A certain computer algorithm executes twice as many operations when it is run with an input of size k as when it is run with an input of size k - 1 (where k is an integer that is greater than 1). When the algorithm is run with an input of size 1, it executes seven operations. How many operations does it execute when it is run with an input of size 26? For each integer n2 1, let s,-, be the number of operations the algorithm executes when it is run with an input of size n. Then so = and s = for each integer k 2 1. Therefore, So, S1, S2 - Select-- v with constant -Select-- V, which is So, for every integer n 2 0, s, = .It follows that for an input of size 26, the number of operations executed by the algorithm is sSelect v which equals

An algorithm takes 0.5ms for input size 100. How long will it take for input size 500 (assuming thatthe lower-order terms are negligible) if the running time of the algorithm is O(1)? What is algorithm is O(n)?

Suppose that you have five different algorithms (A, B, C, D and E) for solving a problem. To solve a problem of size n, the number of operations used by each algorithm is as follows. As n grows, which algorithm uses the most operations? Algorithm number of operations n2+n3 B 2n+1 log n' 26+ log n n(logn)? +1 n100 Algorithm D Algorithm C Algorithm A Algorithm B o Algorithm E

Algorithm

Algorithm 1 [WhoKnows(a1, a2, ..., an: integers)] 1: m := 0 2: for (i := 1 to n – 1) do for (j := i+1 to n) do 3: if (Jai – a;|> m) then m := |ai – a;| 4: 5: 6: end if 7: end for 8: end for 9: return m (a) What does the algorithm compute? (b) Find the exact number of comparisons performed for an input of size n? (c) What is the complexity of the algorithm in terms of big-O notation for the size of input n? (d) Open question. Can you think of a more efficient algorithm solving the same problem as Algorithm (1)? Discuss the details and the complexity.

Given a program designed to calculate rocket trajectories, the algorithm has an O(2n) complexity and takes 1 second for each iteration, and n = 15. How many hours will it take to calculate the trajectories, worst case?

This issue compares the execution times of the two multiplication algorithms listed below:KindergartenAdd algorithm (a, b)A and B are integers, pre-cond.post-con: Produces a and b.

Give the exact number of additions performed in the following segment of certain algorithm assuming a1, a2, ..., an are positive real numbers and n = 20. for i=1 to n for j=1 to n Sij = ai + aj + 2;