Dynamic Programming

Advent of Code 2023 has just kicked off, and I'm going to try something a bit different this year, I'm going to try and share useful concepts and patterns that play a role in solving each day's puzzle.

Today, we're looking at how you can use dynamic programming to save yourself a lot of computation, and how I spot and reason my way towards solutions in this space.

Dynamic Programming

Dynamic programming was always a sore point for me, one of those patterns that I could identify when I saw it, but which I consistently struggled to reason my way towards using. Speaking with others, that's a pretty common experience, and with Advent of Code 2023 Day 4 being a great case for using it, I thought I'd share what it is, and how you can start to use it too.

So, let's start out with recursion. "Recursion?!" you ask incredulously, "I thought this was about dynamic programming?". Okay fine, you caught me, but it turns out that dynamic programming is really recursion in a heavy coat. So, as I was saying, let's start out with recursion.

Recursion

Recursion is a pattern that most of us learn early on in our programming careers, and it's a particularly useful pattern for decomposing complex problems into simpler ones. Let's take calculating the Fibonacci sequence for example.

TIP

The Fibonacci sequence is a sequence of numbers where each number is the sum of the previous two numbers in the sequence. The first two numbers in the sequence are 0 and 1, and the sequence continues indefinitely, going 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

So, let's define that recursively:

fn fibonacci(n: u32) -> u32 {
    match n {
        0 => 0,
        1 => 1,
        _ => fibonacci(n - 1) + fibonacci(n - 2),
    }
}

Okay, great, we're done! We can easily calculate the Fibonacci sequence for any number we want. Here, let me show you:

0 took 0.0ms
1 took 0.0ms
2 took 0.0ms
...
20 took 1.5ms
...
30 took 210ms
...
40 took 27s

Huh, okay that's not working out so well. I'm going to need to update my statement to say that we can easily calculate the Fibonacci sequence for any number we want, so long as you are willing to wait a very long time for the result.

So what's happening here? Well, if you look at the call graph for running the method above, you'll see something like the following:

You'll notice that because of the recursive nature of the algorithm, we end up calling functions at the bottom of the tree multiple times. In fact, if you look at the call patterns, you'll see that we end up that each additional number results in a fibonacci number of increased calls to the next function down.

So, how can we fix this? Well, we can use a technique called memoization to cache the results of previous calls to the function, essentially trading increased memory usage for reduced computation time.

Memoization

Memoization is a technique that involves caching the results of previous calls to a function, and returning the cached result if the function is called again with the same arguments. In general you'll find it implemented as a wrapper around the function you want to memoize and it is often (but not always) implemented using some kind of dictionary.

struct Fibonacci {
    cache: HashMap<u32, u32>,
}

impl Fibonacci {
    fn new() -> Self {
        Self {
            cache: HashMap::new(),
        }
    }

    fn fibonacci(&mut self, n: u32) -> u32 {
        match n {
            0 => 0,
            1 => 1,
            _ => {
                *self.cache.entry(&n).or_insert_with(|| {
                    self.fibonacci(n - 1) + self.fibonacci(n - 2)
                })
            }
        }
    }
}

Now when we run our code, we store the results of each call in our cache and can shortcut the need to re-compute them in future. The result is that each additional number in the sequence adds a constant amount of additional computational work, rather than an exponential amount. Effectively, we've gone from an O(n^2) algorithm to an O(n) algorithm.

The trouble that we're going to run into, however, is that as we start to get to larger numbers we're going to see our call stack grow beyond our stack size limit. We could configure our application to have a larger stack, but eventually this just doesn't scale.

Instead, we'd really rather convert this from a recursive algorithm (which relies on these extra call stacks) to an iterative algorithm (which doesn't, and can therefore be much faster).

Iteration

When it comes to converting a recursive algorithm to an iterative one, the main trick is to spot where the recursion results in an extra step being performed and look at how we can inline that. Regardless of what we do, we're going to end up with a loop of some kind, and for trivial recursion (where you call the same function only once), it's usually a trivial exercise.

Our Fibonacci example isn't one of these cases, you'll notice that we call ourselves twice and that immediately throws a spanner in the works. But if we look carefully, you'll see that if we write out the order of calls, you get something like the following:

That's a nice sequence of incrementing calls, which looks an awful lot like a loop. The only trick is that we need to have filled in the right hand side values before the left hand side gets there, and then we can use those. So let's try that:

fn fibonacci(n: u32) -> u32 {
    let mut cache = HashMap::new();

    cache.insert(0, 0);
    cache.insert(1, 1);

    for i in 2..n+1 {
        cache.insert(i, cache.get(&(i - 1)).unwrap() + cache.get(&(i - 2)).unwrap());
    }

    *cache.get(&n).unwrap()
}

Which brings us neatly back to our original topic, dynamic programming.

Dynamic Programming

When it comes to dynamic programming, we're usually going to find ourselves converting a recursive problem into a linear one, and then taking advantage of the linear steps to act as cache keys. The beauty of using your steps as the cache keys is that they are linear increasing integer values, making them work really well with arrays.

TIP

Just because dictionaries and arrays both cost O(1) for lookups, doesn't mean that the constant cost is the same, in fact dictionary lookups are almost universally slower than array lookups by several orders of magnitude, so if you're able to use an array, you really should.

Looking to our Fibonacci example, we can write the dynamic programming example as follows:

fn fibonacci(n: usize) -> u32 {
    if n < 2 {
        return n;
    }

    let mut cache = vec![0; n + 1];
    cache[1] = 1;

    for i in 2..n+1 {
        cache[i] = cache[i - 1] + cache[i - 2];
    }

    cache[n]
}

Something interesting that you'll note about this example is that we're initializing our cache with its full size from the start. When it come to high performance algorithms, re-allocating memory is one of the biggest sources of latency as the operating system needs to find a block of memory that's large enough to fit your request, and we then need to copy any existing data into the new block. By allocating the full size of the cache up front, we avoid this problem entirely (paying the malloc cost once, rather than a logarithmic number of times as a dynamic vector expands).

Optimizing Further

At this point, you've got an extremely fast dynamic programming solution to the Fibonacci problem, but we can take this a step further by leveraging the insight that we only ever need the last two values. As a result, storing the full cache is a waste of memory, and we can refactor our code to instead only keep the most recent two values in memory.

fn fibonacci(n: u32) -> u32 {
    if n < 2 {
        return n;
    }

    let mut last = 0;
    let mut current = 1;

    for _ in 2..n+1 {
        let next = current + last;
        last = current;
        current = next;
    }

    current
}

Not only does this approach save a bit of memory, it also saves some pressure on the system memory management unit and allows the compiler to optimize the code such that both the last and current variables are stored in registers. While in this specific case, the system memory management unit is likely to cache the recently written values and avoid the round-trip-time to system memory (which is tens of thousands of times slower than a register), removing the extra instructions still helps this implementation grab the lead.

In practice, for most problems, you're likely to find that the look-back windows are larger, or that these kinds of micro-optimization are not worth the effort, but it's useful to be aware of how different approaches result in different system performance characteristics and benefits.

Conclusion

I always found it hard to wrap my head around decomposing a problem into one that worked well with Dynamic Programming, but by following the process of converting a recursive algorithm to a linear one, and introducing memoization in the form of an array, I've found it much easier to spot these opportunities and implement the code for them.

I hope you'll find this useful as you take on Advent of Code 2023 Day 4 (and presumably future days too), and that it helps you avoid spinning CPU cycles on problems that can be solved much more efficiently.