algorithm analysis give an efficient algorithm to determine if there exists an integer such that ai in an array of integers an what is the running time of your algorithm write an alternative gcd algorithm based on the following observations (arrange so that ) gcd(ab gcd( / / if and are both even gcd(abgcd( / bif is even and is odd gcd(abgcd(ab/ if is odd and is even gcd(abgcd(( )/ ( )/ if and are both odd give efficient algorithms (along with running time analysesto find the minimum subsequence sum find the minimum positive subsequence sum find the maximum subsequence product an important problem in numerical analysis is to find solution to the equation ( for some arbitrary if the function is continuous and has two points low and high such that (lowand (highhave opposite signsthen root must exist between low and high and can be found by binary search write function that takes as parameters flowand high and solves for zero what must you do to ensure termination the maximum contiguous subsequence sum algorithms in the text do not give any indication of the actual sequence modify them so that they return in single object the value of the maximum subsequence and the indices of the actual sequence write program to determine if positive integernis prime in terms of nwhat is the worst-case running time of your program(you should be able to do this in onc let equal the number of bits in the binary representation of what is the value of bd in terms of bwhat is the worst-case running time of your programe compare the running times to determine if -bit number and -bit number are prime is it more reasonable to give the running time in terms of or bwhythe sieve of eratosthenes is method used to compute all primes less than we begin by making table of integers to we find the smallest integerithat is not crossed outprint iand cross out iwhen nthe algorithm terminates what is the running time of this algorithmshow that can be computed with only eight multiplications write the fast exponentiation routine without recursion give precise count on the number of multiplications used by the fast exponentiation routine (hintconsider the binary representation of programs and are analyzed and found to have worst-case running times no greater than log and respectively answer the following questionsif possible