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Chapter: Compilers - Principles, Techniques, & Tools : Machine-Independent Optimizations

Region-Based Analysis

1 Regions 2 Region Hierarchies for Reducible Flow Graphs 3 Overview of a Region-Based Analysis 4 Necessary Assumptions About Transfer Functions 5 An Algorithm for Region-Based Analysis 6 Handling Nonreducible Flow Graphs 7 Exercises for Section 9.7

Region-Based Analysis

 

1 Regions

2 Region Hierarchies for Reducible Flow Graphs

3 Overview of a Region-Based Analysis

4 Necessary Assumptions About Transfer Functions

5 An Algorithm for Region-Based Analysis

6 Handling Nonreducible Flow Graphs

7 Exercises for Section 9.7

 

T he iterative data-flow analysis algorithm we have discussed so far is just one approach to solving data-flow problems. Here we discuss another approach called region-based analysis. Recall that in the iterative-analysis approach, we create transfer functions for basic blocks, then find the fixedpoint solution by repeated passes over the blocks. Instead of creating transfer functions just for individual blocks, a region-based analysis finds transfer functions that summa-rize the execution of progressively larger regions of the program. Ultimately, transfer functions for entire procedures are constructed and then applied, to get the desired data-flow values directly.

 

While a data-flow framework using an iterative algorithm is specified by a semilattice of data-flow values and a family of transfer functions closed un-der composition, region-based analysis requires more elements. A region-based framework includes both a semilattice of data-flow values and a semilattice of transfer functions that must possess a meet operator, a composition oper-ator, and a closure operator. We shall see what all these elements entail in Section 9.7.4.

 

A region-based analysis is particularly useful for data-flow problems where paths that have cycles may change the data-flow values. The closure operator allows the effect of a loop to be summarized more effectively than does iterative analysis. The technique is also useful for interprocedural analysis, where trans-fer functions associated with a procedure call may be treated like the transfer functions associated with basic blocks.

 

For simplicity, we shall consider only forward data-flow problems in this section. We first illustrate how region-based analysis works by using the familiar example of reaching definitions. In Section 9.8 we show a more compelling use of this technique, when we study the analysis of induction variables.

 

1. Regions

 

In region-based analysis, a program is viewed as a hierarchy of regions, which are (roughly) portions of a flow graph that have only one point of entry. We should find this concept of viewing code as a hierarchy of regions intuitive, because a block-structured procedure is naturally organized as a hierarchy of regions. Each statement in a block-structured program is a region, as control flow can only enter at the beginning of a statement. Each level of statement nesting corresponds to a level in the region hierarchy.

 

Formally, a region of a flow graph is a collection of nodes N and edges E such that

 

 

1. There is a header h in N that dominates all the nodes in N.

 

2. If some node m can reach a node n in N without going through h, then m is also in N.

3. E is the set of all the control flow edges between nodes n1 and n2 in N, except (possibly) for some that enter h.

 

Example 9 . 5 0 : Clearly a natural loop is a region, but a region does notnecessarily have a back edge and need not contain any cycles. For example, in Fig. 9.47, nodes B1 and B2, together with the edge B1 —> B2: form a region; so do nodes Bi,B2, and B3 with edges Bx B2, B2 ->• B3, and Bi -> B3.

 However, the subgraph with nodes B2 and #3 with edge B2 —> B3 does not form a region, because control may enter the subgraph at both nodes B2 and B3. More precisely, neither B2 nor B3 dominates the other, so condition (1) for a region is violated. Even if we picked, say, B2 to be the "header," we would violate condition (2), since we can reach B3 from B1 without going through B2, and J5i is not in the "region."


 

2. Region Hierarchies for Reducible Flow Graphs

 

In what follows, we shall assume the flow graph is reducible. If occasionally we must deal with nonreducible flow graphs, then we can use a technique called "node splitting" that will be discussed in Section 9.7.6.

 

To construct a hierarchy of regions, we identify the natural loops. Recall from Section 9.6.6 that in a reducible flow graph, any two natural loops are either disjoint or one is nested within the other. The process of "parsing" a reducible flow graph into its hierarchy of loops begins with every block as a region by itself. We call these regions leaf regions. Then, we order the natural loops from the inside out, i.e., starting with the innermost loops. To process a loop, we replace the entire loop by a node in two steps:

 

 

1. First, the body of the loop L (all nodes and edges except the back edges to the header) is replaced by a node representing a region R. Edges to the header of L now enter the node for R. An edge from any exit of loop L is replaced by an edge from R to the same destination. However, if the edge is a back edge, then it becomes a loop on R. We call R a body region.

 

2. Next, we construct a region R' that represents the entire natural loop L. We call R' a loop region. The only difference between R and R' is that the latter includes the back edges to the header of loop L. Put another way, when R' replaces R in the flow graph, all we have to do is remove the edge from R to itself.

 

 

We proceed this way, reducing larger and larger loops to single nodes, first with a looping edge and then without. Since loops of a reducible flow graph are nested or disjoint, the loop region's node can represent all the nodes of the natural loop in the series of flow graphs that are constructed by this reduction process.

 

Eventually, all natural loops are reduced to single nodes. At that point, the flow graph may be reduced to a single node, or there may be several nodes remaining, with ho loops; i.e., the reduced flow graph is an acyclic graph of more than one node. In the former case we are done constructing the region hierarchy, while in the latter case, we construct one more body region for the entire flow graph.

 

 

E x a m p l e 9 . 5 1 : Consider the control flow graph in Fig. 9.48(a). There is one back edge in this flow graph, which leads from B4 to B2. The hierarchy of regions is shown in Fig. 9.48(b); the edges shown are the edges in the region flow graphs. There are altogether 8 regions:

 

            Regions Ri,... ,R& are leaf regions representing blocks B1 through B$, respectively. Every block is also an exit block in its region.

 

Body region RQ represents the body of the only loop in the flow graph; it consists  of regions  R2,R3, and R4 and three  interregion edges: B2     -»  B3, B2    -» BA,  and B3    -> J54 . It has two exit blocks, B3    and £?4 , since they both have outgoing edges not contained in the region. Figure 9.49(a) shows the flow graph with R$ reduced to a single node. Notice that although the edges R3 -> R5 and R4 -» -R5 have both been replaced by edge RQ -» #5, it is important to remember that the latter edge represents the two former edges, since we shall have to propagate transfer functions across this edge eventually, and we need to know that what comes out of both blocks B3 and B4 will reach the header of R$.

3. Loop region R7   represents the entire natural loop.  It includes one subregion, RQ, and one back edge B4    -> B2. It has also two exit nodes, again B3    and B4.  Figure 9.49(b) shows the flow graph after the entire natural loop is reduced to R7.      

4. Finally, body region R8    is the top region. It includes three regions, R±, R7,  R5    and three interregion edges, B1 -> B2,  B3    -» B5,  and  B4 B5. When we reduce the flow graph to R8, it becomes a single node. Since there are no back edges to its header, B±, there is no need for a final step reducing this body region to a loop region.



To summarize the process of decomposing reducible flow graphs hierarchically, we offer the following algorithm.

 

Algorithm 9 . 5 2 : Constructing a bottom-up order of regions of a reducible flow graph.

 

INPUT : A reducible flow graph G.

 

OUTPUT : A list of regions of G that can be used in region-based data-flow problems.

 

 

METHOD :

 

            Begin the list with all the leaf regions consisting of single blocks of G, in any order.

 

 

            Repeatedly choose a natural loop L such that if there are any natural loops contained within L, then these loops have had their body and loop regions added to the list already. Add first the region consisting of the

 

body of L (i.e., L without the back edges to the header of L), and then the loop region of L.

 

            If the entire flow graph is not itself a natural loop, add at the end of the list the region consisting of the entire flow graph.

 

3. Overview of a Region-Based Analysis

For each region R, and for each subregion R' within R, we compute a transfer function /RJNIJ?'] that summarizes the effect of executing all possible paths

 

Where "Reducible" Comes From

 

We now see why reducible flow graphs were given that name. While we shall not prove this fact, the definition of "reducible flow graph" used in this book, involving the back edges of the graph, is equivalent to several definitions in which we mechanically reduce the flow graph to a single node. The process of collapsing natural loops described in Section 9.7.2 is one of them. Another interesting definition is that the reducible flow graphs are all and only those graphs that can be reduced to a single node by the following two transformations:

 

 

T11    Remove an edge from a node to itself.

 

T2: If node n has a single predecessor m, and n is not the entry of the flow graph, combine m and n.

 

 

 

 

leading from the entry of R to the entry of R', while staying within R. We say that a block B within R is an exit block of region R if it has an outgoing edge to some block outside R. We also compute a transfer function for each exit block B of R, denoted / f t , o U T [ . B ] > that summarizes the effect of executing all possible paths within R, leading from the entry of R to the exit of B.

 

We then proceed up the region hierarchy, computing transfer functions for progressively larger regions. We begin with regions that are single blocks, where is,IN[13]  is J u s t fn e   identity function and  / B 5 0UT [ B ]  i s   the transfer function for  the block B itself.  As we move up the hierarchy,

 

•  If R  is  a body region,  then  the  edges belonging to  R form  an  acyclic graph on the subregions of R.  We may proceed to compute the transfer functions in a topological order of the subregions.

            If R is a loop region, then we only need to account for the effect of the back edges to the header of R.

 

Eventually, we reach the top of the hierarchy and compute the transfer functions for region Rn that is the entire flow graph. How we perform each of these computations will be seen in Algorithm 9.53.

 

The next step is to compute the data-flow values at the entry and exit of each block. We process the regions in the reverse order, starting with region Rn and working our way down the hierarchy. For each region, we compute the data-flow values at the entry. For region Rn, we apply /#n ) iN[.R] (IN[ENTRY]) to get the data-flow values at the entry of the subregions R in Rn. We repeat until we reach the basic blocks at the leaves of the region hierarchy.

 

4. Necessary Assumptions About Transfer Functions

 

In order for region-based analysis to work, we need to make certain assumptions about properties of the set of transfer functions in the framework. Specifically, we need three primitive operations on transfer functions: composition, meet and closure; only the first is required for data-flow frameworks that use the iterative algorithm.

 

 

 

Composition

 

The transfer function of a sequence of nodes can be derived by composing the functions representing the individual nodes. Let /i and f2 be transfer functions of nodes n1 and n2. The effect of executing n1 followed by n2 is represented by f2 o /]_. Function composition has been discussed in Section 9.2.2, and an example using reaching definitions was shown in Section 9.2.4. To review, let gerii and kilk be the gen and kill sets for Then:


respectively. The same idea works for any transfer function of the gen-kill form. Other transfer functions may also be closed, but we have to consider each case separately.

 

Meet

 

Here, the transfer functions themselves are values of a semilattice with a meet operator A/. The meet of two transfer functions f1 and f2, /i A/ f2, is defined by  (/i  A/ fi){x)  = fi(x)  A f2(x), where  A  is  the  meet  operator for  data-flow values. The meet operator on transfer functions is used to combine the effect of alternative paths of execution with the same end points. Where it is not am-biguous, from now on, we shall refer to the meet operator of transfer functions also as A. For the reaching-definitions framework, we have


 

That  is,  the  gen  and  kill  sets for  f1 A f2 are gen1 U gen2 and  killi  n  kill2, respectively. Again, the same argument applies to any set of gen-kill transfer functions.

 

Closure

 

If / represents the transfer function of a cycle, then fn represents the effect of going around the cycle n times. In the case where the number of iterations is not known, we have to assume that the loop may be executed 0 or more times. We represent the transfer function of such a loop by /*, the closure of /, which is defined by

 

 


Note that /° must be the identity transfer function, since it represents the effect of going zero times around the loop, i.e., starting at the entry and not moving. If we let / represent the identity transfer function, then we can write


and so on: any fn(x) is gen U (x — kill). That is, going around a loop doesn't affect the transfer function, if it is of the gen-kill form. Thus,


That is, the gen and kill sets for /* are gen and 0, respectively. Intuitively, since we might not go around a ioop at all, anything in x will reach the entry to the loop. In all subsequent iterations, the reaching definitions include those in the gen set.

 

5. An Algorithm for Region-Based Analysis

T he following algorithm solves  a forward data-flow-analysis problem on  a reducible flow graph, according to some framework that satisfies the assumptions of Section 9.7.4.  Recall that  / R J N ^ ' ]  and /j^ouTps] refer  to   transfer functions that transform data-flow values at the entry to region R into the correct value at the entry of subregion R' and the exit of the exit block B, respectively.

 

Algorithm 9 . 5 3 :  Region-based analysis.   

INPUT : A data-flow framework with the properties outlined in Section 9.7.4 and a reducible flow graph G.

OUTPUT : Data-flow values m[B] for each block B of G.

 

METHOD :

1. Use Algorithm 9.52 to construct the bottom-up sequence of regions of G, say Ri, R2,... ,Rn, where Rn is the topmost region.

2. Perform the bottom-up analysis to compute the transfer functions sum-marizing the effect of executing a region. For each region Ri, R2,... ,Rn, in the bottom-up order, do the following:

(a)  If R is a leaf region corresponding to block B, let /RJN[ . B] — a  n  d

/ H , OUT [ B ]  =    / B ,  the transfer function associated with block B.

 

            If R is a body region, perform the computation of Fig. 9.50(a).

 

            If R is a loop region, perform the computation of Fig. 9.50(b).

 

            Perform the top-down pass to find the data-flow values at the beginning of each region.

 

 

(a)        w[Rn]  =   IN[ENTRY] .

 

(b)  For each region R in  {Ri,... Rn-i},  in the top-down order,  compute

 

I N [ J  R ] - / ^ J  I N  [  J  R  ]  ( I N [ J  R ' ] ) ,

 

where R' is the immediate enclosing region of R.

 

Let us first look at the details of how the bottom-up analysis works. In line (1) of Fig. 9.50(a) we visit the subregions of a body region, in some topolog-ical order. Line (2) computes the transfer function representing all the possible paths from the header of R to the header of S; then in lines (3) and (4) we com-pute the transfer functions representing all the possible paths from the header of R to the exits of R — that is, to the exits of all blocks that have successors outside S. Notice that all the predecessors B' in R must be in regions that precede S in the topological order constructed at line (1). Thus, /#( OUT[jB'] will have been computed already, in line (4) of a previous iteration through the outer loop.

 

For loop regions, we perform the steps of lines (1) through (4) in Fig. 9.50(b) Line (2) computes the effect of going around the loop body region S zero or more times. Lines (3) and (4) compute the effect at the exits of the loop after one or more iterations.

 

In the top-down pass of the algorithm, step 3(a) first assigns the boundary condition to the input of the top-most region. Then if R is immediately con-tained in R', we can simply apply the transfer function /#'5 IN[JR] to the data-flow value w[R'] to compute m[R].

1)  for (each subregion S immediately contained in R, in

    topological order) {

2)  IR,W[S]  = Apredecessors B in R of the header of S /fl.OUTps] 5

  /* if S  is the header of region R, then / R J N [ S ] is the   meet over nothing, which is the identity function */

3)  for (each exit block B in S)

4)  /R,OVT[B]    -   / s , O U T [ B ]   0       / R , I N [ S ] ; }

 

(a)  Constructing transfer functions for a body region R

 

1) let S be the body region immediately nested within R] that is, S is R without back edges from R to the header of R;

 

2)  fR,W[S)  =  (Apredecessors B in R of the header of S /s,0UTp3])  5

 

3)  for (each exit block B in R)

 

4)  /fl,OUT[B] = /s ,OUTp3] ° /fl,IN[S];

 

(b)  Constructing transfer functions for a loop region R'

 

Figure 9.50:  Details of region-based data-flow computations

 

Example 9.54 : Let us apply Algorithm 9.53 to find reaching definitions in the flow graph in Fig. 9.48(a). Step 1 constructs the bottom-up order in which the regions are visited;  this order will be the numerical order  of their subscripts, R±,R2,...  ,  Rn.           

The values of the gen  and kill  sets for the five blocks are summarized below:


• To take the meet of transfer functions, take the union of the pen's and the intersection of the kiWs.

 

• To compose transfer functions, take the union of both the pen's and the kill's. However, as an exception, an expression that is generated by the first function, not generated by the second, but killed by the second is not in the gen of the result.

 

• To take the closure of a transfer function, retain its gen and replace the kill by 0.

 

 

The first five regions Ri,... , R5 are blocks B1,... , B5, respectively. For 1 < i < 5, /Ri ( iN[B.] * s *he identity function, and / / j i ( ouT [ Bi ] * s the transfer function for block Bi: 


Figure 9.51: Computing transfer functions for the flow graph in Fig. 9.48(a), using region-based analysis

 

The rest of the transfer functions constructed in Step 2 of Algorithm 9.50 are summarized in Fig. 9.51. Region Re, consisting of regions R2, R3, and R4,

represents the loop body and thus does not include the back edge B4 B2.  The order of processing these  regions will be  the only topological order:  R2,R3,R4.  First, R2    has no predecessors within R6; remember that the edge B4    -» B2 goes outside R6.  Thus, / R 6 ) T N [ B 2 ] i s  t  n  e   identity function,1 1    and /fl6 ,oUTp32 ] i s the transfer function for block B2    itself. 

The header of region  B3     has  one  predecessor within  R6:    namely  R2. The transfer function to its entry is simply the transfer function to the exit of B2, / f i 6 , O U T [ B 2 ] ' which has already been computed.  We compose this function with the transfer function of B3 within its own region to compute the transfer function to the exit of B3.

Last, for the transfer function to the entry of R4, we must compute

 


 

because both B2 and B3 are predecessors of B4, the header of R4. This transfer function is composed with the transfer function /i? 4 ,oUT [B 4 ] to get the desired function /ij6 ,0UT[B4]- Notice, for example, that d3 is not killed in this transfer function, because the path B2 -» B4 does not redefine variable a.

 

Now, consider loop region R7. It contains only one subregion RQ which represents its loop body. Since there is only one back edge, B4 —»• B2, to the header of RQ , the transfer function representing the execution of the loop body 0  or  more times  is just o u T [ s 4 ] :t  n  e 9 e  n s e t  is {d4,d^,do}  and  the kill set is 0.  There are two exits out of region R7, blocks B3    and B4.  Thus, this transfer function is composed with each of the transfer functions of RQ to get the corresponding transfer functions of R7. Notice, for instance, how d6 is in the gen  set for fn.7,B3 because of paths like B2 ->• B4     ->•  B2 -» B3, or even

B2 —> B3   —y B4 B2    —> B3.       

Finally, consider R8, the entire flow graph. Its subregions are Ri, R7, and R5, which we shall consider in that topological order. As before, the transfer function /fj8) iN[jBi] is simply the identity function, and the transfer function /.Rs.OUTpBi] is Just / R I , OUT [ B ! ] , which in turn is fBl.

 

The header of R7, which is B2, has only one predecessor, Bi, so the transfer function to its entry is simply the transfer function out of B1 in region R$. We compose / ^ 8 ) o u T [ B i ] with the transfer functions to the exits of B3 and B4 within R7 to obtain their corresponding transfer functions within R8. Lastly, we consider R5. Its header, B5, has two predecessors within R8,  namely B3 and B4.

Therefore, we compute fRs,0VT[B3) A  fR8,o1JT[B4]  to get /fl 8 ) iN [BB ] - SINCE THE  transfer function of block B5 is the identity function, / R 8 ) OUT [ B6 ] — fR8,lN[B5]-

 

Step 3 computes the actual reaching definitions from the transfer functions. In step 3(a), m[R8] = 0 since there are no reaching definitions at the beginning of the program. Figure 9.52 shows how step 3(b) computes the rest of the data-flow values.  The step starts with the subregions of R8. Since the transfer function from the start  of R8 to the  start of each of its subregion has been computed, a single application of the transfer function finds the data-flow value at the start each subregion. We repeat the steps until we get the data-flow values of the leaf regions, which are simply the individual basic blocks. Note that the data-flow values shown in Figure 9.52 are exactly what we would get had we applied iterative data-flow analysis to the same flow graph, as must be the case, of course.


 

6. Handling Nonreducible Flow Graphs

 

If nonreducible flow graphs are expected to be common for the programs to be processed by a compiler or other program-processing software, then we recom-mend using an iterative rather than a hierarchy-based approach to data-flow analysis. However, if we need only to be prepared for the occasional nonre-ducible flow graph, then the following "node-splitting " technique is adequate.

 

Construct regions from natural loops to the extent possible. If the flow graph is nonreducible, we shall find that the resulting graph of regions has cycles, but no back edges, so we cannot parse the graph any further. A typical situation is suggested in Fig. 9.53(a), which has the same structure as the nonreducible flow graph of Fig. 9.45, but the nodes in Fig. 9.53 may actually be complex regions, as suggested by the smaller nodes within.

 

We pick some region R that has more than one predecessor and is not the header of the entire flow graph. If i? has k predecessors, make k copies of the entire flow graph R, and connect each predecessor of i?'s header to a different copy of R. Remember that only the header of a region could possibly have a predecessor outside that region. It turns out, although we shall not prove it, that such node splitting results in a reduction by at least one in the number of regions, after new back edges are identified and their regions constructed. The resulting graph may still not be reducible, but by alternating a splitting phase with a phase where new natural loops are identified and collapsed to regions, we eventually are left with a single region; i.e., the flow graph has been reduced.


Example 9 . 5 5 :   The splitting  shown  in  Fig.  9.53(b)  has  turned  the  edge R2b -> R3 into a back edge, since f?3    now dominates R2b-  These two regions may thus be  combined into one. The  resulting three regions — Ri, R2a    and the new region — form an acyclic graph, and therefore may be combined into a single body region. We thus have reduced the entire flow graph to a single region. In general, additional splits may be necessary, and in the worst case, the total number of basic blocks could become exponential in the number of blocks in the original flow graph. •

 

 

We must also think about how the result of the data-flow analysis on the split flow graph relates to the answer we desire for the original flow graph. There are two approaches we might consider.

1.                    Splitting regions may be beneficial for the optimization process, and we can simply revise the flow graph to have copies of certain blocks. Since each duplicated block is entered along only a subset of the paths that reached the original, the data-flow values at these duplicated blocks will tend to contain more specific information than was available at the orig-inal. For instance, fewer definitions may reach each of the duplicated blocks that reach the original block.

2.                    If we wish to retain the original flow graph, with no splitting, then after analyzing the split flow graph, we look at each split block B, and its  corresponding set of blocks B1: B2,... ,Bk. We may compute m[B] — m[Bi] A IN[B2 ] A • • • A lN[Bf c ], and similarly for the OUT's.

 

 

7. Exercises for Section 9.7

 

Exercise 9 . 7 . 1 : For the flow graph of Fig. 9.10 (see the exercises for Section 9.1):

 

 

i. Find all the possible regions. You may, however, omit from the list the regions consisting of a single node and no edges.

 

ii.  Give the set of nested regions constructed by Algorithm 9.52.

 

Hi. Give a T1-T2 reduction of the flow graph as described in the box on "Where 'Reducible' Comes From" in Section 9.7.2.

 

Exercise 9 . 7 . 2 :  Repeat Exercise 9.7.1 on the following flow graphs:

 

a)          Fig. 9.3.

 

b)         Fig. 8.9.

 

c)          Your flow graph from Exercise 8.4.1.

 

d)         Your flow graph from Exercise 8.4.2.

 

Exercise 9 . 7 . 3 :  Prove that every natural loop is a region.

 

!! Exercise 9 . 7 . 4: Show that a flow graph is reducible if and only it can be transformed to a single node using:

 

a)  The operations Ti and T2    described in the box in Section 9.7.2.

 

b)  The region definition introduced in Section 9.7.2.

 

! Exercise 9 . 7 . 5 : Show that when you apply node splitting to a nonreducible flow graph, and then perform T1-T2 reduction on the resulting split graph, you wind up with strictly fewer nodes than you started with.

 

! Exercise 9 . 7 . 6 : What happens if you apply node-splitting and T1-T2 reduc-tion alternately, to reduce a complete directed graph of n nodes?


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