[Verse 1]
Backpack full of treasures, but space is running thin
Got diamonds, gold, and silver, can't fit everything in
Unlike zero-one decisions where you take it all or none
Fractional lets you slice it up, grab portions, have some fun
Calculate the ratio, value over weight divine
Sort descending by that number, greediest design
Platinum worth a hundred, weighs just ten pounds flat
Ten dollars per pound ratio, that's where the magic's at
[Chorus]
Greedy by the ratio, slice and dice with care
Value over weight ratio, profits everywhere
Fill it up completely, never leave space bare
Fractional knapsack algorithm, optimal we declare
Sort by ratio, grab the best first
Fill until capacity, quench that greedy thirst
[Verse 2]
Scan through sorted items, grab the heavyweight champs
Those with highest ratios get priority stamps
Whole items fit inside? Toss them in with glee
Partial item breaks the limit? Calculate what percentage we need
If backpack holds fifty pounds, item weighs twenty-three
But only ten pounds left inside, take ten twenty-thirds, you see
Point four three four seven eight, that's our fraction clean
Multiply by item value for the cash we glean
[Chorus]
Greedy by the ratio, slice and dice with care
Value over weight ratio, profits everywhere
Fill it up completely, never leave space bare
Fractional knapsack algorithm, optimal we declare
Sort by ratio, grab the best first
Fill until capacity, quench that greedy thirst
[Bridge]
Linear time complexity once the sorting's done
N log N for the ratio ranking, then we run
Greedy choice at every step, locally optimal
Proves globally optimal too, mathematically radical
No dynamic programming needed here
Simple greedy conquers fear
[Verse 3]
Initialize remaining as capacity complete
While items left and space exists, the loop we repeat
Current item fits entirely? Add value, subtract weight
Exceeds the space remaining? Calculate fractional fate
Remaining over item weight gives fraction decimal
Multiply by item value, add to total tactical
Algorithm terminates when backpack reaches max
Fractional knapsack mastery, no mathematical tax
[Outro]
Ratio sorting, greedy taking
Fractional portions, profit making
Optimal solution, no mistake
Fractional knapsack, for efficiency's sake