Q12b

Consider the following problem. Given $n$ positive integers $a_{1}, a_{2}\dots a_n,$ it is required to partition them in to two parts $A$ and $B$ such that, $\displaystyle{}\left|\sum_{i \in A} a_{i} - \sum_{i \in B} a_{i}\right|$ is minimised Consider a greedy algorithm for solving this problem. The numbers are ordered so that $a_{1} \geq a_{2} \geq \dots a_{n},$ and at $i^{th}$ step, $a_i$ is placed in that part whose sum in smaller at that step. Give an example with $n=5$ for which the solution produced by the greedy algorithm is not optimal.