a) The maximum subsequence sum problem is defined as follows: If

Question

a) The maximum subsequence sum problem is defined as follows: If $a_1,a_2,...,a_n$ are in Z, find the maximum value $\sum_{k=i}^{j}a_i,$ for all $i,\, j,1\leq i\leq j\leq n.$ We assume that the answer is 0 if all the $a_i$ are negative or if the sum is empty. The following algorithm finds a solution to the problem. Here, we assume that $a_i$ s are stored in the array A.

Maximum-Subsequence $(A,MaxSum)$

1 $Sum\leftarrow 0,MaxSum\leftarrow0$

2 $for\, i\leftarrow1\: to\: n$

3 $do$

4 $Sum=Sum+A[i]$

5 $if$ $Sum>MaxSum$

6 $then$ $MaxSum\leftarrow\, Sum$

7 $else\: if\, Sum<0$

8 $then\, Sum=0$

State precisely a loop invariant for the for loop in line 2–8. Prove that your loop invariant holds and hence conclude that the algorithm works.

b) Analyse the algorithm and find an upper bound for the run time of the above algorithm.

08 Mar 2024

Answer :

Word Count : 424

### a) Loop Invariant Loop Invariant: At the beginning of each iteration of the loop (from line 2 to line 8), the variable `MaxSum` holds the maximum sum of any subsequence ending at the current index `i` or earlier, and `Sum` holds the sum of the subsequence that ends at the current index `i`. #### Proof of the Loop Invariant 1. Initialization: Before the loop starts (when `i = 1`), we set `Sum = 0` and `MaxSum = 0`. At this point, no subsequences have been considered, so `MaxSum` correctly represents the maximum subsequence sum as 0 (as it is assumed that an empty subsequence has _____ _______ __________ ___ __________ __________.
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