CS738: Advanced Compiler Optimizations. Flow Graph Theory. Amey Karkare

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CS738: Avance Compiler Optimizations Flow Graph Theory Amey Karkare karkare@cse.

1 CS738: Avance Compiler Optimizations Flow Graph Theory Amey Karkare Department of CSE, IIT Kanpur

2 Agena Speeing up DFA Depth of a flow graph Natural Loops

3 Acknowlegement Rest of the slies base on the material at w06.html

4 Speeing up DFA Proper orering of noes of a flow graph spees up the iterative algorithms: epth-first orering.

5 Speeing up DFA Proper orering of noes of a flow graph spees up the iterative algorithms: epth-first orering. Normal flow graphs have a surprising property reucibility that simplifies several matters.

6 Speeing up DFA Proper orering of noes of a flow graph spees up the iterative algorithms: epth-first orering. Normal flow graphs have a surprising property reucibility that simplifies several matters. Outcome: few iterations normally neee.

7 Depth-First Search Start at entry.

8 Depth-First Search Start at entry. If you can follow an ege to an unvisite noe, o so.

9 Depth-First Search Start at entry. If you can follow an ege to an unvisite noe, o so. If not, backtrack to your parent (noe from which you were visite).

10 Depth-First Spanning Tree (DFST) Root = Entry.

11 Depth-First Spanning Tree (DFST) Root = Entry. Tree eges are the eges along which we first visit the noe at the hea.

12 DFST Example

13 DFST Example

14 DFST Example

15 DFST Example

16 DFST Example

17 Depth-First Noe Orer The reverse of the orer in which a DFS retreats from the noes.

18 Depth-First Noe Orer The reverse of the orer in which a DFS retreats from the noes. Alternatively, reverse of postorer traversal of the tree.

19 DF Orer Example

20 Four Kin of Eges 1. Tree eges.

21 Four Kin of Eges 1. Tree eges. 2. Forwar eges: noe to proper escenant.

22 Four Kin of Eges 1. Tree eges. 2. Forwar eges: noe to proper escenant. 3. Retreating eges: noe to ancestor.

23 Four Kin of Eges 1. Tree eges. 2. Forwar eges: noe to proper escenant. 3. Retreating eges: noe to ancestor. 4. Cross eges: between two noe, niether of which is an ancestor of the other.

24 A Little Magic Of these eges, only retreating eges go from high to low in DF orer.

25 A Little Magic Of these eges, only retreating eges go from high to low in DF orer. Most surprising: all cross eges go right to left in the DFST.

26 A Little Magic Of these eges, only retreating eges go from high to low in DF orer. Most surprising: all cross eges go right to left in the DFST. Assuming we a chilren of any noe from the left.

27 Example: Non-Tree Eges

28 Example: Non-Tree Eges Retreating

29 Example: Non-Tree Eges Retreating 1 Forwar

30 Example: Non-Tree Eges Retreating 1 Forwar Cross

31 Roamap Normal flow graphs are reucible.

32 Roamap Normal flow graphs are reucible. Dominators neee to explain reucibility.

33 Roamap Normal flow graphs are reucible. Dominators neee to explain reucibility. In reucible flow graphs, loops are well efine, retreating eges are unique (an calle back eges).

34 Roamap Normal flow graphs are reucible. Dominators neee to explain reucibility. In reucible flow graphs, loops are well efine, retreating eges are unique (an calle back eges). Leas to relationship between DF orer an efficient iterative algorithm.

35 Dominators Noe ominates noe n if every path from the Entry to n goes through.

36 Dominators Noe ominates noe n if every path from the Entry to n goes through. [Exercise] A forwar-intersection iterative algorithm for fining ominators.

37 Dominators Noe ominates noe n if every path from the Entry to n goes through. [Exercise] A forwar-intersection iterative algorithm for fining ominators. Quick observations:

38 Dominators Noe ominates noe n if every path from the Entry to n goes through. [Exercise] A forwar-intersection iterative algorithm for fining ominators. Quick observations: Every noe ominates itself.

39 Dominators Noe ominates noe n if every path from the Entry to n goes through. [Exercise] A forwar-intersection iterative algorithm for fining ominators. Quick observations: Every noe ominates itself. The entry ominates every noe.

40 Example: Dominators Noe Dominators

41 Example: Dominators Noe Dominators

42 Example: Dominators Noe Dominators ,

43 Example: Dominators Noe Dominators , 2 3 1, 2,

44 Example: Dominators Noe Dominators , 2 3 1, 2, 3 4 1,

45 Example: Dominators Noe Dominators , 2 3 1, 2, 3 4 1, 4 5 1, 5 5 3

46 Common Dominator Cases The test of a while loop ominates all blocks in the loop boy.

47 Common Dominator Cases The test of a while loop ominates all blocks in the loop boy. The test of an if-then-else ominates all blocks in either branch.

48 Back Eges An ege is a back ege if its hea ominates its tail.

49 Back Eges An ege is a back ege if its hea ominates its tail. Theorem: Every back ege is a retreating ege in every DFST of every flow graph.

50 Back Eges An ege is a back ege if its hea ominates its tail. Theorem: Every back ege is a retreating ege in every DFST of every flow graph. Proof? Discuss/Exercise

51 Back Eges An ege is a back ege if its hea ominates its tail. Theorem: Every back ege is a retreating ege in every DFST of every flow graph. Proof? Discuss/Exercise Converse almost always true, but not always.

52 Example: Back Eges Noe Dominators , 2 3 1, 2, 3 4 1, 4 5 1, 5 5 3

53 Example: Back Eges Noe Dominators , 2 3 1, 2, 3 4 1, 4 5 1, 5 5 3

54 Reucible Flow Graphs A flow graph is reucible if every retreating ege in any DFST for that flow graph is a back ege.

55 Reucible Flow Graphs A flow graph is reucible if every retreating ege in any DFST for that flow graph is a back ege. Testing reucibility: Take any DFST for the flow graph, remove the back eges, an check that the result is acyclic.

56 Example: Remove Back Eges Noe Dominators , 2 3 1, 2, 3 4 1, 4 5 1, 5 5 3

57 Example: Remove Back Eges Noe Dominators , 2 3 1, 2, 3 4 1, 4 5 1, Remaining graph is acyclic.

58 Why Reucibility? Folk theorem: All flow graphs in practice are reucible. Fact: If you use only while-loops, for-loops, repeat-loops, if-then(-else), break, an continue, then your flow graph is reucible.

59 Example: Nonreucible Graph A B C

60 Example: Nonreucible Graph A B C In any DFST, one of these eges will be a retreating ege.

61 Example: Nonreucible Graph A A B C B C In any DFST, one of these eges will be a retreating ege.

62 Example: Nonreucible Graph A A A B C C B C B In any DFST, one of these eges will be a retreating ege.

63 Why Care About Back/Retreating Eges? Proper orering of noes uring iterative algorithm assures number of passes limite by the number of neste back eges.

64 Why Care About Back/Retreating Eges? Proper orering of noes uring iterative algorithm assures number of passes limite by the number of neste back eges. Depth of neste loops upper-bouns the number of neste back eges.

65 DF Orer an Retreating Eges Suppose that for a RD analysis, we visit noes uring each iteration in DF orer.

66 DF Orer an Retreating Eges Suppose that for a RD analysis, we visit noes uring each iteration in DF orer. The fact that a efinition reaches a block will propagate in one pass along any increasing sequence of blocks.

67 DF Orer an Retreating Eges Suppose that for a RD analysis, we visit noes uring each iteration in DF orer. The fact that a efinition reaches a block will propagate in one pass along any increasing sequence of blocks. When arrives along a retreating ege, it is too late to propagate from OUT to IN.

68 Example: DF Orer Noe 2 generates efinition

69 Example: DF Orer 1 Noe 2 generates efinition. Other noes empty w.r.t

70 Example: DF Orer 1 Noe 2 generates efinition. Other noes empty w.r.t.. Does reach noe 4?

71 Example: DF Orer 1 Noe 2 generates efinition. Other noes empty w.r.t.. Does reach noe 4?

72 Example: DF Orer 1 Noe 2 generates efinition. Other noes empty w.r.t.. Does reach noe 4?

73 Example: DF Orer 1 Noe 2 generates efinition. Other noes empty w.r.t.. Does reach noe 4?

74 Example: DF Orer 1 Noe 2 generates efinition. Other noes empty w.r.t.. Does reach noe 4?

75 Example: DF Orer 1 Noe 2 generates efinition. Other noes empty w.r.t.. Does reach noe 4?

76 Example: DF Orer 1 Noe 2 generates efinition. Other noes empty w.r.t.. Does reach noe 4?

77 Example: DF Orer 1 Noe 2 generates efinition. Other noes empty w.r.t.. Does reach noe 4?

78 Example: DF Orer Noe 2 generates efinition. Other noes empty w.r.t.. Does reach noe 4? 2 3

79 Example: DF Orer Noe 2 generates efinition. Other noes empty w.r.t.. Does reach noe 4? 2 3

80 Example: DF Orer Noe 2 generates efinition. Other noes empty w.r.t.. Does reach noe 4? 2 3

81 Depth of a Flow Graph The epth of a flow graph is the greatest number of retreating eges along any acyclic path.

82 Depth of a Flow Graph The epth of a flow graph is the greatest number of retreating eges along any acyclic path. For RD, if we use DF orer to visit noes, we converge in epth+2 passes.

83 Depth of a Flow Graph The epth of a flow graph is the greatest number of retreating eges along any acyclic path. For RD, if we use DF orer to visit noes, we converge in epth+2 passes. Depth+1 passes to follow that number of increasing segments.

84 Depth of a Flow Graph The epth of a flow graph is the greatest number of retreating eges along any acyclic path. For RD, if we use DF orer to visit noes, we converge in epth+2 passes. Depth+1 passes to follow that number of increasing segments. 1 more pass to realize we converge.

85 Example: Depth = 2

86 Example: Depth = 2 increasing

87 Example: Depth = 2 increasing retreating

88 Example: Depth = 2 retreating increasing increasing

89 Example: Depth = 2 retreating retreating increasing increasing

90 Example: Depth = 2 retreating retreating increasing increasing increasing

91 Similarly... AE also works in epth+2 passes.

92 Similarly... AE also works in epth+2 passes. Unavailability propagates along retreat-free noe sequences in one pass.

93 Similarly... AE also works in epth+2 passes. Unavailability propagates along retreat-free noe sequences in one pass. So oes LV if we use reverse of DF orer.

94 Similarly... AE also works in epth+2 passes. Unavailability propagates along retreat-free noe sequences in one pass. So oes LV if we use reverse of DF orer. A use propagates backwar along paths that o not use a retreating ege in one pass.

95 In General... The epth+2 boun works for any monotone bit-vector framework, as long as information only nees to propagate along acyclic paths. Example: if a efinition reaches a point, it oes so along an acyclic path.

96 Why Depth+2 is Goo? Normal control-flow constructs prouce reucible flow graphs with the number of back eges at most the nesting epth of loops. Nesting epth tens to be small.

97 Example: Neste Loops 3 neste while loops.

98 Example: Neste Loops 3 neste while loops. epth = 3.

99 Example: Neste Loops 3 neste while loops. 3 neste o-while loops. epth = 3.

100 Example: Neste Loops 3 neste while loops. epth = 3. 3 neste o-while loops. epth = 1.

101 Natural Loops The natural loop of a back ege a b is {b} plus the set of noes that can reach a without going through b.

102 Natural Loops The natural loop of a back ege a b is {b} plus the set of noes that can reach a without going through b. Theorem: two natural loops are either isjoint, ientical, or neste.

103 Natural Loops The natural loop of a back ege a b is {b} plus the set of noes that can reach a without going through b. Theorem: two natural loops are either isjoint, ientical, or neste. Proof: Discuss/Exercise

104 Example: Natural Loops

105 Example: Natural Loops Natural loop 3 2

106 Example: Natural Loops Natural loop 5 1 Natural loop 3 2

107 Reaing Assignment New Dragon Book (Aho, Lam, Sethi, Ullman) Chapter 9

Audiological Bulletin no. 31

Audiological Bulletin no. 31 Auiological Bulletin no. 31 The effect - an introuction News from Auiological Research an Communication 9 502 1043 001 / 05-07 Introuction Venting in earmouls has been use for many years to control the