Control Flow Analysis

1
Control Flow Analysis

• Loops
• Reducibility
• Ordering Nodes of CFG
• Control Dependence

2
Loops

• A subflowgraph G ( N, E,s) is a loop with
  entry point s iff
• for each (n,m) in E and m in N, then either n
  in N or m s
• for each pair of nodes n, m in N, there are
  non-trivial paths from n to m and from m to n
• The first condition ensures a single entry point.
  The second condition ensures the graph is
  strongly connected and non-trivial.

3
Loop Entry Points
We can use the dominance relationship to identify
all the loops in a procedure

• A backward edge is one whose direction is
  inverse to the direction of dominance
• So it goes from a node to a dominator of that
  node
• A loop containing a backward edge to the node
  that is the entry point.
• In fact, this characterizes loop entry points

A node s in N is the entry point of a loop in G
iff there is a node n in N such that (n,s)
in E and s lt n (backward
edge)
4
Loop Entry Points

• Once we have the entry point, we can identify the
  nodes in the loop
• The maximum loop with entry point s is
• the subflowgraph G with initial node s that is
  generated by the set N
• N m in N there is a path from m to s
  that only contains nodes dominated by s

5
Algorithm to Find Loops

• First step
• compute the dominance relation for flowgraph.
• Second step
• find set of nodes s in flowgraph for which there
  is a node n such that
• s lt n in dominance relationship and
• (n,s) is an edge in flowgraph.

Each such node s is a loop entry point and (n,s)
is a backward edge
6
Algorithm to Find Loops

• Third (construction) step
• construct natural loop associated with s and
  (n,s). It is characterized by its nodes N N
• Enter entry point s and node n at the other end
  of the backward edge into N.
• Then include the predecessors of n.
• Continue by adding all the predecessors of nodes
  that are in the loop, except for the predecessors
  of the entry point, until there are no more such
  nodes.

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Loop Example
B1
1. StmtA 2. I 1 3. if ( I gt N) goto 7 4.
StmtB 5. I I 1 6. goto 3 7. StmtC
B1
B1 B2 B3 B4
B2
B2
B3
Graph has a backward edge (B3,B2) B2 is entry
point
Dominator tree
Flowgraph
B4
8
Limitations on Data Flow Analysis

• Many data flow methods work only with reducible
  graphs
• Informally, a reducible graph is one where we can
  partition the edges into a set of forward edges
  and a set of backward edges.

b doesnt dominate c, c doesnt dominate b
An Irreducible Graph
9
Reducibility

• In practice we can overcome this problem
• replace an irreducible graph by another graph in
  which some nodes have been duplicated
• node-splitting

a
c
b
c
10
Reducibility

• b 6
• if ( ) goto prev
• else goto next
• prev if ( a 0 ) goto next
• my .. a if ( ) return
• goto prev
• next if ( a 1 ) goto prev
• my a if ( ) return
• goto next

11
Reducibility
Node-splitting can help

• b 6
• if ( ) goto prev
• else goto next
• prev if ( a 0 ) goto othernext
• my .. a if ( ) return
  goto prev
• next if ( a 1 ) goto prev
• my a if ( ) return goto
  next
• othernext if ( a 1 ) goto prev
• my a if ( ) return goto
  othernext

12
Traversing Flowgraphs

• In order to deal with data flow problems we must
  generally process the nodes of a flowgraph,
  sometimes repeatedly.
• Sometimes the order in which we process nodes is
  important for correctness.
• Mostly it is important for efficiency
• the order in which nodes are visited may strongly
  affect the execution efficiency.
• We give an example of one useful order,
  rPostorder.
• It is not unique there is more than one
  rPostorder for a given flowgraph.

13
Depth-first Spanning Tree

• To construct rPostorder for a graph, we first
  compute a depth-first spanning tree for it.
• Depth-first spanning tree
• a partial subgraph of G containing all nodes of G
  that is also a tree
• obtained by eliminating edges of G
• Use this to visit descendants of a node before
  nodes that are not descendants (bottom-up)
• Or to visit nodes before any of their descendants
  (top-down)

• Eliminate the edge from 4 to 2 to obtain a tree.

14
Depth-first Spanning Tree
We select a subset of edges of G (N, E, s) to
form DFST
Initialization For each node m in N, set mark(m)
to false. Initialize E, set of edges in tree,
to empty set set i to N.

• Begin with s, the root mark it.
• Search for unmarked successors.
• If there are unmarked successors, then no edge in
  tree reaches these nodes.
• Select one and add the edge from the current node
  to this successor to the tree. Mark the selected
  successor.
• This is a depth-first search, so now apply the
  search procedure to the selected node.
• When there are no unmarked successors, number the
  current node with value of i and decrement i.

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start node
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blue ? marked
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edge added
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choose one of successors
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visited
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now choose a successor
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depth first, so go here next
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no successor. number node
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no successor. number node
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Visit successor. Number it.
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Number this one.
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Now visit successor of 2, number it.
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Node Orderings

• Different analyses require us to visit nodes in
  different orders
• Sometimes we need to visit nodes before any of
  the successors are visited
• pre-order
• Sometimes we visit all immediate successors of a
  node before visiting any other successors
• breadth-first search

27
Control Dependence

• If the execution of statement S1 determines
  whether or not statement S2 will be executed,
• then S2 is control dependent on S1.
• Example-
• if ( a b ) then c 5
• execution of the assignment c 5 depends on the
  outcome of the conditional, i.e. upon the result
  of evaluating ab

28
Control Dependence

• Control dependencies must be respected whenever
  we reorder executions, so optimizations must take
  them into account.
• In order to define and compute control
  dependencies, we must first create a single exit
  flowgraph.
• This has a unique end node, or sink.
• It enables us to specify a relationship between
  nodes and their predecessors.
• If necessary, we artificially create a unique
  exit node.

29
Single-Exit Flowgraph

• Definition-
• A single-exit flowgraph is a quadruple G
  (N,E,s,t) where (N,E,s) is a flowgraph, t in N is
  the exit node of G, and there is a path from
  every node of G to t.
• Note that the inverse flowgraph can be created by
  reversing the order of each edge.

30
Post-Dominators

• Let G ( N, E, s,t ) be a single-exit
  flowgraph, and let n, m be nodes.
• Definition n post-dominates m iff every path
  from m to t contains n
• Definition n directly post-dominates m iff n
  post-dominates m and there is no node n such
  that n post-dominates n and n post-dominates m.
• n post-dominates m. Note that n does not post
  dominate itself.

31
Post-Dominators

• Let G ( N, E, s,t ) be a single-exit
  flowgraph, and let n, p be nodes.
• Definition p is control dependent on n iff
• there is a non-trivial path from n to p such
  that every node n in the path (where n ! n,p)
  is post-dominated by p, and
• n is not post-dominated by p

p

• Condition 2 ensures that n has two successor
  nodes, and (at least) two distinct paths to the
  exit node, such that p is not in one of them.

32
1
2
There is a non-trivial path from 5 to 6. But 6
does not post-dominate 5. So 6 is
control-dependent on 5.
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a
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Post-Dominance
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There is a non-trivial path from 5 to 6. But 6
does not post-dominate 5.
a
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34
Control Dependence

• In order to find loops in a flowgraph, we compute
  the dominance tree.
• In order to compute control dependencies, we
  construct the post-dominator tree.
• It is relatively easy to impose an order on
  nodes so that we do not visit them randomly