Fractional Knapsack Problem in DSA C - Execution Trace

Concept Flow - Fractional Knapsack Problem

Start with empty knapsack

↓

Sort items by value/weight ratio

↓

For each item in sorted order

↓

Can full item fit?

No→Take fraction to fill knapsack

|Yes↓

Add full item to knapsack

↓

Knapsack full?

No→Next item

Yes↓

Done

We sort items by value per weight, then add as much as possible from each until the knapsack is full.

Execution Sample

DSA C

items = [(60, 10), (100, 20), (120, 30)]; capacity = 50;
// Sort by value/weight ratio
// Take items fully or fractionally until capacity fills.

This code picks items or fractions to maximize value within capacity.

Execution Table

Step	Operation	Item Taken	Weight Added	Value Added	Remaining Capacity	Knapsack State (weight:value)
1	Sort items by value/weight	-	-	-	50	-
2	Take full item	(60,10)	10	60	40	10:60
3	Take full item	(100,20)	20	100	20	30:160
4	Take fraction of item	(120,30)	20 (fraction)	80	0	50:240
5	Knapsack full	-	-	-	0	50:240

💡 Knapsack capacity reached 0, no more items can be added.

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	Final
Remaining Capacity	50	40	20	0	0
Total Value	0	60	160	240	240
Knapsack Weight	0	10	30	50	50

Key Moments - 3 Insights

Why do we sort items by value/weight ratio before picking?

Sorting ensures we pick items giving the most value per weight first, maximizing total value as shown in Step 1 of execution_table.

Why do we take only a fraction of the last item?

When the remaining capacity is less than the item's weight (Step 4), we take only the fraction that fits to fill the knapsack exactly.

What happens when the knapsack capacity reaches zero?

The process stops immediately (Step 5) because no more weight can be added, ensuring we don't exceed capacity.

Visual Quiz - 3 Questions

Look at the execution_table, what is the remaining capacity after Step 3?

A 20

B 30

C 40

D 10

💡 Hint

Check the 'Remaining Capacity' column at Step 3 in execution_table.

At which step do we add a fractional part of an item?

A Step 2

B Step 3

C Step 4

D Step 5

💡 Hint

Look for 'Take fraction of item' in the Operation column of execution_table.

If the knapsack capacity was 60 instead of 50, what would change in the knapsack state after Step 4?

A Knapsack weight would be 50

B Knapsack weight would be 60

C Knapsack value would be 240

D Knapsack value would be less than 240

💡 Hint

Check how weight added accumulates in variable_tracker and execution_table.

Concept Snapshot

Fractional Knapsack Problem:
- Sort items by value/weight ratio descending
- Pick full items while capacity allows
- Take fraction of next item if needed
- Stop when capacity is full
- Maximizes total value with fractional items allowed

Full Transcript

The Fractional Knapsack Problem involves filling a knapsack with items to maximize value without exceeding capacity. We first sort items by their value-to-weight ratio to prioritize the most valuable per weight. Then, we add items fully if they fit, or fractionally if only part fits, until the knapsack is full. This approach ensures maximum total value. The execution table shows each step: sorting, adding full items, adding a fraction, and stopping when full. Variables like remaining capacity and total value update accordingly. Key points include why sorting is essential, why fractions are taken, and stopping when full. The quiz checks understanding of capacity changes and fractional addition.