To solve these problems you will use dynarnic proyramming method. Dynamic programming method

avoids repeated computation of an optimal solution for the same subset of inputs by storing in memory

an optimal solution obtained for it for the first time; this is known as space$-$time tradeoff or memory$-$time

tradeoff. This can be done using top$-$down recursive algorithms or bottom$-$up iterative algorithms. Let us

use a small rod$-$cutting problem to illustrate a bottom$-$up method. The Table $1$ below shows the price of a

rod for a given length. Unit does not matter as long as we have prices and reverues are for the same unit of

length.

Table $1$: Price table for rods.

The BOTTON$-$UP$-$CUT$-$ROD $($P$,$N$)$ algorithm is used for creating Table $2$ shown below. Two rows ia and ib

are for iteration i the outer$-$loop. Rows i show symbolic values and ib show the numerical values.

Thable $2$: Trace of the bottum$-$up algorithm.

$1)$ For the rod prices shown in Table $3$ create a computation table similar to Table $2.$

Table $3$: Price table for rods and corresponding maximum revenue for a given length.

$2)$ Find an optimal parenthesization of a matrix$-$chain product whose sequence of dimensions is

Show both m and s tables.

Tables are in the image. Thank you.