[Verse 1]
Backpack heavy, treasure scattered on the ground
Gold and diamonds, precious stones all around
But here's the twist - I can break them into pieces
Value per pound, that's where profit increases
Greedy algorithm, take the best ratio first
Fill that bag until capacity bursts
Not zero-one constraint, I can slice and dice
Fractional portions, maximize the price
[Chorus]
Sort by value density, take the cream
Grab the highest ratios, live the dream
Fractional knapsack, slice what you need
Optimal solution guaranteed
Value over weight, that's the key
Greedy choice leads to victory
[Verse 2]
Calculate the ratios, sort descending order
Pack the premium items past the border
When the next item cannot fit complete
Take a fraction, make the puzzle neat
Linear time sorting if pre-arranged
O of n log n when ratios exchanged
Proof by contradiction shows it's tight
Greedy stays optimal, burning bright
[Chorus]
Sort by value density, take the cream
Grab the highest ratios, live the dream
Fractional knapsack, slice what you need
Optimal solution guaranteed
Value over weight, that's the key
Greedy choice leads to victory
[Bridge]
Unlike zero-one where dynamic programming rules
Fractional version breaks traditional tools
No complex subproblems, no memoization
Just sort and grab - pure optimization
Continuous relaxation of the discrete case
Greedy algorithm sets the perfect pace
[Verse 3]
Implementation flows like water downstream
Initialize capacity, sort the scheme
Iterate through items, check available space
Add full items, then the fractional trace
Remaining capacity divided by weight
Multiply by value, seal the fate
Return total profit from the calculated sum
Fractional knapsack algorithm - done
[Chorus]
Sort by value density, take the cream
Grab the highest ratios, live the dream
Fractional knapsack, slice what you need
Optimal solution guaranteed
Value over weight, that's the key
Greedy choice leads to victory
[Outro]
When the bag is packed and profits maximized
Greedy proved its worth, no compromise
Fractional freedom breaks the binary chain
Optimal harvest from algorithmic rain