$(15\%)$ Bill is having dinner at a bar, where he will order many small plates. There are $n$ plates of

food on the menu, where information for plate $i$ is given by a triple of non$-$negative integers $,$

$s_i$ : the plate's volume $v_i$ calories $c_i$ and sweetness $s_{i}in\{0,1\}($the plate is sweet if $s_i=1$ and not

sweet if $s_i=0.$ Now, he wants to eat as much as possible, but no more than $k$ calories this time. He

also wants to order exactly $s$ sweet plates, without purchasing the same dish twice. Please

solve this problem using dynamic programming.

$(a)$ Identify the set $of$ subproblems. $[5\%]$

$(b)$ Identify the relationship between subproblems. $[5\%]$

$(c)$ Briefly discuss the time complexity $of$ your $DP$ algorithm. $[5\%]$