Knapsack

1. a memoized, top-down solution, and

   Initialize a 2D array memo of size n by m+1 with null values
   Define function knapsack(i, w):
     # Base Case
     if i < 0 or w == 0:
   	 return 0
     if memo[i][w] is not null:
   	 return memo[i][w]
     if weights[i] > w:
   	 memo[i][w] = knapsack(i-1, w)
     else:
   	memo[i][w]=max(knapsack(i-1,w),profits[i]+knapsack(i-1,w-weights[i]))
    return memo[i][w]

2. an efficient, bottom-up solution

   Initialize a 2D array dp of size n+1 by m+1 with zeros
   
   for i from 1 to n:
     for w from 1 to m:
       if weights[i-1] <= w:
         dp[i][w] = max(dp[i-1][w],profits[i-1]+dp[i-1][w-weights[i-1]])
       else:
         dp[i][w] = dp[i-1][w]
   return dp[n][m]

Test case 1: 
Weights = [1, 2, 3]
Values = [60, 100, 120]
Max Weight = 5
Top-Down: Result - 220, Time - 0.000007 seconds
Bottom-Up: Result - 220, Time - 0.000007 seconds

Test case 2: 
Weights = [4, 2, 3, 1, 5]
Values = [10, 40, 30, 50, 35]
Max Weight = 7
Top-Down: Result - 120, Time - 0.000010 seconds
Bottom-Up: Result - 120, Time - 0.000011 seconds

Test case 3: 
Weights = [4, 5, 6, 2, 2]
Values = [420, 570, 650, 120, 200]
Max Weight = 10
Top-Down: Result - 1070, Time - 0.000009 seconds
Bottom-Up: Result - 1070, Time - 0.000014 seconds

Test case 4: 
Weights = [2, 3, 4, 5, 9, 7, 8, 11, 1, 2]
Values = [150, 35, 200, 60, 750, 300, 150, 400, 10, 20]
Max Weight = 20
Top-Down: Result - 1250, Time - 0.000051 seconds
Bottom-Up: Result - 1250, Time - 0.000051 seconds

Test case 5: 
Weights = [5, 7, 8, 9, 6, 3, 4, 2, 10, 3, 2, 1, 7, 4, 6, 5, 9]
Values = [10, 5, 15, 7, 6, 18, 3, 5, 8, 17, 2, 1, 4, 3, 7, 2, 9]
Max Weight = 30
Top-Down: Result - 75, Time - 0.000157 seconds
Bottom-Up: Result - 75, Time - 0.000122 seconds

When I look at solving the knapsack problem with both the top-down and bottom-up methods, I notice a few things about how fast they work and how they handle big problems:

1. Small Problems: For smaller problems, I don't really see a big difference in speed between top-down and bottom-up. It feels like either method could work just fine.

2. Big Problems: As the problems get bigger, I start to see that the bottom-up method tends to be quicker. I think it's because it doesn't have to keep track of what it's already done in the same way top-down does, which can slow top-down down.

3. Very Big Problems: But when the problems are very large, there are times the top-down method actually runs faster than I expected. This makes me think that top-down can still be a good option for certain kinds of big problems.

In simple terms, if I'm faced with a big problem, I lean towards using the bottom-up method for its speed. However, the top-down method can also be pretty good for some big problems and it's often easier for me to wrap my head around.

When I worked on the Knapsack problem solutions, here's what I found:

• For the Top-Down approach, which uses recursion:
  1. I mainly struggled with memoization, which is about making sure I don't solve the same problem multiple times.
  2. It was difficult to make sure I remembered which problems I already solved so I wouldn't do unnecessary work.
  3. This approach took me longer because I had to be careful not to repeat work, which added extra steps.

• For the Bottom-Up approach, which builds the solution step by step:
  1. I had fewer issues with this one. The main challenge was figuring out how to solve the problem by filling in a table, one step at a time.
  2. The hardest part was understanding the step-by-step process to build up the solution.
  3. Generally, it was quicker to complete once I understood how to lay out the solution step by step.

In the end, I found the top-down method harder because I had to keep track of a lot to avoid redoing work. The bottom-up method felt more straightforward once I got the hang of the steps.

When I compare the top-down and bottom-up approaches for solving the problem Knapsack, here's what I find:

• Top-Down:
  • It feels more natural to me because it's like breaking down a big problem into smaller ones, which is how I usually think.
  • It can be slower if I'm not careful about remembering which problems I've already solved. Also, if the problem is really big, it might use too much memory because of all the reminders.
  • Sometimes, it takes me longer because I need to make sure I'm not solving the same thing over and over again.

• Bottom-Up:
  • It can be faster and use less memory because I'm building up the solution step by step, which is efficient.
  • At first, it's harder for me to see how to build up the solution. I need to understand the problem really well from the start.
  • Once I get how the steps fit together, it's usually quicker for me to put together.

Overall, choosing between top-down and bottom-up depends on what feels more natural to me for the problem I'm solving, how big the problem is, and whether I'm more worried about speed or using less memory.