Question

Consider n indivisible objects with distinct types i = 1, 2, 3,…, n. We are given positive integer weights W = {w1,w2...wn} and positive integer prices V = {v1,v2...vn} for the objects and a knapsack of weight capacity (m). Our problem is to find the maximum profit possible by including a subset of the objects into the knapsack with total weight of at most m.

This form of the 0/1 Knapsack Problem can be solved by a Decrease and Conquer approach as follows:

• Assume that we have tried objects of types 1,2,..., i -1 to fill the knapsack up to a total weight capacity (j) with a maximum profit of P(i-1,j)

• If j ≥ wi then P(i-1, j - wi) is the maximum profit if we remove the equivalent weight wi of the object of type (i).

• By trying to add object of type (i), we expect the maximum profit to change to P(i-1, j - wi) + vi

• If this change is better, we do it, otherwise we leave things as they were, i.e.,

P(i , j) = max { P(i-1, j) , P(i-1, j - wi) + vi } for j ≥ wi ,

and P(i , j) = P(i-1, j) for j < wi

• Notice that P(0 , j) = 0 (No objects are taken)

The above approach leads to a recursive Decrease and Conquer algorithm.

You are required to:

• Write a pseudocode of a Decrease & Conquer algorithm for solving the above problem

• Find the worst-case number of calls T(n) of this algorithm in terms of n.

Answer 1

The psuedo-code for this is as follows:

1. Create a matrix of size .
2. For , do
   1. ​​​​​​​
3. For ​​​​​​​
   1. For
      1. ​​​​​​​
   2. For ​​​​​​​
      1. ​​​​​​​
4. Output .

The psuedo-code simply follows what the instructions are.

For time analysis, note that the main loop is around instruction 3. This loop runs for times, while the inner loops inside instruction 3 run for time. Hence, the total time for this is ​​​​​​​.