Why Fractional Knapsack Problem in DSA Python?

The Big Idea

What if you could carry the most valuable parts of your items instead of all or nothing?

The Scenario

Imagine you have a backpack with limited weight capacity, and you want to fill it with items to get the most value. You try to pick whole items one by one, but some items are too heavy or you can't carry them all.

The Problem

Picking items one by one without breaking them means you might miss out on getting the best total value. You waste space and can't carry parts of valuable items, making your backpack less useful.

The Solution

The Fractional Knapsack Problem lets you take parts of items instead of whole ones. This way, you fill your backpack with the most valuable pieces first, maximizing total value without wasting space.

Before vs After

✗ Before

items = [(weight1, value1), (weight2, value2)]
capacity = W
for item in items:
    if item[0] <= capacity:
        # take whole item
        capacity -= item[0]
    else:
        # skip item

✓ After

items = [(weight1, value1), (weight2, value2)]
capacity = W
items.sort(key=lambda x: x[1]/x[0], reverse=True)
for item in items:
    taken_weight = min(item[0], capacity)
    capacity -= taken_weight

What It Enables

You can maximize the total value you carry by smartly choosing fractions of items, not just whole ones.

Real Life Example

Imagine filling a suitcase for a trip where you can pack liquids or powders in smaller amounts to save space and carry more valuable things.

Key Takeaways

Manual picking wastes space and value.

Fractional Knapsack allows taking parts of items.

This leads to the best total value within weight limits.

Practice

1. What is the main goal of the Fractional Knapsack Problem?

easy

A. Maximize total value by taking fractions of items within weight limit

B. Minimize total weight by selecting whole items only

C. Maximize number of items regardless of weight

D. Select items randomly without considering value or weight

Solution

1. Step 1: Understand problem objective

   The fractional knapsack problem aims to maximize the total value of items taken, allowing fractions of items to fit within the weight limit.

2. Step 2: Compare options

   Only Maximize total value by taking fractions of items within weight limit correctly describes maximizing value with fractional items. Others describe incorrect goals.

3. Final Answer:

   Maximize total value by taking fractions of items within weight limit -> Option A

4. Quick Check:

   Goal = Maximize value with fractions [OK]

Hint: Remember: fractional means partial items allowed [OK]

Common Mistakes:

• Thinking only whole items can be taken
• Confusing value maximization with weight minimization
• Ignoring the fractional part of items

2. Which of the following Python code snippets correctly sorts items by value-to-weight ratio for the fractional knapsack?

easy

A. items.sort(key=lambda x: x[0]/x[1], reverse=True)

B. items.sort(key=lambda x: x[1]/x[0], reverse=True)

C. items.sort(key=lambda x: x[1]+x[0], reverse=True)

D. items.sort(key=lambda x: x[0]-x[1], reverse=True)

Solution

1. Step 1: Identify ratio for sorting

   Value-to-weight ratio is value divided by weight, so ratio = x[1]/x[0].

2. Step 2: Check sorting code

   items.sort(key=lambda x: x[1]/x[0], reverse=True) sorts items by this ratio descending, which is correct for fractional knapsack.

3. Final Answer:

   items.sort(key=lambda x: x[1]/x[0], reverse=True) -> Option B

4. Quick Check:

   Sort by value/weight descending [OK]

Hint: Sort by value divided by weight, descending [OK]

Common Mistakes:

• Reversing numerator and denominator
• Sorting by sum or difference instead of ratio
• Not using reverse=True for descending order

3. What is the output of this code snippet for fractional knapsack?

items = [(10, 60), (20, 100), (30, 120)]
capacity = 50
items.sort(key=lambda x: x[1]/x[0], reverse=True)
total_value = 0
for w, v in items:
    if capacity >= w:
        capacity -= w
        total_value += v
    else:
        total_value += v * (capacity / w)
        break
print(round(total_value, 2))

medium

A. 240.00

B. 220.00

C. 180.00

D. 200.00

Solution

1. Step 1: Sort items by value/weight ratio

   Ratios: (60/10)=6, (100/20)=5, (120/30)=4. Sorted order: (10,60), (20,100), (30,120).

2. Step 2: Add items until capacity is full

   Take full 10 weight: value=60, capacity=40 left.
   Take full 20 weight: value=60+100=160, capacity=20 left.
   Take fraction of last item: fraction=20/30=2/3, value=160 + 120*(2/3)=160 + 80=240.
   But code prints rounded total_value after break, so total_value=240.

3. Final Answer:

   240.00 -> Option A

4. Quick Check:

   Sum values with fractions = 240.00 [OK]

Hint: Add full items first, then fraction of last item [OK]

Common Mistakes:

• Not sorting items by ratio
• Adding fractions incorrectly
• Rounding errors

4. Find the error in this fractional knapsack code snippet:

def fractional_knapsack(items, capacity):
    items.sort(key=lambda x: x[0]/x[1], reverse=True)
    total_value = 0
    for w, v in items:
        if capacity >= w:
            capacity -= w
            total_value += v
        else:
            total_value += v * (capacity / w)
            break
    return total_value

print(fractional_knapsack([(10, 60), (20, 100), (30, 120)], 50))

medium

A. Function returns total_value before loop

B. Loop variables w and v are swapped

C. Capacity is not updated after taking fraction

D. Sorting ratio is reversed; should be value/weight, not weight/value

Solution

1. Step 1: Check sorting key

   Sorting uses x[0]/x[1] which is weight/value, but ratio should be value/weight (x[1]/x[0]).

2. Step 2: Verify other parts

   Loop variables and capacity update are correct; function returns after loop.

3. Final Answer:

   Sorting ratio is reversed; should be value/weight, not weight/value -> Option D

4. Quick Check:

   Sort by value/weight, not weight/value [OK]

Hint: Sort by value divided by weight, not the other way [OK]

Common Mistakes:

• Using weight/value instead of value/weight for sorting
• Swapping weight and value in loop
• Not updating capacity after fraction

5. Given items with weights and values: [(5, 30), (10, 40), (15, 45)], and capacity 20, what is the maximum value achievable using fractional knapsack?

hard

A. 70.0

B. 75.0

C. 85.0

D. 80.0

Solution

1. Step 1: Calculate value/weight ratios and sort

   Ratios: (30/5)=6, (40/10)=4, (45/15)=3.
   Sorted order: (5,30), (10,40), (15,45).

2. Step 2: Add items until capacity full

   Take full 5 weight: value=30, capacity=15 left.
   Take full 10 weight: value=30+40=70, capacity=5 left.
   Take fraction of last item: fraction=5/15=1/3, value=70 + 45*(1/3)=70 + 15=85.

3. Final Answer:

   85.0 -> Option C

4. Quick Check:

   Sum values with fractions = 85.0 [OK]

Hint: Take full items by ratio, then fraction last to fill capacity [OK]

Common Mistakes:

• Not sorting by ratio
• Taking items in wrong order
• Calculating fraction incorrectly