Re: Question about the structure of a program dependence graph

On Tue, 31 May 2011 13:09:58 -0700 (PDT), Douglas do Couto Teixeira
<douglasdocouto@gmail.com> wrote:


> Given a program P in SSA form, let its dependence graph G = (V, E)
>be a graph with a vertex nv for each variable v in P, and an edge
>(na->nb) if b is defined by an instruction that uses a.
>
> If P is a general program with GOTOs, then it is possible to have a
>graph G that is dense, i.e., has O(N^2) edges, where N is the number
>of variables in P.


Not exactly. Recall that a GOTO has a single target label. There is
no more edge fan-out with GOTO than with a conditional or loop
construct.


> However, if P contains only IF and WHILE, then it seems that the
>number of edges in P will be O(N). Could you tell me if that is the
>case? Otherwise, could you provide me a counter example?


It will be between O(N) and O(N^2). Remember that multiple loops may
have the same entry or exit, multiple conditionals may converge, and
as Andreas already has mentioned, case/switch constructs have at least
as many edges as they have cases (remember the default case may not be
explicit and may not change the value).


Additionally, if the language includes pointers to non-local data,
then there will be edges associated with manipulations of both the
pointer and the value.


> If I add a break with a label, or exceptions, like in Java, then I
>am tempted to believe that the number of edges in the dependence graph
>is still O(N). Could you tell me if this assumption is wrong?


Break is the same as GOTO. For exceptions the answer depends on the
particular implementation.


George

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