Leetcode 题解 - Dynamic Programming 1D

题解

State: (n)

State(n): how many unique BST to form with n nodes from [1, n], where i is the selected root

for range [1, i -1], this is calculated range for unique BST 

for range[i, n-i], this is the another range for unique BST

multiply the answer for above two ranges for final result

High Level:

1. Initialize memo with n + 1 slots
2. call dfs(n)

dfs(n):

1. Base case (n <= 1)  -> return 1
2. If memo[n] != null   -> return memo[n]
3. Ask subproblems for answers:

        a. for i in [1,n]:

            i. left = dfs(i-1), right = dfs(n-i)

            ii. result = left * right

    4. Update memo[n] and return result

时间复杂度：O(n**2)

另可以用正向填表：把dfs的flow反过来，从底层子问题开始往大计算，最终计算到顶层问题

High Level 

1. Initialize memo[n + 1]
2. Fill out base case  -> memo[0] = memo[1] = 1
3. for i in [2,n]:

        a. Use Transition Rule  ->  for j in [1, i]:

            i. memo[i] += memo[j-1] * memo[i - j]

    4. return memo[n]

时间复杂度：O(n**2)

思路

High Level

1. Initialize memo

2. Call dfs(s,n)

dfs(s,n)

1. Base case -> n <= 1 return 1
2. If memo[n] != null -> return memo[n]
3. Ask Subproblems for answers

       a. if x is valid (not 0) -> result += memo[n-1]

       b. if xx is valid (<= 26) -> result += memo[n-2]

    4. Update memo and return result