Dynamic Programming

Assume that multiplying a matrix $G_1$ of dimension $p \times q$ with another matrix $G_2$ of dimension $q \times r$ requires $pqr$ scalar multiplications. Computing the product of n matrices $G_1G_2G_3 \ldots G_n$ can be done by parenthesizing in different ways. Define $G_i G_{i+1}$ as an explicitly computed pair for a given paranthesization if they are directly multiplied. For example, in the matrix multiplication chain $G_1G_2G_3G_4G_5G_6$ using parenthesization $(G_1(G_2G_3))(G_4(G_5G_6))$, $G_2G_3$ and $G_5G_6$ are the only explicitly computed pairs. Consider a matrix multiplication chain $F_1F_2F_3F_4F_5$, where matrices $F_1, F_2, F_3, F_4$ and $F_5$ are of dimensions 2×25, 25×3, 3×16, 16×1 and 1×1000, respectively. In the parenthesization of $F_1F_2F_3F_4F_5$ that minimizes the total number of scalar multiplications, the explicitly computed pairs is/are

Consider the weights and values of items listed below. Note that there is only one unit of each item. $$\begin{array}{|c|c|c|} \hline \textbf{Item number} & \textbf{Weight (in Kgs)} & \textbf{Value (in Rupees)} \\ \hline 1 & 10 & 60 \\ \hline 2 & 7 & 28 \\ \hline 3 & 4 & 20 \\ \hline 4 & 2 & 24 \\ \hline \end{array}$$ The task is to pick a subset of these items such that their total weight is no more than 11 Kgs and their total value is maximized. Moreover, no item may be split. The total value of items picked by an optimal algorithm is denoted by $V_{\text{opt}}$. A greedy algorithm sorts the items by their value-to-weight ratios in descending order and packs them greedily, starting from the first item in the ordered list. The total value of items picked by the greedy algorithm is denoted by $V_{\text{greedy}}$. The value of $V_{\text{opt}} - V_{\text{greedy}}$ is _____.