Consider a variant of the array in previous problem. an array a[1: : n] has a property that it rst rises monotonically up to some point i then it monotonically decreases until some point j and then rises again. that is a[k] < a[k + 1] for all k 2 [1: : i 1] [ [j: : n 1] and a[k] > a[k + 1] if i k < j. can we still nd i in o(log n) time? if not what is the minimum time required?

)a grad student comes up with the following algorithm to sort an array a[1..n] that works by first sorting the first 2/3rds of the array, then sorting the last 2/3rds of the (resulting) array, and finally sorting the first 2/3rds of the new array. 1: function g-sort(a, n) . takes as input an array of n numbers, a[1..n] 2: g-sort-recurse(a, 1, n) 3: end function 4: function g-sort-recurse(a, `, u) 5: if u ⒠` ≤ 0 then 6: return . 1 or fewer elements already sorted 7: else if u ⒠` = 1 then . 2 elements 8: if a[u] < a[`] then . swap values 9: temp ↠a[u] 10: a[u] ↠a[`] 11: a[`] ↠temp 12: end if 13: else . 3 or more elements 14: size ↠u ⒠` + 1 15: twothirds ↠d(2 ◠size)/3e 16: g-sort-recurse(a, `, ` + twothirds ⒠1) 17: g-sort-recurse(a, u ⒠twothirds + 1, u) 18: g-sort-recurse(a, `, ` + twothirds ⒠1) 19: end if 20: end function first (5 pts), prove that the algorithm correctly sorts the numbers in the array (in increasing order). after showing that it correctly sorts 1 and 2 element intervals, you may make the (incorrect) assumption that the number of elements being passed to g-sort-recurse is always a multiple of 3 to simplify the notation (and drop the floors/ceilings).

You are consulting for a beverage distributor who is interested in determining the benefits it could achieve from implementing new information systems. what will you advise as the first step?

The most accurate readings that you can take on an analog vom are when the meter's pointer is at the a. center scale. b. extreme right. c. near right. d. extreme left.