List Homomorphisms and Parallelism

A list homomorphism is a function h on lists for which there exists an associative binary operator ⊙ such that

h (x ++ y) = h x ⊙ h y

for any x and y, where ++ denotes list concatenation. Or to put it another way, a function h is a list homomorphism if we can split the input list any way we wish, apply h to the parts independently, and combine the results using some operator ⊙.

Simple examples of list homomorphisms:

Operationally, when computing a list homomorphism we can split the input into any number of chunks, compute a result per chunk, and then combine the results into a final result for the whole list. Each chunk can be processed independently of the others, in parallel. For the purpose of this text, we can consider “list” to mean “array”, which is perhaps more practical. We will not depend on our “lists” having the behaviour of linked lists, and using linked lists would actually inhibit parallelisation.

In principle h need not be defined for empty inputs, but we’ll assume that it is, such that

h [] = e

where e is necessarily an identity element for ⊙. This is strictly not required for a list homomorphism, but it makes the following discussion simpler.

Example of a nontrivial homomorphism

The maximum sum subarray problem (also known as maximum segment sum) is about finding the largest sum of a subarray A[i:j] of some array A. This is not a list homomorphism - knowing mssp x and mssp y is not enough to compute mssp (x++y). Example:

mssp [0, 3,-2]              = 3
mssp [4,-1, 0]              = 4
mssp ([0,3,-2] ++ [4,-1,0]) = 5

But if we extend the domain a bit, we can indeed obtain a homomorphism. This is called a near homomorphism: it computes the result we care about, plus some ancillary information used to combine partial results. In this case, we will compute a tuple with four integer elements:

  1. The maximum subarray sum (i.e., the final result we are actually interested in).

  2. The maximum subarray sum starting from the first element.

  3. The maximum subarray sum ending at the last element.

  4. The sum of the entire array.

Note that the first three must be non-negative, as a subarray can always be empty.

Now define a function f that morally computes such a tuple for single element subarrays:

f x = (max x 0, max x 0, max x 0, x)

Then we define an associative operator for combining our tuples:

(mssx, misx, mcsx, tsx) ⊙ (mssy, misy, mcsy, tsy) =
  (max mssx (max mssy (mcsx + misy)),
   max misx (tsx+misy),
   max mcsy (mcsx+tsy),
   tsx + tsy)

(Proof of associativity left for the reader.) This operator has an identity element:

e = (0, 0, 0, 0)

Now we can define a list homomorphism for solving the MSSP:

h []       = e
h [x]      = f x
h (x ++ y) = h x ⊙ h y

In a real parallel language, we would probably write this as

reduce ⊙ e (map f A)

Why is this the same? Keep reading!

The list homomorphism theorems

The first two list homomorphism theorems were published by Richard S. Bird in 1987, and the third by Gibbons in 1995 (although he notes it had appeared as a “folk theorem” before then). Especially the Gibbons paper (link below) is recommended reading for a precise exposition that clarifies some things I’m leaving fuzzy here.

The first homomorphism theorem

If h is a list homomorphism, then there is an operator ⊙ and function f such that

h xs = reduce ⊙ e (map f xs)

This theorem means that we can represent a list homomorphism as a function f : a -> b, an associative binary operator ⊙ : b -> b -> b, and its identity element e. In many cases f is merely the identity function, which gives us the reduce commonly found in parallel programming systems.

The second homomorphism theorem

If (f,⊙,e) represents a list homormorphism, then

reduce ⊙ e (map f xs) = foldl ⊕ e xs = foldr ⊗ e xs

where

a ⊕ b = f a ⊙ b

and

a ⊗ b = a ⊙ f b

This means that any list homomorphism can be computed with either a left or a right fold using a specialised function derived from ⊙ and f. This is essentially a form of loop fusion, as it allows us to avoid manifesting the result of the map. In a parallel implementation of reduction, we might break the input into a chunk per processor, then use the second list homomorphism theorem to compute an optimised sequential fold for each chunk.

The third homomorphism theorem

If h can be expressed with both a leftwards and rightwards fold, then h is also a list homomorphism. This implies that if we can write a function as both a leftwards and a rightwards fold, then we can write that function as a parallel reduction. This is possible whenever we can find a function g such that

h ∘ g ∘ h = h

That is, g is similar to (but not exactly) an inverse of the homomorphism h. Gibbons’ proof shows that such a g always exists.

Unfortunately, Gibbons’ proof of the theorem does not tell us exactly how to construct the (f, ⊙, e), or g for that matter. We know it must exist, but not what it looks like. It also does not promise that the homomorphism is going to be as asymptotically efficient as any of the original folds. In particular, it would be nice if we could take a fold implementing Kadane’s algorithm and mechanically derive the solution to MSSP shown above. Still, this theorem can inspire us to look for a parallel implementation.

Indeed, the paper Automatic Inversion Generates Divide-and-Conquer Parallel Programs attacks the problem of obtaining g through a program synthesis technique based on the user providing the leftwards and rightwards definitions of h. The technique is at least effective enough to handle mssp. but there is still no guarantee that a correct g can be found. It is however guaranteed that if it is found, it is going to be efficient.

References