What is dynamic programming?

Cashier Change Problem

Input: Integer Money and positive integers coin1,...coind Output: Min number of coins with denominations coin1....coind that changes money exactly

## Greedy Change

Change -> empty collection of coins while money > 0: coin -> largest denomination that does not exceed money add coin to Change money = money - coin return Change

This fails in Tanzia when changing 40 cents, compared to in the US it works

## Recursive Change

if money = 0 return 0

for i from 1 to coins if money >= coin numCoins -> recursiveChange(money - coin[i], coins) if (numCoins+1 < minNumCoins) minNumCoins = numCoins + 1 return min NumCoins

super slow bc so many recursions !

optimal coin combination for 30 cents....computed trillions of times

## Dynamic Programming

Programming in Dynamic Programming -> when Richard Bellman came up with the idea he was trying to hide what he was doing from secretary of defense, so he called it Programming and it wa dynamic so congress couldn't object.

minNumCoins = 0 for m from 1 to money MinNumCoins(m) -> infinitiy

for 1 to coins { if (m >= coin[i]) { NumCoins = MinNumCoins(m-coin[i]) + 1
if (NumCoins < MinNumCoins(m) { minNumCoins(m) = numCoins } } return MinNumCoins(money)

### Edit Distance

5 min video for computing edit distance

output alignment -> backtracking nodes then we can optimally find the distance for the edit distance

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