DP

• like careful brtue force
• dp = subproblem + 'reuse'

Case1 Fib

memorized version

fib(n)
	if n in memo: return memo[n]
    if n <= 2 f=1
    else f = fib(n-1) + fib(n-2)
    	 memo[n] = f // memorized call constant time
    return f;

the cost?

• linear

if we use memorization we need to only calculate fib$_{1}$~fib$_{n}$ once, each memo cost $\theta(1)$, the total is $\theta(n)$

dict version

• bottom-up fib

fib = {} // a dict
for k in range(1, n+1):
	if k <= 2 f =1
    else f = fib[k-1] + fib[k-2]
    fib[k] = f
return fib[n]

• further step

delete the previous memo, put in the new one.

topological sort

Case2 Shortest Path

• Shortest Path: $\delta(s,v)$
• $\delta(s,v) = min(\delta(s,u)) + w(u,v)$
• subproblem needs to be acyclic, or will be infinite time
• cyclic -> acyclic