何时使用自下而上 DP 何时使用自上而下 DP

Question

I have leant that two ways of DP, but I am confused now. How we choose in different condition? And I find that in most of time top-down is more natural for me. Can anyone tell me that how to make the choice.

PS: I have read this post older post but still get confused. Need help. Don't identify my questions as duplication. I have mentioned that they are different. I hope to know how to choose and when to consider problem from top-down or bottom-up way.

Answer 1

To make it simple, I will explain based on my summary from some sources

1. Top-down: something looks like: a(n) = a(n-1) + a(n-2) . With this equation, you can implement with about 4-5 lines of code by making the function a call itself. Its advantage, as you said, is quite intuitive to most developers but it costs more space (memory stack) to execute.
2. Bottom-up: you first calculate a(0) then a(1) , and save it to some array (for instance), then you continuously save a(i) = a(i-1) + a(i-2) . With this approach, you can significantly improve the performance of your code. And with big n , you can avoid stack overflow.

Answer 2

A slightly longer answer, but I have tried to explain my own approach to dynamic programming and what I have come to understand after solving such questions. I hope future users find it helpful. Please do feel free to comment and discuss:

A top-down solution comes more naturally when thinking about a dynamic programming problem. You start with the end result and try to figure out the ways you could have gotten there. For example, for fib(n), we know that we could have gotten here only through fib(n-1) and fib(n-2). So we call the function recursively again to calculate the answer for these two cases, which goes deeper and deeper into the tree until the base case is reached. The answer is then built back up until all the stacks are popped off and we get the final result.

To reduce duplicate calculations, we use a cache that stores a new result and returns it if the function tries to calculate it again. So, if you imagine a tree, the function call does not have to go all the way down to the leaves, it already has the answer and so it returns it. This is called memoization and is usually associated with the top-down approach.

Now, one important point I think for the bottom-up approach is that you must know the order in which the final solution has to be built. In the top-down case, you just keep breaking one thing down into many but in the bottom-up case, you must know the number and order of states that need to be involved in a calculation to go from one level to the next. In some simpler problems (eg. fib(n)), this is easy to see, but for more complex cases, it does not lend itself naturally. The approach I usually follow is to think top-down, break the final case into previous states and try to find a pattern or order to then be able to build it back up.

Regarding when to choose either of those, I would suggest the approach above to identify how the states are related to each other and being built. One important distinction you can find this way is how many calculations are really needed and how a lot might just be redundant. In the bottom up case, you have to fill an entire level before you go to the next. However, in the top down case, an entire subtree can be skipped if not needed and in such a way, a lot of extra calculations can be saved.

Hence, the choice obviously depends on the problem, but also on the inter-relation between states. It is usually the case that bottom-up is recommended because it saves you stack space as compared to the recursive approach. However, if you feel the recursion isn't too deep but is very wide and can lead to a lot of unnecessary calculations by tabularization, you can then go for top-down approach with memoization.

For example, in this question: https://leetcode.com/problems/partition-equal-subset-sum/ , if you see the discussions, it is mentioned that top-down is faster than bottom-up, basically, the binary tree approach with a cache versus the knapsack bottom up build-up. I leave it as an exercise to understand the relation between the states.

Answer 3

Bottom-up and Top-down DP approaches are the same for many problems in terms of time and space complexity. Difference are that, bottom-up a little bit faster, because you don't need overhead for recursion and, yes, top-down more intuitive and natural.

But, real advantage of Top-bottom approach can be on some small sets of tasks, where you don't need to calculate answer for all smaller subtasks. And you can reduce time complexity in this cases.

For example you can use Top-down approach with memorization for finding N-th Fibonacci number, where the sequence is defined as a[n]=a[n-1]+a[n-2] So, you have both O(N) time for calculating it (I don't compare with O(logN) solution for finding this number). But look at the sequence a[n]=a[n/2]+a[n/2-1] with some edge cases for small N. In botton up approach you can't do it faster than O(N) where top-down algorithm will work with complexity O(logN) (or maybe some poly-logarithmic complexity, I am not sure)

Answer 4

To add on to the previous answers,

1. Optimal time:

 if all sub-problems need to be solved → bottom-up approach else → top-down approach

2. Optimal space: Bottom-up approach

The question Nikhil_10 linked (ie https://leetcode.com/problems/partition-equal-subset-sum/ ) doesn't require all subproblems to be solved. Hence the top-down approach is more optimal.

Answer 5

If you like the top-down natural then use it if you know you can implement it. bottom-up is faster than the top-down one. Sometimes Bottom-ups are easier and most of the times the bottom-up are easy. Depending on your situation make your decision.