficult to track down a (L,R)- consistent city…
- Correct answer will be upvoted else downvoted. Computer science. way from block u to impede v is an arrangement u=x0→x1→x2→⋯→xk=v, where there is a street from block xi−1 to obstruct xi for each 1≤i≤k. The length of a way is the amount of lengths over all streets in the way. Two ways x0→x1→⋯→xk and y0→y1→⋯→yl are unique, if k≠l or xi≠yi for some 0≤i≤min{k,l}. Subsequent to moving to another city, Homer just recollects the two unique numbers L and R yet fails to remember the numbers n and m of squares and streets, separately, and how squares are associated by streets. Be that as it may, he accepts the number of squares ought to be no bigger than 32 (on the grounds that the city was little). As the dearest companion of Homer, if it's not too much trouble, let him know whether it is feasible to view as a (L,R)- constant city or not. Input The single line contains two integers L and R (1≤L≤R≤106). Output In case it is difficult to track down a (L,R)- ceaseless city…Let pn(x) be the probability of selling the house to the highest bidder when there are n people, and you adopt the Look-Then-Leap algorithm by rejecting the first x people. For all positive integers x and n with x < n, the probability is equal to p(n(x))= x/n (1/x + 1/(x+1) + 1/(x+2) + … + 1/(n-1)) If n = 100, use the formula above to determine the integer x that maximizes the probability n = 100 that p100(x). For this optimal value of x, calculate the probability p100(x). Briefly discuss the significance of this result, explaining why the Optimal Stopping algorithm produces a result whose probability is far more than 1/n = 1/100 = 1%.There are n people who want to carpool during m days. On day i, some subset si ofpeople want to carpool, and the driver di must be selected from si . Each person j hasa limited number of days fj they are willing to drive. Give an algorithm to find a driverassignment di ∈ si each day i such that no person j has to drive more than their limit fj. (The algorithm should output “no” if there is no such assignment.) Hint: Use networkflow.For example, for the following input with n = 3 and m = 3, the algorithm could assignTom to Day 1 and Day 2, and Mark to Day 3. Person Day 1 Day 2 Day 3 Limit 1 (Tom) x x x 2 2 (Mark) x x 1 3 (Fred) x x 0
- There are n people who want to carpool during m days. On day i, some subset ???? of people want to carpool, and the driver di must be selected from si . Each person j has a limited number of days fj they are willing to drive. Give an algorithm to find a driver assignment di ∈ si each day i such that no person j has to drive more than their limit fj. (The algorithm should output “no” if there is no such assignment.) Hint: Use network flow. For example, for the following input with n = 3 and m = 3, the algorithm could assign Tom to Day 1 and Day 2, and Mark to Day 3. Person Day 1 Day 2 Day 3 Driving Limit 1 (Tom) x x x 2 2 (Mark) x x 1 3 (Fred) x x 0Correct answer will be upvoted else Multiple Downvoted. Computer science. Jason discovered that alarms can sing one of the n+1 tunes, which have the accompanying construction: let si (0≤i≤n) be the I-th tune and t be a line of length n, then, at that point, for each i<n: si+1=sitisi. All in all i+1-st tune is the connection of I-th tune, I-th letter (0-recorded) of t and the I-th melody. Luckily, he likewise knows s0 and t. Jason considers how frequently a mariner's name is referenced in a specific tune. Answer q inquiries: given the mariner's name (w) and the file of a tune (I) output the number of events of w in si as a substring. As this number can be very huge, output its remaining portion modulo 109+7. Input In the main line of input there are two integers n, q ( 1≤n,q≤105) implying that there are n+1 tunes and q questions. In the following two lines strings s0 and t follow (1≤|s0|≤100,|t|=n). Next q lines depict the questions; every one contains an integer k…We are given three ropes with lengths n₁, n2, and n3. Our goal is to find the smallest value k such that we can fully cover the three ropes with smaller ropes of lengths 1,2,3,...,k (one rope from each length). For example, as the figure below shows, when n₁ = 5, n₂ 7, and n3 = 9, it is possible to cover all three ropes with smaller ropes of lengths 1, 2, 3, 4, 5, 6, that is, the output should be k = 6. = Devise a dynamic-programming solution that receives the three values of n₁, n2, and n3 and outputs k. It suffices to show Steps 1 and 2 in the DP paradigm in your solution. In Step 1, you must specify the subproblems, and how the value of the optimal solutions for smaller subproblems can be used to describe those of large subproblems. In Step 2, you must write down a recursive formula for the minimum number of operations to reconfigure. Hint: You may assume the value of k is guessed as kg, and solve the decision problem that asks whether ropes of lengths n₁, n2, n3 can be covered by…
- For all positive numbers aaand bbwith a>ba>b, ln(a−b)=ln(a)/ln(b)ln(a−b)=ln(a)/ln(b) True or falseConsidering the function f(x) = x – cos(x), what is the value of x7 after performing fixed point iteration. Assume an initial guess of 1. Use the equation form that will seem fit according to the choices provided.Group of answer choices 0.72210 0.71537 0.76396 0.72236An electrician has wired n lights, all initially on, so that: 1) light 1 can always be turned on/off, and 2) for k > 1, light k cannot be turned either on or off unless light k – 1 is on and all preceding lights are off for k > 1. The question we want to explore is the following: how many moves are required to turn all n lights off? For n = 5, a solution sequence has been worked out below. Fill in the missing entries. The lights are counted from left to right, so the first bit is the first light, and so on. 11111 01111 11011 10011 00010 10010 11010