|Version 18 (modified by chak, 5 years ago) (diff)|
Preventing space blow-up due to replicate
The vectorisation transformation lifts scalar computations into vector space. In the course of this lifting, scalar constants are duplicated to fill an array, using the function 'replicateP'. Array computations are lifted in a similar manner, which leads to array constants being replicated to form arrays of arrays, which are represented as a segmented arrays. A simple example is our 'smvm' example code:
smvm :: [:[: (Int, Double) :]:] -> [:Double:] -> [:Double:] smvm m v = [: sumP [: x * (v !: i) | (i,x) <- row :] | row <- m :]
Here the variable 'v' is constant in the array comprehensions and will be replicated while lifting the expression v !: i. In other words, for every single element in a row, lifting implies the allocation of a separate copy of of the entire array v — and this only to perform a single indexing operation on that copy of v. More precisely, in the lifted code, lifted indexing (which we usually denote by (!:^) is applied to a nested array consisting of multiple copies of v; i.e., it is applied to the result of replicatePA (lengthPA row) v.
This is clearly wasted effort and space. However, the situation is even worse in Ben's pathological example:
treeLookup :: [:Int:] -> [:Int:] -> [:Int:] treeLookup table xx | lengthP xx == 1 = [: table !: (xx !: 0) :] | otherwise = let len = lengthP xx half = len `div` 2 s1 = sliceP 0 half xx s2 = sliceP half half xx in concatP (mapP (treeLookup table) [:s1, s2:])
Here table is constant in mapP (treeLookup table) [:s1, s2:]; hence, the entire table gets duplicated on each level of the recursion, leading to space consumption that is exponential in the depth of the recursion.
What's happening here?
Replication of scalars and arrays is always a waste of time and space. However, it is particularly problematic if the replicated structure is consumed by an indexing operation as it can change the asymptotic work complexity of the vectorised program. This holds not only for indexing, but for any operation that consumes only a small part of its input array(s). In other words, if a replicated structure is consumed in its entirety (for example by a fold), the asymptotic work complexity of replication matches that of consuming the structure. For operations that only consume a small part of their input, that is not the case. Hence, lifting, which introduces the replication, does increase asymptotic work.
Fixing the problem: Plan A (Repeated Segments)
Generally, we don't want to copy replicated data — it's a waste of space! We definitely don't want to do it for large data structures, and in particular, we don't want to do it for arrays. After all, that can change the asymptotic work complexity of a program. To keep it simple, we are for the moment focusing on avoiding replicating arrays, as this is were our practical problems are coming from at the moment. (Some cases of replicated scalars are also eliminated by fusion.)
NB: We will have to revisit replication of scalar structures as such scalar structures may be large trees.
A replicated array results is always represented by a segmented array; more precisely, by a segmented array where a contiguous sequence of segments contains the same data. For example, we have
replicatePA 3 [:1, 2, 3:] = [:[:1, 2, 3:], [:1, 2, 3:], [:1, 2, 3:]:] where [:[:1, 2, 3:], [:1, 2, 3:], [:1, 2, 3:]:] = ([:3, 3, 3:], [:1, 2, 3, 1, 2, 3, 1, 2, 3:])
expandPA [:2, 3:] [:[:1, 2:], [:3:]:] = [:[:1, 2:], [:1, 2:], [:3:], [:3:], [:3:]:] where [:[:1, 2:], [:1, 2:], [:3:], [:3:], [:3:]:] = ([:2, 2, 1, 1, 1:], [:1, 2, 1, 2, 3, 3, 3:])
NB: expandPA is lifted replication; expandPA is the name we used in HtM.
Collapse repeated segments
In practice, segments descriptors store more information than just the segment length. They at least additionally store the start position of each segment in the data array. In the conventional representation, an invariant is that the start positions are such that the segments don't overlap. To represent arrays with repeated segments more efficiently, we propose to relax that invariant. Specifically, all start positions of a contiguous sequence of repeated segments are the same; i.e., the segment data is stored only once per sequence of repeated segments.
Then, we have for [:[:1, 2, 3:], [:1, 2, 3:], [:1, 2, 3:]:],
start: [:0, 0, 0:] len: [:3, 3, 3:] data: [:1, 2, 3:])
and for [:[:1, 2:], [:1, 2:], [:3:], [:3:], [:3:]:],
start: [:0, 0, 2, 2, 2:] len: [:2, 2, 1, 1, 1:] data: [:1, 2, 3:])
This is merely a change in the array representation that does not affect vectorisation.
Operations on arrays with repeated segments
As multiple segments overlap in arrays with repeated segments, array consumers need to be adapted to work correctly in this situation.
In the smvm example, a replicated array is consumed by lifted indexing to extract matching elements of the vector for all non-zero elements of the matrix. Using just an length array as a segment descriptor without overlapping segments, lifted indexing might be implemented as follows:
(as_len, as_data) !:^ is = bpermutePA as_data ((prescanPA (+) 0 as_len) +^ is)
With overlapping segments, we have
(as_start, as_len, as_data) !:^ is = bpermutePA as_data (as_start +^ is)
In the case of smvm, where the first argument is produced by replicatePA (lengthPA row) v, we have as_start = replicatePA (lengthPA row) 0 and as-data = v. In other words, lifted indexing draws from a single copy of v, which is what we wanted.
TODO rl's exmaple
Multiple levels of nesting
TODO What if we have segment descriptors on top of one with repeated segments?
Fixing the problem: Plan B (Lifted Application)
The applications of replicatePA and expandPA that introduce segmented arrays with repeated segments in Plan A are in the definition of mapP, specifically replicatePA is used in mapP_S and expandPA in mapP_L (see Page 403 of HtM):
mapP_S :: (a :-> b) -> PA a :-> PA b mapP_S (Clo env _ fl) xss = fl (replicatePA (lengthPA xss) env) xss mapP_L :: PA (a :-> b) -> PA (PA a) -> PA (PA b) mapP_L (AClo env _ fl) xss = unconcatPA xss (fl (expandPA xss env) (concatPA xss))
In both cases, we replicate the environment of a closure before we apply the lifted version of the function represented by the closure. This is important as it guarantees that the consumer of these occurrences of replicatePA and expandPA process (topmost) segment structure in a homomorphic manner (after all, we are implementing a map function here)!
The idea behind Plan A is that replicatePA and expandPA produce a segmented array that encodes the replication without unnecessarily copying data and that the consumer —the lifted function fl— processes segmented arrays with encoded replication in a special manner.
- The work-efficient vectorisation paper by Prins et al. Their approach only works in a first-order language.
- Blelloch's work on the space-behaviour of data-parallel programs.
A plan to fix the problem
Generally, we don't want to copy replicated data — it's a waste of space! We definitely don't want to do it for large data structures, and in particular, we don't want to do it for arrays. After all, that can change the asymptotic work complexity of a program. So, instead of having replicateP allocate and fill a new array with multiple copies of the same data, we use a special array representation that stores the data (once!) together with the replication count. This is surely the most space efficient representation for a replicated array.
The downside of a special representation is that we now also need to modify all consumers of replicated arrays to accept this new representation and to handle it specially. This leads to some code blow up (each array consumer needs to be able to dynamically decide between two input array representations), and we need to be careful not to disturb fusion.
The trouble with indices
Although, a replicated array stores its replicated payload only once, it needs to be handled with care. When indexing into a replicated array, we still use the same indices as if the array data would have been copied multiple times. That can be a problem in examples, such as treeLookup above where the replicated array iteration-space grows exponentially — even 64bit indices will eventually overflow. However, we can circumvent that problem by taking care in code that consumes replicated arrays.
In the treeLookup example, the table is replicated and grows exponentially. But it is a segmented structure (one segment per copy of the original array) and it is accessed in the base case by a lifted index operation. When you look at the input to that application of lifted indexing, its first argument is huge (the replicated table), but the second argument contains the same data as the original value of xx, albeit segmented into an array with one segment per element. So we have effectively got
[:table, table, ...., table:] !^ [:[:xx_1:], [:xx_2:], ..., [:xx_n:]:]
Note how the xx_i are unchanged. It is only in the implementation of (!^) that the xx_i are blown up to index into the data vector of [:table, table, ...., table:] (which is concatP [:table, table, ...., table:]). It is that multiplication of the xx_i with the prescaned segment descriptor of [:table, table, ...., table:] that will overflow. Notice how that is internal to the implementation of (!^). If the first argument of (!^) is a replicated structure, there is no need whatsoever to perform that multiplication (and subsequent division) and the indices never overflow!
Never take the length of a replicated array
Unfortunately, not only indices blow out, the length of replicated arrays may also overflow 64bit integers. Hence, the consuming code must carefully avoid to take the length of such arrays. This is only the case for replicatePs introduced by the vectoriser. It is the responsibility of the DPH user to ensure that replicatePs that are explicit in the user code do not blow out. (We may want to switch to 64bit indices —at least on 64bit builds— anyway.)
Concrete implementation of replicated arrays
The DPH library is built on the vector package (that provides high-performance sequential arrays). This package heavily relies on a cheap representation of sliced arrays — i.e., arrays of which a subarray is extracted. Such array slices are not copied, but represented by a reference to the original array together with markers for the start and end of the slice.
Replicating and slicing
When implementing replicated arrays, we need to take into account that (1) a replicated may be a sliced vector and (b) that the partitioning of a parallel array across multiple threads requires to slice that parallel array.
* Is this really an issue? After all, we don't replicated thread-local slices of parallel arrays (which in turn may be sliced vectors), but replicate parallel arrays (which are already distributed over multiple threads).