Constructing%20the%20Natural%20Loop%20of%20a%20Back%20Edge

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Constructing the Natural Loop of a Back Edge

• Input a flow graph G and a back edge ngd.
• Output the set loop consisting of all nodes in
  the natural loop ngd.
• procedure insert(m)
• if m is not in loop then -- check if
  visited
• loop loop U m -- mark as visited
• push m onto stack -- push for
  predecessor search
• / main program /
• stack empty
• loop d -- mark d as visited
• insert(n) -- first node, mark visited
• while stack is not empty do
• pop m, the first element of stack, off
  stack -- active node
• for each predecessor p of m do insert(p) --
  mark and push

2
Take 8g3 what is the natural loop?

• 1

• Step1
• loop 3 stack empty
• Step 2
• loop 3, 8 stack 8
• Step 3
• loop 3, 8, 7 stack 7
• Step 4
• loop 3, 8, 7, 5, 6, 10 stack 5, 6, 10
• Step 5
• loop 3, 8, 7, 5, 6, 10, 4 stack 4, 6,
  10
• Step 6
• loop 3, 8, 7, 5, 6, 10, 4 stack 4, 10
• Step 7
• loop 3, 8, 7, 5, 6, 10, 4 stack 8
• Step 8
• loop 3, 8, 7, 5, 6, 10, 4 stack
• Hence loop 3, 4, 5, 6, 7, 8, 10

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10
3
Finding Dominators

• Main Idea If p1, ..p2 are all the predecessors
  of node n and
• d ¹ n then d dom n if and only if d dom pi for
  each i.
• Input Flow graph G (N, E, no)
• Output The relation dom.
• Method Compute iteratively D(n), the set of
  dominators of n.
• D(no) no
• for n ÎN \ nodo D(n) N / end of
  initialization /
• while changes to any D(n) occur do
• for n ÎN \ nodo
• D(n) nU Ç D(p)

p Î pred(n)
Notes This is an example of Data-Flow Analysis
algorithm (later) -convergence D(n) cannot get
smaller indefinitely.
4
Inner Loops and Pre-Headers

• Inner-loop A loop that contains no other loops
• Property of natural loops unless two loops have
  the same header
• - they are either disjoint or
• - one is entirely contained (nested within) the
  other
• - when two loops have the same header but
  neither is nested within each other, they are
  treated as a single loop
• Preheader Necessary if we want to move
  statements outside.
• - successor The header
• - predecessors all edges which formerly entered
  the header of L from outside L.

Bo
preheader
header
B1
header
B2
B3
loop L
loop L
Same header loops
5
Reducible Flow Graphs

• Intuition Produced by exclusive use of flow of
  control statements such as if-then-else,
  while-do, continue, break etc.
• Statistics Even programs written using goto
  statements are almost always reducible.
• Main property there are no jumps in the middle
  of loops from outside the only entry to a loop
  is through its header.
• Definition A flow graph G is reducible if and
  only if we can partition the edges into two
  disjoint groups, often called the forward edges
  and back edges with properties
• - forward edges from an acyclic graph in which
  every node can be reached from the initial node
  of G.
• - the back edges consist only of edges whose
  heads dominate their tails.

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Example of Flow Graphs
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Example of Reducible Flow Graphs

• Knowing dom we can find and remove all the back
  edges.
• We get the forward edges.
• Check if they form an acyclic graph

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Example Reducible Flow Graphs
Consider the following flow graph with initial
node 1.

• Intuition This flow graph is not reducible
  because the cycle 2 - 3 can be entered at two
  different places, 2 and 3.
• Key property of reducible flow graphs The set of
  nodes informally regarded as a loop must contain
  a back edge.
• This flow graph has no back edges
• 2 dom 3 and 3 dom 2
• The loop 2 - 3 has two headers.

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Strongly Structured Reducible Flow Graphs

• Key Property Single entry point and single end
  point
• Generated from Structured Programs (grammar)
• S id E S S if E then S else S do S
  while E
• E id id id
• Result They can be analyzed as Abstract Syntax
  Trees (AST)

S1
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Weakly Structured Reducible Flow Graphs
Example i m 1 j n a u1 do
i i 1 j j 1 if e3 then (d6)
a u2 else (d7) i u3
break while e2 Problem Jump at the
end of the loop (break) Conclusion AST
techniques are not naturally handled.
If-then-else DOES NOT have a single end point.

S1
do

e2
S2
if
e1
d6

break
d7
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Intervals of (Reducible) Flow Graphs

• Motivation Apply syntax-directed techniques in
  a more general setting where the syntax does not
  necessarily provide the structure but the flow
  graph does.
• Intuition Divide flow graphs into intervals that
  put a hierarchical structure on the flow graph.
• Intuitive Def A natural loop acyclic dangling
  structure.
• Property The header dominated all nodes in the
  interval.
• Formal Def Given a flow graph G (N, E, n0) and
  a node n ? N, the interval with header n, I(n) is
  defined by
• n is in I(n)
• If predecessors(m) Í I(n) for m ¹ n0 then m ?
  I(n)
• Nothing else is in I(n).