package fix
Algorithmic building blocks for memoization, recursion, and more
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Dune Dependency
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doc/src/fix/DataFlow.ml.html
Source file DataFlow.ml
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(******************************************************************************) (* *) (* Fix *) (* *) (* François Pottier, Inria Paris *) (* *) (* Copyright Inria. All rights reserved. This file is distributed under the *) (* terms of the GNU Library General Public License version 2, with a *) (* special exception on linking, as described in the file LICENSE. *) (* *) (******************************************************************************) open Sigs (* Such a data flow analysis problem could also be solved by the generic least fixed point computation algorithm [Fix.Make.lfp]. However, such an approach would be less efficient, as (1) it would require reversing the graph first, so to have access to predecessors; (2) whenever a dirty node is examined, the contributions of all of its predecessors would be recomputed and joined, whereas the forward data flow analysis algorithm pushes information from a dirty node to its successors, thereby avoiding recomputation along edges whose source is not dirty; (3) the generic algorithm performs dynamic discovery of dependencies, whereas in this situation, all dependencies are explicitly provided by the user. *) (* We require a minimal semi-lattice, equipped with a [leq_join] operation, as opposed to a semi-lattice, which offers separate [leq] and [join] operations. Although [leq_join] is less powerful, it is sufficient for our purposes, and is potentially more efficient than the sequence of [leq] [join]. *) module Run (M : MINIMAL_IMPERATIVE_MAPS) (P : MINIMAL_SEMI_LATTICE) (G : DATA_FLOW_GRAPH with type variable = M.key and type property = P.property) = struct open P type variable = M.key (* A mapping of variables to properties. This mapping is initially empty. *) let properties = M.create() (* A set of dirty variables, whose outgoing transitions must be examined. *) (* The set of dirty variables is represented as a combination of a queue and a map of variables to Booleans. This map keeps track of which variables are in the queue and allows us to avoid inserting a variable into the queue when it is already in the queue. (In principle, a map of variables to [unit] should suffice, but our minimal map API does not offer a [remove] function. Thus, we have to use a map of variables to Booleans.) *) (* A FIFO queue is preferable to a LIFO stack. It leads to a breadth-first traversal. In the absence of cycles, in particular, this guarantees that a node is examined only after its predecessors have been examined. Compared to (say) a depth-first traversal, this allows more accurate properties to be computed in a single iteration, therefore reduces the number of iterations required in order to reach a fixed point. *) let pending : variable Queue.t = Queue.create() let dirty : bool M.t = M.create() let is_dirty (x : variable) = try M.find x dirty with Not_found -> false let schedule (x : variable) = if not (is_dirty x) then begin M.add x true dirty; Queue.add x pending end (* [update x' p'] ensures that the property associated with the variable [x'] is at least [p']. If this causes a change in the property at [x'], then [x'] is scheduled or rescheduled. *) let update (x' : variable) (p' : property) = match M.find x' properties with | exception Not_found -> (* [x'] is newly discovered. *) M.add x' p' properties; (* We assume that the transformation function [foreach_successor] maps the property [bottom] at the source to [bottom] at each successor. Thanks to this assumption, if the newly discovered property [p'] is [bottom] then there is really no need to schedule [x']. However, at present, we have no way of testing whether a property is bottom. Furthermore, [p'] is likely to be non-bottom in practice anyway. So, we do not exploit this assumption; we schedule [x'] always. *) schedule x' | p -> (* [x'] has been discovered earlier. *) let p'' = P.leq_join p' p in if p'' != p then begin (* The failure of the physical equality test [p'' == p] implies that [P.leq p' p] does not hold. Thus, [x'] is affected by this update and must itself be scheduled. *) M.add x' p'' properties; schedule x' end (* [examine] examines a variable that has just been taken out of the queue. Its outgoing transitions are inspected and its successors are updated. *) let examine (x : variable) = (* [x] is dirty, so a property must have been associated with it. *) let p = try M.find x properties with Not_found -> assert false in G.foreach_successor x p update (* Populate the queue with the root variables. *) (* Our use of [update] here means that it is permitted for [foreach_root] to seed several properties at a single root. *) let () = G.foreach_root update (* As long as the queue is nonempty, extract a variable and examine it. *) let () = try while true do let x = Queue.take pending in M.add x false dirty; examine x done with Queue.Empty -> () (* Expose the solution. *) type property = P.property option let solution x = try Some (M.find x properties) with Not_found -> None end module ForOrderedType (T : OrderedType) = Run(Glue.PersistentMapsToImperativeMaps(Map.Make(T))) module ForHashedType (T : HashedType) = Run(Glue.HashTablesAsImperativeMaps(T)) module ForType (T : TYPE) = ForHashedType(Glue.TrivialHashedType(T)) module ForIntSegment (K : sig val n: int end) = Run(Glue.ArraysAsImperativeMaps(K)) (* [ForCustomMaps] is a forward data flow analysis that is tuned for performance. *) module ForCustomMaps (P : MINIMAL_SEMI_LATTICE) (G : DATA_FLOW_GRAPH with type property := P.property) (V : ARRAY with type key := G.variable and type value := P.property) (B : ARRAY with type key := G.variable and type value := bool) : sig end = struct open P open G (* Compared to [Queue], [CompactQueue] is significantly faster and consumes less memory. *) let pending = CompactQueue.create () (* The queue stores a set of dirty variables, whose outgoing transitions must be examined. The map [B] records whether a variable is currently queued. *) (* We assume that the transformation function [foreach_successor] maps the property [bottom] at the source to [bottom] at each successor. In this code, contrary to [Run] above, this assumption *is* used. Indeed, we assume that the map [V] initially maps every variable to [bottom], and in [update], we mark a variable dirty (and insert it into the queue) only if its property has changed, so only if its new property is non-bottom. In other words, as long as a variable is mapped to [bottom], we do not examine its successors. This is acceptable only because we assume that [bottom] at the source of an edge translates to [bottom] at the destination of this edge. *) let schedule (x : variable) = if not (B.get x) then begin B.set x true; CompactQueue.add x pending end (* [update x' p'] ensures that the property associated with the variable [x'] is at least [p']. If this causes a change in the property at [x'], then [x'] is scheduled or rescheduled. *) let update (x' : variable) (p' : property) = let p = V.get x' in let p'' = P.leq_join p' p in if p'' != p then begin (* The failure of the physical equality test [p'' == p] implies that [P.leq p' p] does not hold. Thus, [x'] is affected by this update and must itself be scheduled. *) V.set x' p''; schedule x' end (* [examine] examines a variable that has just been taken out of the queue. Its outgoing transitions are inspected and its successors are updated. *) let examine (x : variable) = let p = V.get x in G.foreach_successor x p update (* Populate the queue with the root variables. *) (* Our use of [update] here means that it is permitted for [foreach_root] to seed several properties at a single root. *) let () = G.foreach_root update (* As long as the queue is nonempty, take a variable and examine it. *) let () = try while true do let x = CompactQueue.take pending in B.set x false; examine x done with CompactQueue.Empty -> () end
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