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Dominators in a Flowgraph Flowgraph: G = ( V, E, r ); each v in V is - PowerPoint PPT Presentation

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Dominators in a Flowgraph Flowgraph: G = ( V, E, r ); each v in V is reachable from r v dominates w if every path from r to w includes v Application areas : Program optimization, VLSI testing, theoretical biology, distributed systems,

  1. The Lengauer-Tarjan Algorithm: Semidominators • For any w  r , idom(w) *  sdom(w) +  w . • sdom(w) = min ( { v | (v, w)  E and v < w }  { sdom(u) | u > w and  (v, w)  E such that u *  v } ). sdom(u) sdom(u) nca(w,v) = v nca(w,v) = w nca(w,v) w u w u v v 39

  2. The Lengauer-Tarjan Algorithm: Semidominators • For any w  r , idom(w) *  sdom(w) +  w . • sdom(w) = min ( { v | (v, w)  E and v < w }  { sdom(u) | u > w and  (v, w)  E such that u *  v } ). • Let w  r and let u be any vertex with min sdom(u) that satisfies sdom(w) +  u *  w . Then idom(w) = idom(u) . sdom(u) sdom(w) idom(w) = idom(u) u w 40

  3. The Lengauer-Tarjan Algorithm: Semidominators • For any w  r , idom(w) *  sdom(w) +  w . • sdom(w) = min ( { v | (v, w)  E and v < w }  { sdom(u) | u > w and  (v, w)  E such that u *  v } ). • Let w  r and let u be any vertex with min sdom(u) that satisfies sdom(w) +  u *  w . Then idom(w) = idom(u) . Moreover, if sdom(u) = sdom(w) then idom(w) = sdom(w) . sdom(u) = sdom(w) idom(w) = sdom(w) u w 41

  4. The Lengauer-Tarjan Algorithm Overview of the Algorithm 1. Carry out a DFS. 2. Process the vertices in reverse preorder. For vertex w , compute sdom(w) . 3. Implicitly define idom(w) . 4. Explicitly define idom(w) by a preorder pass. 42

  5. Evaluating minima on tree paths sdom(u) sdom(u) nca(w,v) = v nca(w,v) nca(w,v) = w u u w w v v If we process vertices in reverse preorder then the sdom values we need are known. 43

  6. Evaluating minima on tree paths Data Structure: Maintain forest F and supports the operations: link( v , w ): Add the edge (v,w) to F . eval( v ): Let r be the root of the tree that contains v in F . If v = r then return v . Otherwise return any vertex with minimum sdom among the vertices u that satisfy r +  u *  v . Initially every vertex in V is a root in F . 44

  7. The Lengauer-Tarjan Algorithm dfs( r ) for all w  V in reverse preorder do for all v  pred(w) do u  eval( v ) if semi(u) < semi(w) then semi(w)  semi(u) done add w to the bucket of semi(w) link( parent(w) , w ) for all v in the bucket of parent(w) do delete v from the bucket of parent(w) u  eval( v ) if semi(u) < semi(v) then dom(v)  u else dom(v)  parent(w) done done for all w  V in reverse preorder do if dom(w)  semi(w) then dom(w)  dom(dom(w)) 45 done

  8. The Lengauer-Tarjan Algorithm: Example r 1 c 2 b 8 e g 7 3 9 f a 11 10 4 h d 12 j 13 l i 5 k 6 46

  9. The Lengauer-Tarjan Algorithm: Example eval( 12 ) = 12 r 1 c 2 b 8 e g 7 3 9 f a 11 10 4 h d 12 j 13 l i 5 [12] k 6 47

  10. The Lengauer-Tarjan Algorithm: Example add 13 to bucket(12) r 1 link( 13 ) c 2 b 8 e g 7 3 9 f a 11 13 10 4 h d 12 j 13 l i 5 [12] k 6 48

  11. The Lengauer-Tarjan Algorithm: Example delete 13 from bucket(12) r 1 eval(13) = 13 c 2 b 8 e g 7 3 9 f a 11 10 4 h d 12 j 13 l i dom(13)=12 5 [12] k 6 49

  12. The Lengauer-Tarjan Algorithm: Example eval(11) = 11 r 1 c 2 b 8 e g 7 3 9 f a 11 10 4 h d 12 j [11] 13 l i dom(13)=12 5 [12] k 6 50

  13. The Lengauer-Tarjan Algorithm: Example eval(8) = 8 r 1 c 2 b 8 e g 7 3 9 f a 11 10 4 h d 12 j [8] 13 l i dom(13)=12 5 [12] k 6 51

  14. The Lengauer-Tarjan Algorithm: Example add 12 to bucket(8) r 1 link(12) 12 c 2 b 8 e g 7 3 9 f a 11 10 4 h d 12 j [8] 13 l i dom(13)=12 5 [12] k 6 52

  15. The Lengauer-Tarjan Algorithm: Example eval(8)=8 r 1 12 c 2 b 8 e g 7 3 9 f a 11 [8] 10 4 h d 12 j [8] 13 l i dom(13)=12 5 [12] k 6 53

  16. The Lengauer-Tarjan Algorithm: Example eval(1)=1 r 1 12 c 2 b 8 e g 7 3 9 f a 11 [1] 10 4 h d 12 j [8] 13 l i dom(13)=12 5 [12] k 6 54

  17. The Lengauer-Tarjan Algorithm: Example 11 add 11 to bucket(1) r 1 link(11) 12 c 2 b 8 e g 7 3 9 f a 11 [1] 10 4 h d 12 j [8] 13 l i dom(13)=12 5 [12] k 6 55

  18. The Lengauer-Tarjan Algorithm: Example 11 delete 12 from bucket(8) r 1 eval(12) = 11 c 2 b 8 e g 7 3 9 f a 11 [1] 10 4 h d 12 j dom(12)=11 [8] 13 l i dom(13)=12 5 [12] k 6 56

  19. The Lengauer-Tarjan Algorithm: Example 11 eval(13) = 11 r 1 c 2 b 8 e g 7 3 9 f a 11 [1] 10 4 h d 12 j dom(12)=11 [1] [8] 13 l i dom(13)=12 5 [12] k 6 57

  20. The Lengauer-Tarjan Algorithm: Example 11 10 9 8 add 8 to bucket(1) r 1 link(8) c 2 b 8 [1] e g 7 3 9 f a 11 [1] [1] 10 4 h d 12 j dom(12)=11 [1] [8] 13 l i dom(13)=12 5 [12] k 6 58

  21. The Lengauer-Tarjan Algorithm: Example 10 9 8 delete 11 from bucket(1) r 1 eval(11) = 11 c 2 b 8 [1] e g 7 3 9 f a 11 dom(11)=1 [1] [1] 10 4 h d 12 j dom(12)=11 [1] [8] 13 l i dom(13)=12 5 [12] k 6 59

  22. The Lengauer-Tarjan Algorithm: Example 9 8 delete 10 from bucket(1) r 1 eval(10) = 10 c 2 b 8 [1] e g 7 3 9 f a 11 dom(11)=1 [1] [1] 10 4 dom(10)=1 h d 12 j dom(12)=11 [1] [8] 13 l i dom(13)=12 5 [12] k 6 60

  23. The Lengauer-Tarjan Algorithm: Example 8 delete 9 from bucket(1) r 1 eval(9) = 9 c 2 b 8 [1] dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [1] [1] 10 4 dom(10)=1 h d 12 j dom(12)=11 [1] [8] 13 l i dom(13)=12 5 [12] k 6 61

  24. The Lengauer-Tarjan Algorithm: Example delete 8 from bucket(1) r 1 eval(8) = 8 c 2 b 8 dom(8)=1 [1] dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [1] [1] 10 4 dom(10)=1 h d 12 j dom(12)=11 [1] [8] 13 l i dom(13)=12 5 [12] k 6 62

  25. The Lengauer-Tarjan Algorithm: Example 6 eval(6) = 6 r 1 c 2 b 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [2] [1] [1] 10 4 dom(10)=1 h d 12 j dom(12)=11 [1] [8] 13 l i dom(13)=12 5 [12] [1] k 6 63 [1]

  26. The Lengauer-Tarjan Algorithm: Example 6 5 add 5 to bucket(1) r 1 link(5) c 2 b 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [2] [1] [1] 10 4 dom(10)=1 h d 12 j dom(12)=11 [1] [8] 13 l i dom(13)=12 5 [12] [1] k 6 64 [1]

  27. The Lengauer-Tarjan Algorithm: Example 6 5 eval(3) = 3 r 1 c 2 b 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [2] [1] [1] 10 4 dom(10)=1 h d 12 j dom(12)=11 [1] [3] [8] 13 l i dom(13)=12 5 [12] [1] k 6 65 [1]

  28. The Lengauer-Tarjan Algorithm: Example 6 5 add 4 to bucket(3) r 1 link(4) c 2 b 8 dom(8)=1 [1] 4 dom(7)=2 dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [2] [1] [1] 10 4 dom(10)=1 h d 12 j dom(12)=11 [1] [3] [8] 13 l i dom(13)=12 5 [12] [1] k 6 66 [1]

  29. The Lengauer-Tarjan Algorithm: Example 6 5 delete 4 from bucket(3) r 1 eval(4) = 4 c 2 b 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [2] [1] [1] 10 4 dom(10)=1 h d 12 j dom(4)=3 dom(12)=11 [1] [3] [8] 13 l i dom(13)=12 5 [12] [1] k 6 67 [1]

  30. The Lengauer-Tarjan Algorithm: Example 6 5 eval(2) = 2 r 1 c 2 b 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [2] [1] [2] [1] 10 4 dom(10)=1 h d 12 j dom(4)=3 dom(12)=11 [1] [3] [8] 13 l i dom(13)=12 5 [12] [1] k 6 68 [1]

  31. The Lengauer-Tarjan Algorithm: Example 6 5 eval(5) = 5 r 1 c 2 b 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [2] [1] [1] [1] 10 4 dom(10)=1 h d 12 j dom(4)=3 dom(12)=11 [1] [3] [8] 13 l i dom(13)=12 5 [12] [1] k 6 69 [1]

  32. The Lengauer-Tarjan Algorithm: Example 6 5 3 add 3 to bucket(1) r 1 link(3) c 2 b 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [2] [1] [1] [1] 10 4 dom(10)=1 h d 12 j dom(4)=3 dom(12)=11 [1] [3] [8] 13 l i dom(13)=12 5 [12] [1] k 6 70 [1]

  33. The Lengauer-Tarjan Algorithm: Example 6 5 3 eval(1) = 1 r 1 c 2 [1] b 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [2] [1] [1] [1] 10 4 dom(10)=1 h d 12 j dom(4)=3 dom(12)=11 [1] [3] [8] 13 l i dom(13)=12 5 [12] [1] k 6 71 [1]

  34. The Lengauer-Tarjan Algorithm: Example 6 5 3 2 add 2 to bucket(1) r 1 link(2) c 2 [1] b 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [2] [1] [1] [1] 10 4 dom(10)=1 h d 12 j dom(4)=3 dom(12)=11 [1] [3] [8] 13 l i dom(13)=12 5 [12] [1] k 6 72 [1]

  35. The Lengauer-Tarjan Algorithm: Example 5 3 2 delete 6 from bucket(1) r 1 eval(6) = 6 c 2 [1] b 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [2] [1] [1] [1] 10 4 dom(10)=1 h d 12 j dom(4)=3 dom(12)=11 [1] [3] [8] 13 l i dom(13)=12 5 [12] [1] dom(6)=1 k 6 73 [1]

  36. The Lengauer-Tarjan Algorithm: Example 3 2 delete 5 from bucket(1) r 1 eval(5) = 5 c 2 [1] b 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f a 11 dom(11)=1 [2] [1] [1] [1] 10 4 dom(10)=1 h d 12 j dom(4)=3 dom(12)=11 [1] [3] [8] 13 l i dom(13)=12 5 dom(5)=1 [12] [1] dom(6)=1 k 6 74 [1]

  37. The Lengauer-Tarjan Algorithm: Example 2 delete 3 from bucket(1) r 1 eval(3) = 3 c 2 [1] b 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f dom(3)=1 a 11 dom(11)=1 [2] [1] [1] [1] 10 4 dom(10)=1 h d 12 j dom(4)=3 dom(12)=11 [1] [3] [8] 13 l i dom(13)=12 5 dom(5)=1 [12] [1] dom(6)=1 k 6 75 [1]

  38. The Lengauer-Tarjan Algorithm: Example delete 2 from bucket(1) r 1 eval(2) = 2 c 2 [1] b dom(2)=1 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f dom(3)=1 a 11 dom(11)=1 [2] [1] [1] [1] 10 4 dom(10)=1 h d 12 j dom(4)=3 dom(12)=11 [1] [3] [8] 13 l i dom(13)=12 5 dom(5)=1 [12] [1] dom(6)=1 k 6 76 [1]

  39. The Lengauer-Tarjan Algorithm: Example dom(12)  semi(12) r 1 set dom(12)=dom(11) c 2 [1] b dom(2)=1 8 dom(8)=1 [1] dom(7)=2 dom(9)=1 e g 7 3 9 f dom(3)=1 a 11 dom(11)=1 [2] [1] [1] [1] 10 4 dom(10)=1 h d 12 j dom(4)=3 dom(12)=1 [1] [3] [8] 13 l i dom(13)=12 5 dom(5)=1 [12] [1] dom(6)=1 k 6 77 [1]

  40. The Lengauer-Tarjan Algorithm Running Time = O(n + m) + Time for n -1 calls to link() + Time for m + n -1 calls to eval() 78

  41. Data Structure for link() and eval() We want to apply Path Compression : v 0 [ l 0 ] v 0 [ l 0 ] eval( v 3 ) v 1 [ l 1 ] v 2 [ l 2 ] v 3 [ l’ 3 ] v 2 [ l’ 2 ] v 1 [ l’ 1 ] v 3 [ l 3 ] l’ 1 = l 1 semi(l’ 2 ) = min { semi(l 1 ) , semi(l 2 ) } semi(l’ 3 ) = min { semi(l 1 ) , semi(l 2 ) , semi(l 3 ) } 79

  42. Data Structure for link() and eval() We maintain a virtual forest VF such that: 1. For each T in F there is a corresponding VT in VF with the same vertices as T . 2. Corresponding trees T and VT have the same root with the same label. 3. If v is any vertex, eval( v , F ) = eval( v , VF ). Representation: ancestor(v) = parent of v in VT . 80

  43. Data Structure for link() and eval() eval( v ): Compress the path r *  v and return the label of v . link( v , w ): Make v the parent of w . VF satisfies Properties 1-3. Time for n -1 calls to link() + Time for m + n -1 calls to eval() = O(m  log 2+  m / n  n) 81

  44. Experimental Results (Small Graphs) 82

  45. Experimental Results (Large Graphs) 83 Running time in ms . Missing values correspond to execution time >1 h.

  46. The Lengauer-Tarjan Algorithm: Correctness Lemma 1:  v , w such that v  w , any path from v to w contains a common ancestor of v and w in T . Follows from Property 1 Lemma 2: For any w  r , idom(w) is an ancestor of w in T . idom(w) is contained in every path from r to w 84

  47. The Lengauer-Tarjan Algorithm: Correctness Lemma 3: For any w  r , sdom(w) is an ancestor of w in T . • (parent(w), w) is an SDOM-path  sdom(w)  parent(w) . • SDOM-path P = ( v 0 = sdom(w) , v 1 , …, v k = w ); Lemma 1  some v i is a common ancestor of sdom(w) and w . We must have v i  sdom(w)  v i = sdom(w) . 85

  48. The Lengauer-Tarjan Algorithm: Correctness Lemma 4: For any w  r , idom(w) is an ancestor of sdom(w) in T . sdom(w) The SDOM-path from sdom(w) to w avoids the proper ancestors of w that SDOM-path are proper descendants of sdom(w) . w 86

  49. The Lengauer-Tarjan Algorithm: Correctness Lemma 5: Let v , w satisfy v *  w . Then v *  idom(w) or idom(w) *  idom(v) . For each x that satisfies idom(v) idom(v) +  x +  v x there is a path P x from r to v that avoids x . idom(w)  P x P x  v *  w is a path from r to w that avoids x  idom(w)  x . v 87 w

  50. The Lengauer-Tarjan Algorithm: Correctness Theorem 2: Let w  r . If sdom(u)  sdom(w) for every u that satisfies sdom(w) +  u *  w then idom(w) = sdom(w) . r Suppose for contradiction sdom(w)  Dom(w)   path P from r to w that x avoids sdom(w) . sdom(w) P x = last vertex  P such that x < sdom(w) y = first vertex  P  sdom(w) *  w y Q = part of P from x to y w Lemma 1  y < u ,  u  Q – { x , y }  sdom(y) < sdom(w) . 88

  51. The Lengauer-Tarjan Algorithm: Correctness Theorem 3: Let w  r and let u be any vertex for which sdom(u) is minimum among the vertices u that satisfy sdom(w) +  u *  w . Then idom(u) = idom(w) . r x Lemma 4 and Lemma 5  idom(w) *  idom(u) . P idom(u) Suppose for contradiction idom(u)  idom(w) . y   path P from r to w that avoids idom(u) . sdom(w) u 89 w

  52. The Lengauer-Tarjan Algorithm: Correctness Theorem 3: Let w  r and let u be any vertex for which sdom(u) is minimum among the vertices u that satisfy sdom(w) +  u *  w . Then idom(u) = idom(w) . r x = last vertex  P such that x < idom(u) . x y = first vertex  P  idom(u) *  w . P idom(u) Q = part of P from x to y . Lemma 1  y < u ,  u  Q – { x , y } y  sdom(y) < idom(u)  sdom(u) . sdom(w) Therefore y  v for any v that satisfies u idom(u) +  v *  u . 90 But y cannot be an ancestor of idom(u) . w

  53. The Lengauer-Tarjan Algorithm: Correctness From Theorem 2 and Theorem 3 we have sdom  idom : Corollary 1: Let w  r and let u be any vertex for which sdom(u) is minimum among the vertices u that satisfy sdom(w) +  u *  w . Then idom(w) = sdom(w) , if sdom(w) = sdom(w) and idom(w) = idom(u) otherwise. We still need a method to compute sdom . 91

  54. The Lengauer-Tarjan Algorithm: Correctness Theorem 4: For any w  r , sdom(w) = min ( { v | ( v , w )  E and v < w }  { sdom(u) | u > w and  ( v , w )  E such that u *  v } ). Let x = min ( { v | ( v , w )  E and v < w }  { sdom(u) | u > w and  ( v , w )  E such that u *  v } ). We first show sdom(w)  x and then sdom(w)  x . 92

  55. The Lengauer-Tarjan Algorithm: Correctness Theorem 4: For any w  r , sdom(w) = min ( { v | ( v , w )  E and v < w }  { sdom(u) | u > w and  ( v , w )  E such that u *  v } ). • sdom(w)  x Assume x = v such that ( v , w )  E and v < w  sdom(w)  x . 93

  56. The Lengauer-Tarjan Algorithm: Correctness Theorem 4: For any w  r , sdom(w) = min ( { v | ( v , w )  E and v < w }  { sdom(u) | u > w and  ( v , w )  E such that u *  v } ). • sdom(w)  x Assume x = sdom(u) such that u > w and ( v , w )  E for some x = sdom(u) descendant v of u in T . P P = SDOM-path from x to u  u P  u *  v  ( v , w ) is an SDOM-path from x to w . w v 94

  57. The Lengauer-Tarjan Algorithm: Correctness Theorem 4: For any w  r , sdom(w) = min ( { v | ( v , w )  E and v < w }  { sdom(u) | u > w and  ( v , w )  E such that u *  v } ). • sdom(w)  x Assume that ( sdom(w) , w )  E  sdom(w)  x . 95

  58. The Lengauer-Tarjan Algorithm: Correctness Theorem 4: For any w  r , sdom(w) = min ( { v | ( v , w )  E and v < w }  { sdom(u) | u > w and  ( v , w )  E such that u *  v } ). • sdom(w)  x Assume that P = ( sdom(w) = v 0 , v 1 , … , v k = w ) is a simple path v i > w , 1  i  k -1. sdom(w) P j = min { i  1 | v i *  v k-1 }. w v j Lemma 1  v i > v j , 1  i  j -1  x  sdom(v j )  sdom(w) . 96 v k-1

  59. The Lengauer-Tarjan Algorithm: Almost-Linear-Time Version We get better running time if the trees in F are balanced (as in Set-Union). F is balanced for constants a > 1, c > 0 if for all i we have: # vertices in F of height i  cn/a i Theorem 5 [Tarjan 1975]: The total length of an arbitrary sequence of m path compressions in an n -vertex forest balanced for a , c is O((m+n)   (m+n, n)) , where the constant depends on a and c . 97

  60. Linear-Time Algorithms There are linear-time dominators algorithms both for the RAM Model and the Pointer-Machine Model. • Based on LT, but much more complicated. • First published algorithms that claimed linear-time , in fact didn’t achieve that bound. RAM: Harel [1985]  Alstrup, Harel, Lauridsen and Thorup [1999] Pointer-Machine: Buchsbaum, Kaplan, Rogers and Westbrook [1998]  G. and Tarjan [2004], 98  Buchsbaum, G., Kaplan, Rogers, Tarjan and Westbrook [2008]

  61. GT Linear-Time Algorithm: High-Level View Partition DFS-tree D into nontrivial microtrees and lines. Nontrivial microtree: Maximal subtree of D of size  g that contains at least one leaf of D . Trivial microtree: Single internal vertex of D . Line: Maximal unary path of 1 trivial microtrees. lines 2 16 3 17 20 27 30 4 5 6 13 18 19 21 28 29 31 7 14 15 22 8 9 23 24 nontrivial 10 25 g =3 microtrees 99 11 12 26

  62. GT Linear-Time Algorithm: High-Level View Partition DFS-tree D into nontrivial microtrees and lines. Nontrivial microtree: Maximal subtree of D of size  g that contains at least one leaf of D . Trivial microtree: Single internal vertex of D . Line: Maximal unary path of 1 trivial microtrees. 2 16 {1} Core C : Tree D – nontrivial microtrees 3 20 {2,3,6,7,9} {16,20,21,22} 6 21 C’ : contract each line of C to C’ a sigle vertex 7 22 9 C 100

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