5. Consider the following problem: {a₁, a2,. an} of positive integers, and an "size" s(a) for each a € A. Question: Is there a subset A' CA such that the sum of the sizes of elements in A' is equal to the sum of the sizes of elements in A - A'? Or in other words, is there is subset A' of A such that the sum of the sizes of elements in A' is half of the total sum of sizes of elements in A? Input: A finite set A = ...9 Examples: For A = {a₁, a2, a3, a4, a5}, s(a₁) = 1, s(a₂) = 9, s(a3) 9, s(a3) = 5, s(a₂) + s(a4) = 26/2 = s(a3) + s(a5). = 5, s(a4) = 3, s(as) s(a5) = 8, the answer is YES because s(a1) + s(a4) = 3, For A = {a1, a2, A3, A4, A5}, s(α₁) = 1, s(a2) = 9, s(a3) = 5, s(a4) = 3, s(a5) = 7, the answer is immediately NO because the sum of size is 25, which is not divisible by 2, so no A' exists. Let us denote by B the sum of sizes of elements of A and assume it is even. For 1 ≤ i ≤ n and 0 ≤ j ≤ B/2, let t(i, j) be the truth value of the statement: "there is a subset of {a₁, a2, ... a;} for which the sum of the sizes of elements is exactly j". (d) Consider a dynamic programming solution using the recurrence above, where you build a table to hold your partial solutions. The rows of the table correspond to i, and columns correspond to j. How many cells does this table have? (e) What is the running time of the dynamic programming algorithm? (f) Is this a polynomial-time algorithm?