[Verse 1]
Started with a pocket full of quarters and some dimes
Customer needs change, algorithm's mine
Dynamic programming breaking down the grind
Minimum coins needed, optimal design
Got denominations stacked like building blocks
Twenty-five, ten, five, ones - unlock the locks
Bottom-up approach, table in my mind
Subproblems cascade, solutions intertwined
[Chorus]
Memo-ize the base case, zero coins for zero
Build up from the bottom, be that algo hero
Take the minimum plus one, check each denomination
Coin change mastery through memoization
Memo-ize the base case, zero coins for zero
Greedy won't work always, dynamic's the real hero
[Verse 2]
Table indexing from zero to amount
Each position holds the minimum coin count
For every value check what came before
Add one coin, find what's less in your store
If amount minus coin exists in the array
Take that count plus one, update right away
Compare all options, keep the smallest number
This is how optimal solutions slumber
[Chorus]
Memo-ize the base case, zero coins for zero
Build up from the bottom, be that algo hero
Take the minimum plus one, check each denomination
Coin change mastery through memoization
Memo-ize the base case, zero coins for zero
Greedy won't work always, dynamic's the real hero
[Bridge]
Time complexity linear with the sum
Space complexity matching what will come
Sometimes no solution exists in sight
Return negative one or infinite might
Top-down recursion with memoization
Same result, different computation
[Verse 3]
Cashier's algorithm breaks with certain sets
Eleven cents with five and seven gets
Greedy takes five-one-one-one-one-one-one
But optimal solution: seven plus four ones done
This is why we need the power of DP
Breaking complex problems systematically
Every subproblem solved just once and stored
Efficiency maximized, performance soared
[Outro]
From vending machines to financial systems
Coin change algorithm solves the problems
Bottom-up table or recursive call
Dynamic programming conquers them all