Introduction: We are now ready to discuss the speed of convergence of iterative algorithms. As discussed in Section 9.3.3, the maximum number of iterations the algorithm may take is the product of the height of the lattice and the number of nodes in the flow graph. For many data-flow analyses, it is possible to order the evaluation such that the algorithm converges in a much smaller number of iterations. The property of interest is whether all events of significance at a node will be propagated to that node along some acyclic path. Among the data-flow analyses discussed so far, reaching definitions, available expressions and live variables have this property, but constant propagation does not. More specifically:

• If a definition d is in IN[.£?], then there is some acyclic path from the block containing d to B such that d is in the iN's and OUT's all along that path.
• If an expression x y is not available at the entrance to block B, then there is some acyclic path that demonstrates that either the path is from the entry node and includes no statement that kills or generates x y, or the path is from a block that kills x y and along the path there is no subsequent generation of x y.
• If x is live on exit from block B, then there is an acyclic path from B to a use of x, along which there are no definitions of x.

We should check that in each of these cases, paths with cycles add nothing. For example, if a use of x is reached from the end of block B along a path with a cycle, we can eliminate that cycle to find a shorter path along which the use of x is still reached from B. In contrast, constant propagation does not have this property. Consider a simple program that has one loop containing a basic block with statements

L: a = b

b = c

c = 1

goto L

The first time the basic block is visited, c is found to have constant value 1, but both a and b are undefined. Visiting the basic block the second time, we find that b and c have constant values 1. It takes three visits of the basic block for the constant value 1 assigned to c to reach a.

If all useful information propagates along acyclic paths, we have an opportunity to tailor the order in which we visit nodes in iterative data-flow algorithms, so that after relatively few passes through the nodes we can be sure information has passed along all the acyclic paths.

Recall from Section 9.6.3 that if a ->• b is an edge, then the depth-first number of b is less than that of a only when the edge is a retreating edge. For forward data-flow problems, it is desirable to visit the nodes according to the depth-first ordering. Specifically, we modify the algorithm in Fig. 9.23(a) by replacing line (4), which visits the basic blocks in the flow graph with for (each block B other than ENTRY, in depth-first order) {

Example: Suppose we have a path along which a definition d propagates, such as

3 5 -)> 19 -» 35 -> 16 - 23 - 45 - 4 -> 10

where integers represent the depth-first numbers of the blocks along the path.

Then the first time through the loop of lines (4) through (6) in the algorithm in Fig. 9.23(a), d will propagate from OUT[3] to IN[5] to OUT[5], and so on, up to OUT[35]. It will not reach IN[16] on that round, because as 16 precedes 35, we had already computed IN[16] by the time d was put in OUT[35]. However, the next time we run through the loop of lines (4) through (6), when we compute IN[16], d will be included because it is in OUT[35]. Definition d will also propagate to OUT[16], IN[23], and so on, up to OUT[45], where it must wait because IN[4] was already computed on this round. On the third pass, d travels to IN[4], OUT[4], IN[10], OUT[10], and IN[17], so after three passes we establish that d reaches block 17.

It should not be hard to extract the general principle from this example. If we use depth-first order in Fig. 9.23(a), then the number of passes needed to propagate any reaching definition along any acyclic path is no more than one greater than the number of edges along that path that go from a higher numbered block to a lower numbered block. Those edges are exactly the retreating edges, so the number of passes needed is one plus the depth. Of course Algorithm 9.11 does not detect the fact that all definitions have reached wherever they can reach, until one more pass has yielded no changes. Therefore, the upper bound on the number of passes taken by that algorithm with depth-first block ordering is actually two plus the depth. A study1 0 has shown that typical flow graphs have an average depth around 2.75. Thus, the algorithm converges very quickly.

In the case of backward-flow problems, like live variables, we visit the nodes in the reverse of the depth-first order. Thus, we may propagate a use of a variable in block 17 backwards along the path

3 -> 5 19 ->• 35 16 23 -> 45 4 -> 10

in one pass to IN [4], where we must wait for the next pass in order to reach OUT[45]. On the second pass it reaches IN[16], and on the third pass it goes from OUT[35] to OUT[3].

In general, one-plus-the-depth passes suffice to carry the use of a variable backward, along any acyclic path. However, we must choose the reverse of depth-first order to visit the nodes in a pass, because then, uses propagate along any decreasing sequence in a single pass.

The bound described so far is an upper bound on all problems where cyclic paths add no information to the analysis. In special problems such as dominators, the algorithm converges even faster. In the case where the input flow graph is reducible, the correct set of dominators for each node is obtained in the first iteration of a data-flow algorithm that visits the nodes in depth-first ordering. If we do not know that the input is reducible ahead of time, it takes an extra iteration to determine that convergence has occurred.

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