Algorithms

$1.$ a$.$ What is the optimal way to multiply $5$ matrices with the following vector of dimensions:

$=[4,10,3,12,20,7]$

B$.$ How many times is the value of $[3,4]$M used during the algorithm?

third. How many times the value, m$\{$a$,$b$\}$ is used for n $>=$ b $>$ a $>=1$

$2.$ Given a series with n different unsorted numbers. Propose a way to find a monotonically increasing subseries of maximal length $($the solution is not necessarily unique$).$

For example if the series of numbers is $3513,21,5,1,7,14,3,$ then the series $3,$

$7,21,35$ is an increasing monotonic sequence of length $4.$

$3.$ a$.$ Given a rod of length $6$ and the array of prices $[109,7,6,3,2,]=$P$.$ Find the price

The maximum at which the rod can be sold.

B$.$ What parts will the rod be divided into to get this maximum price?

third. It was proved $($by induction$)$ that the recursive solution of the problem $($which does not use the dynamic programming method$)$ performs an exponential number of recursive calls

$4.$ Given an array with n numbers. A subset of the numbers must be chosen so that their sum

maximum without selecting a pair of adjacent numbers in the array.

The problem must be solved using the dynamic programming method by the four steps studied.

The algorithm for steps $3$ and $4$ must be written