Question

Write pseudocode for an iterative algorithm which finds the maximum value of a list of integers. Use a loop invariant to prove your algorithm is correct. (Presumably you have already witten a program like this over the course of your studies! The interesting part here is the proof

Answer 1

-> Finds the maximum value of a list of interes Algorithm max(A) : Input: an array A storing nihtegles, . Where na l. Output: the largest element in A.. ma 26 F A CO] ILI n varianti maulama if A[i] > max max < Ali] return man

- Now, lets do the proofi Loop Invariant: At the start of the iteration with Lindex i of the loop, the variable man should contain the manumum of the numbers from the subarray ACO:i]. Initialization. At the start of the first loop, we have i=2. Therefore the loop invariant states: 'At the start of the iteration with indes i of the loop the variable max should contain the monimum of the mimber from the Subarray AcoilJ. which is Alol. This is what man has been set bo Maintenance - Assume that the loop invariant holds at the start of iteration i Then it I must be that main contains the musim.um of numbers in subarray Acoi J.

There are two cases! (i) A[i] = max. From the loop invariant we get that act ] is deurger then the maximum of the hermbes in Aloil. This, Ari] is the maximum of . A co iiti In this case, the algorithm sets answer to Acil thus in this case he loop invariant holds again at the beggining of the next loop. c) Aci z man. That is the monimum . In A Co. i) is at least as large as Acil, thens the maximum of A co jt I is the same as the :همگی ايده The algorithan also doesn't change man thus in this case the loop vana invariant holds again at the beginning of the heat loop. Termination! When the for-doop terminates رامطات .طک 1 Now, the tool hoop invariant gives The variable may contains the maximum of all humbers in Susaaray A coin] - A. This is exactly the value that. the algoritm Should output and which it then outputs These feel the algorithmsnis-Correct