If we choose to represent each genotype as a partial permutation, then provide an algebraic expression, in terms of n, for the size of the genetic search space (the number of possible valid genotypes), in terms of n. Assume that n ≥ 4.

For each subquestion below we will change various values in the recurrence relation T(n) = 81T(³5n) + 192n³ Not that each subproblem should be answered as if its change is the only change made (i.e. you are not "accumulating" the changes of previous subproblems).

Explain, with an example why the following definition, would not be suitable or useful: f(n) is Ω( g(n) ) if and only if there exists n0, such that:forall n ≥ n0, there exists c > 0 such that,f(n) ≥ c g(n)

Consider a sequence of integers defined by the following recurrence: f(0) = 0, f(1) = 1, and f(i) = f(i – 1) + f([i/2]) for i > 2. We would like to compute f(n) using DP. If we use bottom-up approach, what is the running time? Explain how you obtained your answer. (Assume every basic operation such as plus takes O(1) time as usual.) Give a pseudo-code for a top-down approach with memoization.

Let m be a matrix with n rows and n columns whose entries are either 1 or 0. recall that the element of m on row i and column j is denoted by mij . the diagonal entries of m are {mii} for 1 ≤ i ≤ n. We call M permutable if it is possible to swap some of the rows and some of columns so that all diagonal entries are 1. Design a polynomial time algorithm that decides whether a binary matrix M is permutable or not. Note that we can swap any two rows or two columns. Also, the order in which these swaps are done is not important.

Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Assume that T(n) is constant for n ≤ 2. Make your bounds as tight as possible, and justify your answers. T(n)=9 T(n/5)+n^2=θ(n^2) T(n)=T(√n)+4

Let T(n) be defined by the following recurrence relation T(0) = T(1) = 1 T(n) = T(n-1) +T(n − 2) +1 for n ≥2 Show that T(n) = 2Fn1 for n ≥ 0, where Fn is the nth Fibonacci number, i.e., Fo=F₁ = 1; Fn = Fn-1 + Fn-2 for n ≥ 2.

EXM.1.AHL.TZ0.37 IB [Maximum mark: 8] In this part, marks will only be awarded if you show the correct application of the required algorithms, and show all your working. In an offshore drilling site for a large oil company, the distances between the planned wells are given below in metres. 1 2 3 4 5 6 7 8 9 10 2 30 3 40 60 4 90 190 130 5 80 200 10 160 6 70 40 20 40 40 130 7 60 120 50 90 30 60 8 50 140 90 70 70 140 70 40 9 40 170 140 60 50 90 50 70 10 200 200 80 150 110 90 30 190 90 100 11 150 30 200 120 190 120 60 190 150 200 + It is intended to construct a network of paths to connect the different wells in a way that minimises the sum of the distances between them. Use Prim's algorithm, starting at vertex 3, to find a network of paths of minimum total length that can span the whole site. [8]

Prove that the sum of the first n odd positive integers is n2. In other words, show that 1 + 3 + 5 + .... + (2n + 1) = (n + 1)2 for all n ∈ N.

Explain why the Gale-Shapley Algorithm lasts at most n rounds, with at most n^2 total proposals. Also explain why the algorithm must eventually terminate, with every resident and every hospital getting matched.