There is a large class of problems that we can solve by nice mathematical recurrence relations.
The recurrence is usually simple to read, to reason about, and describes the solution with concision. But a naive translation of this recurrence into code will very often lead to a very inefficient algorithm.
When the sub-problems of the recurrence relation form a DAG, the usual trick is to use Dynamic Programming to speed up the computation. But it often results in a complete change of the code, hiding the recurrence, sometimes to the point we cannot recognize the problem anymore.
Thankfully, with open recursion, can have the best of both worlds!
Counting Binary Search Trees
Before introducing the technique, let us first take a nice example whose solution can be described by a simple recurrence relation: counting binary search trees.
We observe that we can cut any collection [X1, X2 .. Xn] in two parts, around one of its element Xi. Then, we can:
- Recursively count all BST on the left part [X1 .. Xi).
- Recursively count all BST on the right part (Xi .. Xn]
- Multiply these values to get the count of BST rooted in Xi.
As we have to do this for all i in [1..N], the recurrence relation becomes:
bstCount :: Int -> Integer bstCount n | n <= 1 = 1 | otherwise = sum [bstCount i * bstCount (n-1-i) | i <- [0..n-1]]
But this algorithm is terribly inefficient: we keep recomputing the same sub-solutions over an over again.
We know the next step, which is to memoize the solutions to the sub-problems. However, as our original goal was to do it without compromising the code readability, let us first introduce open recursion.
Adding open recursion
Open recursion consists of avoiding direct recursion by adding an extra layer of indirection. It typically means transforming our recurrence relation to take a new parameter, a function that will be called instead of recurring.
By doing so, the recurrence formula looses its recursive nature. Here is how it would translate into Haskell:
bstCount :: (Int -> Integer) -> Int -> Integer bstCount rec n | n <= 1 = 1 | otherwise = sum [rec i * rec (n-1-i) | i <- [0..n-1]]
How can we get our recurrence relation back? By introducing a wrapper function that will give itself to the “bstCount” recurrence. Instead of having a direct recursion, we have a two-step recursion. This is best explained by example:
bstCountNaive :: Int -> Integer bstCountNaive = bstCount bstCountNaive
By simple renaming, we can see that it can be expressed as: x = f x. So the naive recursive algorithm we had earlier is effectively the fixed point of the open recurrence relation. Which can be written in Haskell as:
import Data.Function(fix) bstCountNaive :: Int -> Integer bstCountNaive = fix bstCount
We can now exploit this open recursion to insert some memoization in the middle of the recursion. In our specific case, the sub-problems exhibit a simple structure:
- We can compute a vector of the results of each sub-problem 0..N
- Then recurring part consist in indexing into this vector in O(1)
So instead of triggering a recursion, we search the sub-solution result into our memoization vector, which translates into the following Haskell code:
memoBstCount :: Int -> Integer memoBstCount n = Vector.last memo where memo = Vector.generate (n+1) (bstCount (memo Vector.!))
How can it even work? You are witnessing here the magic of Haskell: laziness helps us refer to the item we are computing inside its own computation.
Following this change, at each N we now have N-1 steps to perform, each taking constant time, thanks to memoization. This gives us a quadratic complexity (the result of the sum from 1 to N).
Using open recursion in combination with Haskell laziness has effectively let us decoupled the following aspects:
- The recurrence relation, solution to our problem
- The memoization strategy, indexing into a vector
- The order in which we compute these sub-solutions
As a result, we get all the benefits of having a simple recurrence relation, untainted by the implementation details required to get an efficient implementation.
In the next post, we will see how to apply this trick in a more mainstream language: C++.