Dynamic Programming


Dynamic Programming is a special form of divide and conquer method, it’s very useful when the subproblems overlap each other. By using a tabular method, it stores the computed result to accelerate the computation.

There’re 4 steps to construct a solution to DP problems:

1. Characterize the structure of an optimal solution(How to represent the problem using "last step" and "subproblem")
2. Recursively build the value of an optimal solution(How to use a function to build the relationship)
3. Compute the value of setup 2 typically in a bottom-up fashion(Initial Value[0, INF])
4. Construct the solution from computed information(Bottom up)

Top-down and Bottom-up

Both top-down and bottom-up can be used to solve the DP problems, but they have different though process and coding paradigm.

  1. With Top-down:

    a. you have to start from the final question (F(N))

    b. each level reduce the problem to a smaller size, until you reach the base case and the problem is trivial to solve (F(N-1) or F(N/2) etc.)

    c. return the base case immediately (F(0) or F(1)) (return type depends on the question)

    d. don’t forget to also return at each general case level (Build the relationship at this step)

    With Top-down, you can use memoization to reduce the time complexity.

    Time Complexity: O(N*)

    Space Complexity: O(height)

  2. With Bottom-up:

    a. you have to start from the base solution (F(0) or F(1))

    b. each level build the solution to a larger size, until you reach the final solution (F(N-1) or F(N/2) etc.)

    c. return the final solution immediately (F(N))

    Time Complexity: O(N**2)

    Space Complexity: O(N**2)

  3. Another way to separate the 2 methods is to check their implementation:

    ​ Bottom-up is the Iterative implementation of Top-down

    ​ Top-down is recursive way which uses the stack to hold and wait for the return of F(N-1)

Top-down and Backtracking

  1. They have similar recursive call implementation

  2. They can all be treated as a Root-Leaf Path

  3. They have different way to verify the base case

  4. backtracking usually have one global variable to record curr path