$1.$ A $0-1$ Knapsack problem is below:

$4$ items

Max weight is $5$

Item $1$: weight $2$; benefit $3$

Item $2$: weight $4$ ; benefit $5$

Item $3$: weight $5$ ; benefit $6$

Item $4$: weight $3$ ; benefit $4$

$(1)(10$ points$)$ What is the subproblem of the shaded cell?

$(2)(35$ points$)$ Please solve this $0-1$ Knapsack problem using the given table.

$(3)(15$ points$)$ Please show what items should be placed in the knapsack to maximize the benefit.

$2.$ The longest nondecreasing subsequence of a sequence is the longest subsequence in which all the elements are nondecreasing. This subsequence is not necessarily contiguous, or unique.

For example: 'lamar' has $3$ longest nondecreasing subsequences, $\text{'}$lmr$\text{'},$ 'amr', and 'aar'.

The longest common subsequence can be used to find the longest nondecreasing subsequence of a sequence.

$(1)$ The first sequence is the given original sequence,

$(2)$ The second sequence is a sorted sequence by ordering all elements in the first sequence in nondecreasing$/$alphabetical order.

$(3)$ Apply LCS to the first and second sequences to find the longest nondecreasing subsequence of the given original sequence $($i$.$e$.,$ the first sequence$).$

Please find the longest nondecreasing subsequence of 'algorithm' using the steps above. $(40$ points$)$