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In dynamic programming there are all possibilities and more time as compared to greedy programming. and the Dynamic programming approach always gives the accurate or correct answer. In dynamic programming have to compute only distinct function call because as soon as compute and store in one data structure so that after this reuse afterward if it is needed.

Let, $a_i鈭�a_1{a}_2鈥�鈥�.a_n$

Let the subproblem be$S\left(t-a_i\right).$ So at each subproblem,$S\left(t-a_i\right).$ which is the sum can be made from remaining value of 鈥� $t鈥�$.

We have following condition:

This means for any value of a_i, we can make sum up to 鈥榓nd it鈥檚 possible to obtain$\left(t-a_i\right)$from n-1 integers also. Thus,$S\left(n,t\right)$is True.

This means a_i cannot make value sum to 鈥榯鈥� and we can make value 鈥榯鈥� from$n-1$integers. Thus,

$S\left(n,t\right)$ is True.

This shows that it is not possible to make sum value 鈥� from using some integers.

So, the recursive equation is:

Where In dynamic programming have to compute only distinct function call because as soon as compute and store in one data structure so that after this reuse afterward if it is needed. And it is always solved in$O\left(nt\right)$ time by using dynamic programming paradigm. Here we will first define the recurrence relation and then solve each subproblem recursively.

$p\left[x,y,i\right]=p\left[x,y,i-1\right]鈭�p\left[x-a,y,i-1\right]鈭�p\left[x,y-a,i-1\right];fori,x,y>0$

Analysis of Above Recursive Relation in which each subproblem $S\left(n,t\right)$can be computed in $O\left(1\right)$time that is, in linear time. After computing $S\left(i,x\right)forin\left\{0鈥�.n\right\}andxn\left\{0鈥�.t\right\}.$Thus, the runtime is$O\left(nt\right).$