Interval Partitioning of a Flow Graph

1
Interval Partitioning of a Flow Graph

• Input a flow graph G (N, E, n0).
• Output a partition of G into a set of disjoint
  intervals.
• Method for each n, compute I(n).
• I(n) n
• while there exists a node m ? n0
• all of whose predecessors are in I(n)
• do I(n) I(n) ? m
• The headers of the intervals are chosen as
  follows
• construct I(n0) and select all nodes in that
  interval,
• while there is a node m not yet selected
• but with a selected predecessor
• do construct I(m) and select all nodes in that
  interval.
• Note the order of selection does not affect
  final partitioning

2
Interval Partitioning Example

• Initially I(1) 1
• -1 is selected
• Pick node 2
• I(2) 2 then execute while loop
• I(2) 2 U 3 U 4 U 5
• 2, 3, 4, 5 are selected
• Note After 1, the only node we can pick is 2,
  because it is the only one with selected
  predecessor!

1
Natural loop
2
3
4
5

• Note node 2 dominates nodes 3, 4, 5.
• node 2 is the only entry of the interval
  (header).

3
Interval Graphs

• From the intervals of one flow graph G one may
  construct a new flow graph I(G) by the following
  rules
• Nodes The nodes of I(G) correspond to the
  intervals in the interval partition of G.
• Initial node The initial node of I(G) is the
  interval that contains the initial node of G.
• Edges There is an edge from interval I to
  interval J if and only if there is an edge from
  some node in I to the header of J.
• Note There could not be an edge entering some
  node n of J other than the header, because there
  would be no way n could have added to J in
  interval partitioning algorithm.

4
Interval Graphs (Cont)

• Limit flow graph of G Applying interval
  partitioning and interval graph construction in
  alternation, leads to a sequence of graphs G,
  I(G), I(I(G)) In(G) where the nodes of In(G)
  are in one interval. In(G) is the limit.

Property A flow graph is reducible if and only
if its limit flow graph is a single node
(historic definition).
5
T1-T2 Analysis

• Motivation A convenient way to achieve same
  effect as interval analysis.
• Definition Repeatedly apply two simple trans
  formations to flow graphs
• - T1 If n is a node with a loop, i.e., an edge
  n?n, delete that edge.
• - T2 If there is a node n ? n0 that has a
  unique predecessor, m, then m may consume n by
  deleting n and making all successors of n
  (including m, possibly) be successors of m.
• Facts
• - By applying (T1 T2 )k (G) until no further
  application is possible then a unique flow-graph
  results.
• - The flow graph (T1 T2 )k (G) is the limit
  flow-graph of G.

6
Example of T1 - T2 Analysis
T2
T1
T2
T2
a
a
a
ab
abcd
b
c
b
cd
b
cd
cb
d
7
Example of T1 - T2 Analysis
Regions a set of nodes N with a header
dominating all other nodes. Property While
reducing a flow graph with T1-T2 following holds
all the time
T2
T1
T2
T2
a
a
a
ab
abcd
b
c
b
cd
b
cd
cb
d
T2
T1
T2
T2

• A node represents a region of G.
• An edge from a to b represents a set of edges of
  G. Each is from some node in a to header of b.
• -Each node and edge of G is represented by
  exactly one node or edge of the current graph.

8
Optimizations of Basic Blocks

• Equivalent transformations Two basic block are
  equivalent if they compute the same set of
  expressions.
• -Expressions are the values of the live
  variables at the exit of the block.
• Two important classes of local transformations
• -structure preserving transformations
• common sub expression elimination
• dead code elimination
• renaming of temporary variables
• interchange of two independent adjacent
  statements.
• -algebraic transformations (countlessly many)
• simplify expressions
• replace expensive operations with cheaper ones.

9
The DAG Representation of Basic Blocks

• Directed acyclic graphs (DAGs) give a picture of
  how the value computed by each statement in the
  basic block is used in the subsequent statements
  of the block.
• Definition a dag for a basic block is a directed
  acyclic graph with the following labels on nodes
• leaves are labeled with either variable names or
  constants.
• they are unique identifiers
• from operators we determine whether l- or
  r-value.
• represent initial values of names. Subscript with
  0.
• interior nodes are labeled by an operator symbol.
• Nodes are also (optionally) given a sequence of
  identifiers for labels.
• - interior node ? computed values
• - identifiers in the sequence have that value.

10
Example of DAG Representation

1. t1 4i
2. t2 at1
3. t3 4i
4. t4 bt3
5. t5 t2 t4
6. t6 prod t5
7. prod t6
8. t7 i 1
9. i t7
10. if i lt 20 goto 1

t6, prod

t5

prod
t4
(1)
t2


lt
t1, t3
t7, i


a
b
20
4
i0
1

• Three address code Corresponding DAG
• Utility Constructing a dag from 3AS is a good
  way of determining
• common sub expressions (expressions computed more
  than once),
• which names are used inside the block but
  evaluated outside,
• which statements of the block could have their
  computed value used outside the block.

11
Constructing a DAG

• Input a basic block. Statements (i) x y op z
  (ii) x op y (iii) x y
• Output a dag for the basic block containing
• - a label for each node. For leaves an
  identifier - constants are permitted. For
  interior nodes an operator symbol.
• - for each node a (possibly empty) list of
  attached identifiers - constants not permitted.
• Method Initially assume there are no nodes, and
  node is undefined.
• If node(y) is undefined created a leaf labeled
  y, let node(y) be this node. In case(i) if
  node(z) is undefined create a leaf labeled z and
  that leaf be node(z).
• In case(i) determine if there is a node labeled
  op whose left child is node(y) and right child is
  node(z). If not create such a node, let be n.
  case(ii), (iii) similar.
• Delete x from the list attached to node(x).
  Append x to the list of identify for node n and
  set node(x) to n.