309. Best Time to Buy and Sell Stock with Cooldown

class Solution(object):
    def maxProfit(self, prices):
        """
        :type prices: List[int]
        :rtype: int
        """
        #cash1:cash on hand if i sell today 
        #cash2:cash on hand if i do nothing today 
        #start with 0 cash on hand 
        cash1,cash2=0,0
        for i in xrange(1,len(prices)):
            cash1_yesterday=cash1
            #2 scenarios:
            #1.i just sold the stock yesterday, so i need to buy back yesterday and sell today.(postpone the selling from yesterday to today)
            #2. did nothing yesterday (didnt have a stock), so need to buy today and sell today(equal to do nothing) 
            cash1=max(cash1_yesterday-prices[i-1]+prices[i],cash2)
            cash2=max(cash1_yesterday,cash2)
            
        return max(cash1,cash2)

state machine 
three state
s0: not holding a stock, free to trade. actions: a1 stay put->s0; a2 buy a stock->s1
s1: holding a stock, free to buy. action: a1 stay put ->s1; a2 sell a stock->s2 
s2: not holding a stock, cool down period not able to trade. action: a1 rest ->s0 
s0[i]=max(s0[i-1],s2[i-1])
s1[i]=max(s0[i-1]-p[i],s1[i-1])
s2[i]=s1[i-1]+p[i]
on day i, max profit = max(s0[i],s2[i])
initial condition: 
s0[0]=0
s1[0]=-p[0]
s2[0]=s2 came from s1 by selling, without price information, assign s2[0] as -inf 

class Solution(object):
    def maxProfit(self, prices):
        """
        :type prices: List[int]
        :rtype: int
        """
        #cash1:cash on hand if i sell today 
        #cash2:cash on hand if i do nothing today 
        #start with 0 cash on hand 
        if not prices: return 0
        days=len(prices)
        s0=[0]*days
        s1=s0[:]
        s2=s0[:]
        
        #initialize 
        s1[0]=-prices[0]
        s2[0]=-float('inf') 
        
        for i in xrange(1,days):
            s0[i]=max(s0[i-1],s2[i-1])
            s1[i]=max(s0[i-1]-prices[i],s1[i-1])
            s2[i]=s1[i-1]+prices[i]
        return max(s0[days-1],s2[days-1])