### saytzeff's blog

By saytzeff, history, 11 months ago, ,

You are given N stones, labeled from 1 to N. The i-th stone has the weight W[i]. There are M colors, labeled by integers from 1 to M. The i-th stone has the color C[i] (of course, an integer between 1 to M, inclusive). You want to fill a Knapsack with these stones. The Knapsack can hold a total weight of X. You want to select exactly M stones; one of each color. The sum of the weights of the stones must not exceed X. Since you paid a premium for a Knapsack with capacity X (as opposed to a Knapsack with a lower capacity), you want to fill the Knapsack as much as possible.

Write a program that takes all the above values as input and calculates the best way to fill the Knapsack – that is, the way that minimizes the unused capacity. Output this unused capacity. See the explanation of the sample test cases for clarity.

Input The first line of input contains the integer T, the number of test cases. Then follows the description of T test cases. The first line of each test case contains three integers, N, M and X, separated by singlespace. The next line contains N integers, W[1], W[2], W[3] … W[N], separated by single space. The next line contains N integers C[1], C[2], C[3] … C[N], separated by single space.

Output An optimal way of filling the Knapsack minimizes unused capacity. There may be several optimal ways of filling the Knapsack. Output the unused capacity of the Knapsack (a single integer on a line by itself) for an optimal way. If there is no way to fill the Knapsack, output -1. Output T lines, one for each test case.

Constraints 1 ≤ T ≤ 10 1 ≤ M ≤ 100 M ≤ N ≤ 100 1 ≤ W[i] ≤ 100 1 ≤ C[i] ≤ M 1 ≤ X ≤ 10000

Sample Input 4

9 3 10

2 3 4 2 3 4 2 3 4

1 1 1 2 2 2 3 3 3

9 3 10

1 3 5 1 3 5 1 3 5

1 1 1 2 2 2 3 3 3

3 3 10

3 4 4

1 2 3

3 3 10

3 3 3

1 2 1

Sample Output 0

1

-1

-1

Is this possible using only two states dp? I think it might need three states- for ith element, color c, and knapsack capacity X.

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 » 11 months ago, # |   0 Auto comment: topic has been updated by saytzeff (previous revision, new revision, compare).
 » 11 months ago, # |   0 Auto comment: topic has been updated by saytzeff (previous revision, new revision, compare).
 » 11 months ago, # |   +3 compute dp(i,j) for i ranging from 1 to m and j ranging from 1 to W. It indicates a boolean value implying whether it is possible to achieve weight j such that exactly one element from first i colored objects are taken. Ans = W — imax where imax is the maximum i such that dp(m, i) is true.
•  » » 11 months ago, # ^ |   0 Thanks!
•  » » 11 months ago, # ^ |   0 Btw, how long did it take you to arrive at the solution? How would you rate this question in div2?
•  » » » 11 months ago, # ^ |   +3 Around 20 minutes. I would say Div2 C