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DA5. The Knapsack Problem

Statement

In your own words describe the ‘knapsack problem’. Further, compare and contrast the use of a brute force and a dynamic programming algorithm to solve this problem in terms of the advantage and disadvantages of each. An analysis of the asymptotic complexity of each is required as part of this assignment.

Include one or two examples to explain your thought process to show what is occurring and how the methodology works. Use APA citations and references for any sources used.

Answer

Introduction

The knapsack problem involves finding the optimal combination of items to include in a knapsack, given a set of items with different weights and values, and a knapsack with a maximum weight capacity. The goal is to maximize the total value of carried items without exceeding the weight capacity of the knapsack (Shaffer, 2011).

This text will start by an introduction to the knapsack problem. Then, it will discuss both the brute force and dynamic programming approaches to solving it, and finally conclude with a comparison of the two approaches. There are two types of knapsack problems: the first allows for multiple copies of each item to be taken, while the second allows only for one item of each type to be carried, and our discussion will focus on the second type.

The problem is a classic optimization problem that has many real-world applications, such as resource allocation, financial portfolio management, project scheduling, and many more. The problem has been studied extensively using different algorithm patterns, including brute force, dynamic programming, and greedy approaches (Hristakeva & Shrestha, 2005).

We will use the following example to illustrate the problem: given a knapsack with a weight capacity of 10 units, and a set of items with weights and values as follows:

Item Weight Value
1 2 6
2 3 5
3 5 4
4 6 3

Brute Force Approach

The brute force approach is straightforward: generate all possible combinations of items that can be carried; calculate the total weight and value of each combination; and then search the solution space for the combination with the most value and weights within the sack’s capacity. The combinations are usually generated by an array of 0/1 values, where 0 indicates the item is not included, and 1 indicates the item is included.

The brute force approach has the advantage of being simple to implement and understand while guaranteeing the most optimal solution. However, it has a significant disadvantage in terms of time complexity, as the generated combinations cost exponentially both in space and time to process them. The time complexity of the brute force approach is O(2^n), where n is the number of items, making it impractical for large problem instances (Hristakeva & Shrestha, 2005).

Below is a table summarizing the possible generated combinations for the example problem, with each column representing a portion of work; notice that the optimal solution is to carry items 1,2 and 3, with a total weight of 10 and a total value of 15 (the table was slashed for brevity):

S1: combination S2: weight S3: value S4: valid? S5: best?
0 0 0 0 0 0 No
0 0 0 1 6 3 Yes
0 0 1 0 5 4 Yes
0 0 1 1 11 7 No
… … … … …
0 1 1 1 14 12 No
1 0 1 1 13 13 No
1 1 0 1 11 14 No
1 1 1 0 10 15 Yes Yes ✅
1 1 1 1 16 18 No

Dynamic Programming Approach

The dynamic programming approach to the knapsack problem involves breaking down the problem into smaller subproblems and solving them only once; and then memoize the results to solve larger problems. The key idea is to build a table to store the maximum value that can be achieved with a given weight capacity and a subset of items. The table is filled iteratively, starting with the smallest subproblem and building up to the full problem (Hristakeva & Shrestha, 2005).

The dynamic programming approach has the advantage of being more efficient than the brute force approach, as it eliminates redundant computations caused by the un-used irrelevant combinations storing and reusing the results of subproblems. The time complexity of the dynamic programming approach is O(nW), where n is the number of items and W is the weight capacity of the knapsack, making it more practical for larger problem instances. The space complexity is O(n^2) to store the two dimensional array that holds the results of the subproblems (Hristakeva & Shrestha, 2005).

Below is a table summarizing the dynamic programming approach for the example problem, with first column representing the possible items, and the first row representing the maximum weight capacity; hence, each cell represents the maximum value that can be achieved with the given capacity (column) and items ids (row):

Items(i) Capacity(j) 0 1 2 3 4 5 6 7 8 9 10
0 0 0 0 0 0 0 0 0 0 0
1 0 0 6 6 6 6 6 6 6 6 6
1,2 0 0 6 6 6 11 11 11 11 11 11
1,2,3 0 0 6 6 6 11 11 11 11 11 15
1,2,3,4 0 0 6 6 6 11 11 11 11 11 15 ✅

Yo do not re-compute everything when filling each cell, instead you will simply use the value stored in the previous cell and weight of the current item to determine the value of the cell, according to the following formula (Hristakeva & Shrestha, 2005):

n = 4 # number of items
c = 10 # capacity of the knapsack
weights = [0, 2, 3, 5, 6] # weights of the items, added the 0 because we are 1-indexing
values = [0, 6, 5, 4, 3] # values of the items, added the 0 because we are 1-indexing
table = [[0 for _ in range(c+1)] for _ in range(n+1)] # initialize the table with zeros
for i in range(1, n+1):
    for j in range(1, c+1):
        if weights[i] > j:
            table[i][j] = table[i-1][j]
        else:
            table[i][j] = max(table[i-1][j], table[i-1][j-weights[i]] + values[i])

Conclusion

In conclusion, the knapsack problem is a classic optimization problem that has many real-world applications. The brute force approach is simple to implement but has exponential time and space complexity; thus, it is not recommended for large problems. The dynamic programming approach, on the other hand, is more efficient, both in terms of time and space complexity, although its implementation is more complex.

Looking at the generated tables, we can see that the dynamic approach’s table is less dense in space, and computation is simply relies on checking the previous cell; hence, it is preferred over the brute force approach as the problem size becomes large enough.

References