create MIT dp memo

Author: dingjy94

algorithm/DP/DP_Note_MIT.md

@@ -0,0 +1,83 @@
+# Dynamic Programming
+This is the not for the DP part (4 lectures) of [MIT 6.006](https://www.youtube.com/watch?v=moPtwq_cVH8&list=PLUl4u3cNGP61Oq3tWYp6V_F-5jb5L2iHb&index=23).
+
+## What is DP
+~= "careful brute force"
+~= "subproblems" + "reuse" + guessing
+
+## 5 steps
+1. define subproblems
+2. guess (part of solution)
+   - in step2(& 3): which subprob to use to solve larger subprob
+   - in step1: add more subprob to guess/remember more (remember more information about past) (for complicate DP)
+3. relate subproblem solution
+4. recurse & memoize -> acyclic -> topo order OR build DP table bottom-up
+   - For memoize: use array OR parent pointer (remember which guess was best)
+5. solve original problem (time = number of subprob * time/subprob)
+
+## For subproblems of strings / sequences
+- suffixes x[i:] O(n)
+- prefixes x[:i] O(n)
+- substrings x[i:j] O(n^2)
+
+## Examples
+### Fibonacci number
+1. subproblems: prefixes fib(n - 1), if memo O(1) each
+2. guess: sum of fib(n - 1) and fib(n - 2)
+3. fib(n) = fib(n - 1) + fib(n - 2)
+4. Easily turn to a DAG, each node indegrees is last two nodes.
+5. fib(n)
+```python
+# Recursive
+if n in memo: return memo[n]
+if n <= 2: f = 1
+else: f = fib(n - 1) + fib(n - 2)
+memo[n] = f
+return f
+
+# Bottom up
+fib = {}
+for k in range(k):
+    if k <= 2: f = 1
+    else: f = fib[k - 1] + fib[k - 2]
+    fib[k] = f
+return fib[n]
+```
+### Shortest paths
+### Text justification
+Split text into "good" lines, badness(i, j): how bad the line start with i, end with j - 1. Should minimize the total badnesss value.
+1. subproblems: suffixes words[i:], each sub O(n)
+2. guess: where to start the next line
+3. `DP(i) = min(i, j)(DP(j) + badness(i, j)), DP(n) = 0`
+4. topoOrder: n, n - 1, n - 1, ..., 0
+   total time: O(n ^ 2)
+5. original prob: DP(0)
+### Perfect information blackjack
+- Deck: c0, c1, ..., cn-1
+- 1 player vs dealer
+- $1 bet/hand
+- Max income
+1. suffix ci
+2. guess: how many hits?
+3.
+```
+BJ(i) = max({-1, 1, 0} + BJ(j - 1) for j in range(i + 4, n) if valid game)
+```
+5. O(n^3)
+### Parenthesization
+Optimal evealuation of associative expression
+1. subprob: optimal evealuation of Ai*...*Aj-1 (substring)
+2. guess: outermost/last multiplation
+3.
+```
+DP(i, j) = min(DP(i, k) + DP(k, j) + cost of (Aik * Akj) for k in range(i + 1, j))
+DP(i, i + 1) = 0
+```
+time/sub = O(n)
+4. topoorder: increasing substring size
+5. DP(0, n)
+### Edit distance
+### Knapsack
+### Piano/Guitar fingering
+### Tetris training
+### Super Mario Bros I