[Verse 1]
Cashier counting quarters, got a problem to unpack
Customer needs seventy-three cents, what coins should I stack back?
Greedy algorithm says grab the biggest first
Twenty-five cent pieces, but is that the best or worst?
Dynamic programming whispers there's a smarter way
Building up solutions from the bottom every day
Subproblems repeating like an echo in the vault
Memoization saves us when recursion hits a halt
[Chorus]
Bottom-up or top-down, pick your poison for the count
Minimum coins needed, that's what we're trying to amount
DP table filling, every value gets its turn
Optimal substructure, that's the pattern we must learn
Overlapping subproblems, cache the answers that we find
Coin change optimization, efficiency by design
[Verse 2]
Start with denomination one, base case always true
Add each coin type slowly, see what combinations brew
For each amount from zero up to target that we seek
Check every coin that's smaller, let the recurrence speak
Take the minimum plus one from all the valid choices
Previous computed answers become our guiding voices
Space complexity crushing when we only need the row
Time complexity tamed at O of n times m to go
[Chorus]
Bottom-up or top-down, pick your poison for the count
Minimum coins needed, that's what we're trying to amount
DP table filling, every value gets its turn
Optimal substructure, that's the pattern we must learn
Overlapping subproblems, cache the answers that we find
Coin change optimization, efficiency by design
[Bridge]
Unbounded knapsack cousin, infinite supply in hand
Greedy fails with certain sets, but DP takes command
Reconstruction possible, backtrack the chosen path
Array indices revealing the algorithmic math
[Verse 3]
Initialize with infinity, except position zero
Mark impossible amounts when no solution can be hero
Iterate through target range, for each coin in the set
Update minimum if valid, no regrets and no debt
Linear scan optimization, rolling array technique
Memory footprint shrinking when efficiency we seek
[Outro]
From pocket change to algorithms, patterns never lie
Dynamic programming conquers what brute force can't satisfy
Polynomial time solution where exponential once ruled
Coin change problem conquered, optimization schooled