就时间复杂度而言，自下而上的 DP 解决方案是否优于自上而下？

Question

Are there any Dynamic Programming Problems where the Bottom-Up solution has better time complexity than Top-Down (Memoization) solution?

Answer 1

You might be able to make one up, but generally no.

When both kinds of solutions are available, the worst case time complexities are the same.

Bottom-up solutions are often faster in the worst case, in absolute terms (not asymptotic complexity) because memoization is a relatively expensive operation.

Top-down solutions are often faster in best or special cases, because they evaluate only the subproblems that are needed for each problem instance.

Answer 2

Asymptotically both DP and Memoization give the same Time Complexity. However, at run-time, the DP solution outpaces Memoization technique by some constant factor.
This is because of the fact that in the case of DP we just need to look into the table for results of sub-problems while in case of Memoization we need to call recursion and then it returns the result whether by directly checking into hashtable/array(whatever you used) if its already computed or by computing it, which in turn consumes more CPU cycle.

In some cases, it might happen that our sub-problem space is very large but we do not require all sub-problems to get our answer then, Memoization could be lucrative because it solves only inevitable problems instead of all. But, this case is very seldom to occur because Many a time we optimise our DP code only in such a way that it doesn't iterate for all sub-problems instead only for required sub-problems.

Hence, generally speaking, Bottom-Up approach ie, DP solution always runs faster than its corresponding Memoization solution.

Note: I'm using the word runs faster instead of time complexity because time complexity is asymptotic which is same for both.