Prove the time complexity of the Binary search algorithm using Masters theorem, identify all the terms in Masters theorem clearly in the proof def binarySearch(arr, I, r, x): if r >= 1: mid=1 + (r - 1) // 2 if arr[mid] == X: return mid elif arr[mid] > x: return binarySearch(arr, I, mid-1, x) else: return binarySearch(arr, mid + 1, r, x) else: return -1

Question 2: Recurrences (a) Below is the pseudo-code for two algorithms: Practicel (A,s,f) and Practice2(A,s,f), which take as input a sorted array A, indexed from s to f. The algorithms make a call to Bsearch(A,s,f,k) which we saw in class. Determine the worst-case runtime recurrece for each algorithm: T1(n) and T2(n). Show that T(n) is O((log n)²) and T2(n) is O(n log n). Practicel (A,s,f) if s< f ql = L(s+ f)/2] if BSearch(A,s,ql,1) = true Practice2(A,s,f) if s < f if BSearch(A,s+1,f,1) = true return true return true else else return Practice2(A, s, f-1) return Practicel(A, ql+1, f) else else return false return false

Implement algorithm GetExp (s, i) pre-cond: s is a string of tokens, andi is an index that indicates a starting point within s. post-cond: The output consists of a parsing p of the longest substring s[i], s[i + 1], ... , s[j − 1] of s that starts at index i and is a valid expression. The output also includes the index j of the token that comes immediately after the parsed expression.

Interchange Base Points AlgorithmOutput: a base B = [131,132..... 13j-1, 13L.+1, 13j, 13j+2, 13j+3..... 13k] for G; a strong generating set relative to B; procedure interchange(var B: sequence of points; vat S: set of elements; j: 1..k-1);Interchange Base Points AlgorithmInput: a group G; a base [131,132..... 13k] for G and a strong generating set; an integer j between 1 and k-l; Output: a base [131,132..... 13k] for G and a strong generating set.