mirror of
https://github.com/mfocko/blog.git
synced 2024-11-15 02:17:33 +01:00
159 lines
5.4 KiB
Markdown
159 lines
5.4 KiB
Markdown
---
|
||
id: bottom-up-dp
|
||
slug: /recursion/pyramid-slide-down/bottom-up-dp
|
||
title: Bottom-up DP solution
|
||
description: |
|
||
Bottom-up DP solution of the Pyramid Slide Down.
|
||
tags:
|
||
- java
|
||
- dynamic-programming
|
||
- bottom-up-dp
|
||
last_update:
|
||
date: 2023-08-17
|
||
---
|
||
|
||
# Bottom-up dynamic programming
|
||
|
||
If you try to think in depth about the top-down DP solution, you might notice
|
||
that the _core_ of it stands on caching the calculations that have been already
|
||
done on the lower “levels” of the pyramid. Our bottom-up implementation will be
|
||
using this fact!
|
||
|
||
:::tip
|
||
|
||
As I have said in the _top-down DP_ section, it is the easiest way to implement
|
||
DP (unless the cached function has complicated parameters, in that case it might
|
||
get messy).
|
||
|
||
Bottom-up dynamic programming can be more effective, but may be more complicated
|
||
to implement right from the beginning.
|
||
|
||
:::
|
||
|
||
Let's see how we can implement it:
|
||
|
||
```java
|
||
public static int longestSlideDown(int[][] pyramid) {
|
||
// In the beginning we declare new array. At this point it is easier to just
|
||
// work with the one dimension, i.e. just allocating the space for the rows.
|
||
int[][] slideDowns = new int[pyramid.length][];
|
||
|
||
// Bottom row gets just copied, there's nothing else to do… It's the base
|
||
// case.
|
||
slideDowns[pyramid.length - 1] = Arrays.copyOf(pyramid[pyramid.length - 1],
|
||
pyramid[pyramid.length - 1].length);
|
||
|
||
// Then we need to propagate the found slide downs for each of the levels
|
||
// above.
|
||
for (int y = pyramid.length - 2; y >= 0; --y) {
|
||
// We start by copying the values lying in the row we're processing.
|
||
// They get included in the final sum and we need to allocate the space
|
||
// for the precalculated slide downs anyways.
|
||
int[] row = Arrays.copyOf(pyramid[y], pyramid[y].length);
|
||
|
||
// At this we just need to “fetch” the partial results from “neighbours”
|
||
for (int x = 0; x < row.length; ++x) {
|
||
// We look under our position, since we expect the rows to get
|
||
// shorter, we can safely assume such position exists.
|
||
int under = slideDowns[y + 1][x];
|
||
|
||
// Then we have a look to the right, such position doesn't have to
|
||
// exist, e.g. on the right edge, so we validate the index, and if
|
||
// it doesn't exist, we just assign minimum of the ‹int› which makes
|
||
// sure that it doesn't get picked in the ‹Math.max()› call.
|
||
int toRight = x + 1 < slideDowns[y + 1].length
|
||
? slideDowns[y + 1][x + 1]
|
||
: Integer.MIN_VALUE;
|
||
|
||
// Finally we add the best choice at this point.
|
||
row[x] += Math.max(under, toRight);
|
||
}
|
||
|
||
// And save the row we've just calculated partial results for to the
|
||
// “table”.
|
||
slideDowns[y] = row;
|
||
}
|
||
|
||
// At the end we can find our seeked slide down at the top cell.
|
||
return slideDowns[0][0];
|
||
}
|
||
```
|
||
|
||
I've tried to explain the code as much as possible within the comments, since it
|
||
might be more beneficial to see right next to the “offending” lines.
|
||
|
||
As you can see, in this approach we go from the other side[^1], the bottom of
|
||
the pyramid and propagate the partial results up.
|
||
|
||
:::info How is this different from the _greedy_ solution???
|
||
|
||
If you try to compare them, you might find a very noticable difference. The
|
||
greedy approach is going from the top to the bottom without **any** knowledge of
|
||
what's going on below. On the other hand, bottom-up DP is going from the bottom
|
||
(_DUH…_) and **propagates** the partial results to the top. The propagation is
|
||
what makes sure that at the top I don't choose the best **local** choice, but
|
||
the best **overall** result I can achieve.
|
||
|
||
:::
|
||
|
||
## Time complexity
|
||
|
||
Time complexity of this solution is rather simple. We allocate an array for the
|
||
rows and then for each row, we copy it and adjust the partial results. Doing
|
||
this we get:
|
||
|
||
$$
|
||
\mathcal{O}(rows + 2n)
|
||
$$
|
||
|
||
Of course, this is an upper bound, since we iterate through the bottom row only
|
||
once.
|
||
|
||
## Memory complexity
|
||
|
||
We're allocating an array for the pyramid **again** for our partial results, so
|
||
we get:
|
||
|
||
$$
|
||
\mathcal{O}(n)
|
||
$$
|
||
|
||
:::tip
|
||
|
||
If we were writing this in C++ or Rust, we could've avoided that, but not
|
||
really.
|
||
|
||
C++ would allow us to **copy** the pyramid rightaway into the parameter, so we
|
||
would be able to directly change it. However it's still a copy, even though we
|
||
don't need to allocate anything ourselves. It's just implicitly done for us.
|
||
|
||
Rust is more funny in this case. If the pyramids weren't used after the call of
|
||
`longest_slide_down`, it would simply **move** them into the functions. If they
|
||
were used afterwards, the compiler would force you to either borrow it, or
|
||
_clone-and-move_ for the function.
|
||
|
||
---
|
||
|
||
Since we're doing it in Java, we get a reference to the _original_ array and we
|
||
can't do whatever we want with it.
|
||
|
||
:::
|
||
|
||
# Summary
|
||
|
||
And we've finally reached the end. We have seen 4 different “solutions”[^2] of
|
||
the same problem using different approaches. Different approaches follow the
|
||
order in which you might come up with them, each approach influences its
|
||
successor and represents the way we can enhance the existing implementation.
|
||
|
||
---
|
||
|
||
:::info source
|
||
|
||
You can find source code referenced in the text
|
||
[here](pathname:///files/algorithms/recursion/pyramid-slide-down.tar.gz).
|
||
|
||
:::
|
||
|
||
[^1]: definitely not an RHCP reference :wink:
|
||
[^2]: one was not correct, thus the quotes
|