Question

You're walking along the beach and you stub your toe on something in the sand. You dig around it and find that it is a treasure chest of n golden bricks of different (integral) weight. Your backpack can only carry up to Wmax. ounces before it breaks apart. You want to put as much in it as possible without going over, but you cannot break the golden bricks up. For the following problems, assume the arrays V=[v1,v2,V3.......Vn ] and W=[w1 ,w 2,…,wn] describe the value and weight of each golden brick. That is, the i-th picking

a) golden bricks of the highest value, until reaching the limit of your backpack. Give an example of golden bricks for which this strategy results in suboptimal solution. Make sure to list the weights and values of golden bricks, state the value in which the above strategy results, and state the optimal (larger) value that you could take away.
(b) Prove that the problem of picking the golden bricks of maximum total value (given the contraints of your backpack) exhibits optimal substructure

Answer 1

The student's question involves the Knapsack problem, where a simple high-value-first strategy can be suboptimal when there's a weight capacity. An example demonstrates this, and the problem's optimal substructure is explained by showing how solutions to smaller problems help construct the solution to the larger one.

Step-by-step explanation:

The subject question deals with a classic optimization problem often studied in mathematics and computer science: the Knapsack problem. Specifically, it asks for an example where simply choosing the golden bricks with the highest value first may not lead to the optimal solution when considering the weight capacity of a backpack (Wmax).

Here's an example:

• Golden Brick A: Weight = 1 oz, Value = $1
• Golden Brick B: Weight = 4 oz, Value = $3
• Golden Brick C: Weight = 3 oz, Value = $4

Let's assume the backpack can carry Wmax = 4 oz. Using the strategy of picking the highest value bricks would lead us to choose Golden Brick C for a total value of $4. However, the optimal solution is to pick Golden Bricks A and B, yielding a value of $4 ($1 + $3) but utilizing the full capacity of the backpack.

For part (b), the Knapsack problem exhibits optimal substructure because the optimal solution to the entire problem contains within it the optimal solutions to subproblems. For any subset of the n items and any weight capacity W, if you know the optimal solution for this subproblem, you can extend it optimally by choosing whether to include or exclude the next item, considering its value and weight, to yield an optimal solution for the increased problem size.