MLRISC
MLRISC
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Basic Compiler Graphs


Basic Compiler Graphs
Introduction
-Dominator/Post-dominator Trees
-Control Dependence Graph
-Dominance Frontiers
-Iterated Dominance Frontiers
-Loop Nesting Tree
-Static Single Assignment
-IDEFS/IUSE sets

Introduction

In this section we describe the set of core compiler specific graphs and algorithms implemented in MLRISC. Mostly of these algorithms are parameterized with respect to the actual intermediate representation, and as such they do not provide many facilities that are provided by higher abstraction layers, such as in MLRISC IR, or in SSA.

Dominator/Post-dominator Trees

Dominance is a fundamental concept in compiler optimizations. Node A dominates B iff all paths from the start node to B intersects A. A dual notion is the concept of post-dominance: A post-dominates B iff all paths from B to the stop node intersects A. A (post-)dominator tree can be used to summarize the dominance/post-dominance relationship.

  functor DominatorTree
     (GraphImpl : GRAPH_IMPLEMENTATION) : DOMINATOR_TREE
 
The functor implements dominator analysis and creates a dominator/post-dominator tree from a graph G. A dominator tree is implemented as a graph with the following definition:
  signature DOMINATOR_TREE = sig
     exception Dominator
     datatype 'n dom_node =
        DOM of { node : 'n, level : int, preorder : int, postorder : int }
     type ('n,'e,'g) dom_info
     type ('n,'e,'g) dominator_tree = ('n dom_node,unit,('n,'e,'g) dom_info) graph
     type ('n,'e,'g) postdominator_tree = ('n dom_node,unit,('n,'e,'g) dom_info) graph
 
We annotated each node in a dominator tree with three extra fields of information, which is useful for other algorithms:
  • level is the nesting level of the tree. The root node has level 0, children of the root has level 1 and so on.
  • preorder is the preorder numbering of a node
  • preorder is the postorder numbering of a node.

To create a dominator tree and a postdominator tree from a graph, the following function should be called.

  val dominator_trees : ('n,'e,'g) graph ->
          ('n,'e,'g) dominator_tree * ('n,'e,'g) postdominator_tree
 
We use the algorithm of Tarjan and Lengauer, which runs in time O(|V+E|α(|V+E|)) where α is the functional inverse of the Ackermann function.

To perform many common queries on a dominator tree, we first call the function methods to obtain a method object.

 
   val methods : ('n,'e,'g) dominator_tree -> dominator_methods
 
The methods are packed into the following type:
    type dominator_methods =
          { dominates              : node_id * node_id -> bool,
            immediately_dominates  : node_id * node_id -> bool,
            strictly_dominates     : node_id * node_id -> bool,
            postdominates          : node_id * node_id -> bool,
            immediately_postdominates : node_id * node_id -> bool,
            strictly_postdominates : node_id * node_id -> bool,
            control_equivalent     : node_id * node_id -> bool,
            idom         : node_id -> node_id, $(* ~1 if none *)$
            idoms        : node_id -> node_id list,
            doms         : node_id -> node_id list,
            ipdom        : node_id -> node_id, $(* ~1 if none *)$
            ipdoms       : node_id -> node_id list,
            pdoms        : node_id -> node_id list,
            dom_lca      : node_id * node_id -> node_id,
            pdom_lca     : node_id * node_id -> node_id,
            dom_level    : node_id -> int,
            pdom_level   : node_id -> int,
            control_equivalent_partitions : unit -> node_id list list
          }
 
The query methods are as follows:
dominates(a,b)
returns true iff a dominates b
immediately_dominates(a,b)
returns true iff a immediately dominates b
strictly_dominates(a,b)
returns true iff a strictly dominates b
postdominates(a,b)
returns true iff a post-dominates b
immediately_postdominates(a,b)
returns true iff a immediately post-dominates b
strictly_postdominates(a,b)
returns true iff a strictly post-dominates b
control_equivalent(a,b)
returns true iff a dominates b and vice versa
idom(a)
returns the immediate dominator of a, or -1 if none exists
idoms(a)
returns all nodes that a immediately dominates
doms(a)
returns all nodes that a dominates (including a itself)
ipdom(a)
returns the immediate post-dominator of a, or -1 if none exists
ipdoms(a)
returns all nodes that a immediately post-dominates
pdoms(a)
returns all nodes that a post-dominates (including a itself)
dom_lca(a,b)
returns the least common ancestor of a and b in the dominator tree
pdom_lca(a,b)
returns the least common ancestor of a and b in the post-dominator tree
dom_level(a)
returns the nesting level of a in the dominator tree
pdom_level(b)
returns the nesting level of a in the post-dominator tree
control_equivalent_partitions
partitions the graph into a set of control equivalent nodes.
The methods dom_lca, pdom_lca and control_equivalent_partitions executes in O(n) time, where n is the size of the dominator tree. The other methods run in O(1) time.

Control Dependence Graph

Given two nodes A and B in a control flow graph G, we say that B is control dependent on A iff
  • B post-dominates a successor of A
  • B does not strictly post-dominates A
Intuitively, B is control dependent on A means that some path in the program that goes through A can by-passed B, and furthermore, A is the point in which this divergence can occur. Control dependence is used to various kinds of analysis and optimizations in a compiler, such as code motion and global scheduling[bernstein-rodeh].

To build a control dependence graph, the functor ControlDependenceGraph can be used:

  signature CONTROL_DEPENDENCE_GRAPH = sig
     type ('n,'e,'g) cdg = ('n,'e,'g) graph
 
     val control_dependence_graph :
           ('e -> bool) ->
           ('n,'e,'g) dominator_tree *
           ('n,'e,'g) postdominator_tree ->
           ('n,'e,'g) cdg
  end
  functor ControlDependenceGraph
     (structure Dom : DOMINATOR_TREE
      structure GraphImpl : GRAPH_IMPLEMENTATION
     ) : CONTROL_DEPENDENCE_GRAPH
 
The control depedence graph is a subcomponent of the program dependence graph commonly used in modern compiler optimizations.

Dominance Frontiers

Many algorithms involving the notion of control dependence or dominance can be rephrased in terms of dominance frontiers. A node A is in the dominance frontiers of B iff B dominates a predecessor of A but B does not strictly-dominate A. We denote this as A in DF(B). The dual notion of post-dominance frontiers can be defined analogously using the post-dominator tree\footnote{Control dependence can be defined in terms of post-dominance frontiers.}.

   functor DominanceFrontiers(Dom : DOMINATOR_TREE) : DOMINANCE_FRONTIERS
 
The functor DominanceFrontiers can be used to compute all the dominance frontiers of all the nodes in a graph. It has the following signature.

  signature DOMINANCE_FRONTIERS = sig
    structure Dom : DOMINATOR_TREE
    type dominance_frontiers = node_id list array
    val DFs : ('n,'e,'g) Dom.dominator_tree -> dominance_frontiers
  end
 

Iterated Dominance Frontiers

Iterated dominance frontiers (denoted as DF+) are defined as the least fixed point of iterating the operation DF. Formally, define the dominance frontiers on a set S as follows:
DF(S) = \UnionA in S DF(A)
Define iteration of DF, denoted as DFn, as follows:
DF1(S) = DF(S)
DF^{n+1}(S) = DF(S \union DFn(S))
The iterated dominance frontiers DF+(S) on a set S are defined as the limit:
DF+(S) = limn \to \infty DFn(S)
Iterated dominance frontiers of a set S can be computed in time O(|S|+|V|+|E|) using the algorithm by Sreedhar and Gao[linear-time-IDF]\footnote{ In practice it is often sub-linear in |V|+|E|.}.

   functor DJGraph(Dom : DOMINATOR_TREE) : DJ_GRAPH
 
The functor DJGraph implements this algorithm. It satisfies the signature below:
  signature DJ_GRAPH = sig
     structure Dom : DOMINATOR_TREE
     type ('n,'e,'g) dj_graph = ('n,'e,'g) Dom.dominator_tree
     val dj_graph : ('n,'e,'g) dj_graph ->
         {  DF   : node_id -> node_id list,
            IDF  : node_id -> node_id list,
            IDFs : node_id list -> node_id list
         }
  end
 
The function dj_graph takes a dominator tree and returns three query methods for computing dominance and iterated dominance frontiers. Method DF computes DF(v) for a single node v. Method IDF computes the DF+(v), and method IDFs computes DF+(S) when given a set of node ids. The dominator tree must not be updated while these operations are being performed.

Sreedhar's original algorithm is phrased in terms of the DJ-graph, which is a fusion of the dominator tree with its underlying flowgraph. Our variant operates on the dominator tree and the flowgraph at the same time, without building an intermediate data structure.

Iterated dominance frontiers are used in many algorithms that deal with the notion of dominance. For example, our SSA construction algorithm uses iterated dominance frontiers to identify confluent points in the program where phi-functions are to be placed.

Loop Nesting Tree

A natural loop L in a graph is a maximal strongly connected component such that all nodes in L are dominated by a single node h, called the loop header. Loops tend to form good optimization candidates and consequently loop detection is an essential task in a compiler. The functor
  functor structure.sml" target=code>LoopStructure 
   (structure GraphImpl : GRAPH_IMPLEMENTATION
    structure Dom       : DOMINATOR_TREE
   ) : LOOP_STRUCTURE 
 
recognizes all natural loops in a graph and built a loop nesting tree that describes the loop nesting relationship between graphs.

  signature structure.sig" target=code>LOOP_STRUCTURE = sig
    structure Dom : DOMINATOR_TREE
    datatype ('n,'e,'g) loop =
       LOOP of { nesting    : int,
                 header     : node_id,
                 loop_nodes : node_id list,
                 backedges  : 'e edge list,
                 exits      : 'e edge list
               }
 
    type ('n,'e,'g) loop_info
    type ('n,'e,'g) loop_structure = (('n,'e,'g) loop,unit, ('n,'e,'g) loop_info) graph
 
    val loop_structure : ('n,'e,'g) Dom.dominator_tree -> ('n,'e,'g) loop_structure
    val nesting_level : ('n,'e,'g) loop_structure -> node_id array
    val header        : ('n,'e,'g) loop_structure -> node_id array
  end
 
Our algorithm computes the loop nesting tree in time O((|V|+|E|)α(|V|+|E|)). Each node in this tree represents a loop in the flowgraph, except the root of the tree, which represents the entire graph. Given a flowgraph G, the root of the loop nesting tree is defined to be the sole vertex in #entry G. Other nodes in the tree are indexed by the loop header node ids.

Loop detection classifies each loop and for each loop L, the following information is obtained:

  • An integer nesting. The root of the tree has nesting depth 0. The top level loops have nesting depth 1, etc.
  • The node id of the loop header h.
  • A set of loop_nodes. Loop nodes are nodes that are in the strongly connected component L, but excluding the header h and all nodes that are part of any nested loops. Thus all nodes are uniquely partitioned in header nodes and loop nodes, and loop nodes are further partitioned into different sets according to which headers they are immediately nested under.
  • A set of backedges. A back-edge is an edge that targets the header h and originates from a loop node in L.
  • A set of loop exits. An exit-edge is an edge that originates from a loop node within L targets a node outside of L. Note that this set does not include any exit-edges contained in loops nested in L but target a node out of L.

Static Single Assignment

An SSA construction algorithm based on[SSA,Briggs-SSA,linear-time-IDF] is implemented in the following functor:
   functor StaticSingleAssignmentForm
      (Dom : DOMINATOR_TREE) : STATIC_SINGLE_ASSIGNMENT_FORM
 
SSA-based optimizations in MLRISC are actually implemented on top of a high-level SSA layer described in Section SSA Optimizations. So it is not necessary to use this module directly. Nevertheless, there can be situations in which this module can be specialized in other ways; for example, in the construction of sparse evaluation graphs.

  signature STATIC_SINGLE_ASSIGNMENT_FORM = sig
    structure Dom : DOMINATOR_TREE
    type var     = int 
    type phi  = var * var * var list $(* orig def/def/uses *)$
    type renamer = {defs : var list, uses: var list} ->
                   {defs : var list, uses: var list}
    type copy    = {dst : var list, src: var list} -> unit
 
    val compute_ssa : 
        ('n,'e,'g) Dom.dominator_tree ->
        { max_var      : var,  
          defs         : 'n node -> var list,
          is_live      : var * int -> bool,
          rename_var   : var -> var,
          rename_stmt  : {rename:renamer,copy:copy} -> 'n node -> unit,
          insert_phi   : {block    : 'n node,
                          in_edges : 'e edge list,
                          phis     : phi list 
                         } -> unit
        } -> unit
  end
 
This module defines the function compute_ssa, which constructs an SSA graph. It requires the following information from the client:
  • A dominator tree of the flowgraph.
  • max_var -- the maximum variable id (integer) that exists in the flowgraph. All variables are assumed to be indexed by non-negative integers.
  • defs(X) -- a function that returns defs(X), i.e.~the set of variable names defined in block X. If a minimal SSA form is desired, this set should include all the definitions in X. If a pruned SSA form is required, this set should include only the set of names that are live-out in X.
  • is_live(v,X) -- a function that determines if variable v is live-in into block X. If not, a φ-function will not be placed in X. For example, to compute the minimal-SSA form, this function should always return true.
  • rename_var(v) -- a function that returns a new unique name for variable v.
  • rename_stmt -- a function of type \sml{{rename:renamer,copy:copy} -> 'n node -> unit} where
        type renamer = {defs : var list, uses: var list} ->
                       {defs : var list, uses: var list}
        type copy    = {dst : var list, src: var list} -> unit
     
    Function rename_stmt is called for each block in the flowgraph in the order of the dominator tree, and is responsible for renaming all the variables in X by calling the functions renamer or copy. Function renamer renames all definitions and uses of a statement, while function copy renames of a set of parallel assignments
  • insert_phi(X,es,phis) -- a function that inserts a set of φ-definitions phis in block X, where es is the list of control flow edges that merge into block X.

IDEFS/IUSE sets

Reif and Tarjan define the following useful notions for computing approximate birth-points for expressions, which in turn can be used to drive other optimizations. Given a node X, let idom(X) denote the immediate dominator of X. Let def(X) (use(X)) denote all the definitions (uses) in X. Given a path p == v1... vn, define def(p) (use(p)) as
def(v1... vn) == \unioni in 1 ... n def(vi)
use(v1... vn) == \unioni in 1 ... n use(vi)
Let P(X) denotes all the paths from idom(X) to X that does not cross idom(X) internally. Then define idef(X) (iuse(X)) as:
idef(X) == \Unionidom(X) v1 ... vn X in P(X) def(v1... vn)
iuse(X) == \Unionidom(X) v1 ... vn X in P(X) use(v1... vn)
The sets ipostdef(X) and ipostuse(X) are defined analogously using the postdominator tree.

  signature IDEFS = sig
    type var = int
    val compute_idefs : 
        {def_use : 'n Graph.node -> var list * var list,
         cfg     : ('n,'e,'g) Graph.graph
        } ->
        { idefuse      : unit -> (RegSet.regset * RegSet.regset) Array.array,
          ipostdefuse  : unit -> (RegSet.regset * RegSet.regset) Array.array
        }
  end
  structure IDefs : IDEFS
 
Structure IDefs implements the function comput_idefs for computing the idef, iuse, ipostdef and ipostuse sets of a control flow graph. It takes as arguments a flowgraph and a function def_use, which takes a graph node and returns the def/use sets of the node. It returns two functions idefuse and ipostdefuse which compute the idef/iuse and ipostdef/ipostuse sets. These sets are returned as arrays indexed by node ids.
Lal George
Allen Leung
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Last modified: Thu Jan 9 19:38:15 EST 2003 by leunga@slinky