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MinFill_Presentation
- 1. Minimum Fill-In Anna Lasota Berlin, December 2003
- 2. Overview Introduction • What is Fill-In ? • Perfect Elimination and Chordal Graphs • The Minimum Fill-In Problem Heuristics • Minimum Degree • Minimum Deficiency • Nested Dissection • Hybrid Algorithm Approximation Algorithm
- 3. Sparse Systems of Linear Equations Ax = b sparse, symmetric, positive definite matrices arise frequently in physical applications such as • finite element method • electrical network analysis • analysis of structural systems famous direct methods to solve a linear system Ax = b are • Gauss Elimination • Cholesky Factorization
- 4. Example Finite Element Method - Bath Tub
- 5. Solving Ax = b with Gauss a11 . . . a1n ... ... ... an1 ann x1 ... xn = b1 ... bn eliminate first column a11x1 + · · · + a1nxn = b1 ⇐⇒ x1 = (−1 (a12x2 + · · · + a1nxn) + b1)/a11 substitute x1 in subsequent equations a21x1 + . . . + a2nxn = b2 ... ... ... ... an1x1 + . . . + annxn = bn
- 6. Introduction of Fill on a 3 × 3 Matrix a11 a12 a13 a21 a22 a31 a33 x1 x2 x3 = b1 b2 b3 eliminate first column a11x1 + a12x2 + a13x3 = b1 ⇐⇒ x1 = (−1 (a12x2 + a13x3) + b1)/a11 substitute x1 in subsequent equations a21x1 + a22x2 + = b2 a31x1 + + a33x3 = b3 =⇒ a22 a23 a32 a33 x2 x3 = b2 b3 new non-zero elements a23 and a32 are called Fill-In.
- 7. Gauss Elimination Process and the Graph Theoretic View 1 2 3 4 × × × ×××× × × × 2 3 4 ××× × × × ×• • • • • • 3 4 ××× × × × × •• • • 4 ××× × × × × • ••
- 8. Disadvantages of Fill-In Fill-In increases the storage requirement Fill-In increases the arithmetic operations needed to solve the system Questions that arise: 1. Are there instances where no fill is introduced? 2. Is there a way to minimize the Fill-In introduced during elimination ?
- 9. Modelling the Elimination Process on a Graph Let the graph G = (V, E) correspond to A and let the vertex vi correspond to xi. Theorem: Elimination Graph [Parter 1961] Eliminating xi from the subsequent equations, the new graph Gvi of the remaining system is formed by: 1. eliminating the vertex vi that corresponds to xi from Gvi−1 2. pair-wise connecting the vertices of N(vi). v1 v2 v2 v3 v3 v3 v4v4 v4v4 G Gv1 Gv2 Gv3 →→ →
- 10. Modelling the Elimination Process on a Graph Proof: assume aii = 0, then xi = −1/aii k=i aikxk + yi/aii if aij = 0 no substitution need be made if aij = 0 and aik = 0 then • introduce xj to the kth equation and • introduce xk to the jth equation
- 12. Solving Ax = b with Cholesky We can decompose A into A = LLT We know aij =< li, lj > constructing L • Algorithm at blackboard A = LLT = LU, where LT = U
- 13. Cholesky Factorization and Fill-In × × × × × × × ××× A = 1 2 3 4 × × × × ××× 1 2 3 4 × × × × ××× •→ 1 2 3 4 × × × × ××× • •→ 1 2 3 4 × × × ×× ×× • ••→ 1 2 3 4 × × × × ××× • •• U =
- 14. Overview Introduction • What is Fill-In ? • Perfect Elimination and Chordal Graphs • The Minimum Fill-In Problem Heuristics • Minimum Degree • Minimum Deficiency • Nested Dissection • Hybrid Algorithm Approximation Algorithm
- 15. Perfect Elimination Definition: [Rose 1972] A is a perfect elimination matrix if there exists a permutation matrix P such that ˜A = PAPT = LLT and aij = 0 ⇒ lij = 0, i < j. Definition: For a graph G = (V, E) with |V | = n an ordering of V is a bijection σ : {1, 2, . . . , n} → V . 1 1 22 3 3 4 4 v1v1 v1v1 v2v2 v2v2 v3v3 v3v3 v4 v4v4 v4 × × × × × × × ×× × × × × × × × × × × × × × ×× × × × ×× × × ××× • •• → LT = LT =
- 16. Chordal Graph (Hanjal und Suranyi [1958]) Definition: A Graph G = (V, E) is chordal if every cycle of length ≥ 4 has a chord. A chord is an edge connecting two nonconsecutive vertices of the cycle. G1 chordal G2 not chordal G3 not chordal The class of chordal graphs is the class of perfect elimination graphs.
- 17. Definition: The ordered graph Gσ is monotone transitive when ∀v ∈ V (v, u1) ∈ E and (v, u2) ∈ E =⇒ (u1, u2) ∈ E, with σ−1 (v) < σ−1 (u1) and σ−1 (v) < σ−1 (u2) Theorem: (Rose [1972]) For G = (V, E) the following statements are equivalent: A is a perfect elimination matrix. ∃ an ordering σ of V such that Gσ is monotone transitive σ is a perfect elimination order G is chordal
- 18. What happens if G is not chordal ? v1 v2 v3 v4 v5 v6 G
- 19. Elimination Graph v1 v2 v3 v4 v5 v6 G
- 20. Elimination Graph v1 v2 v3 v4 v5 v6 G
- 21. Elimination Graph v2 v3 v4 v5 v6 Gv1
- 22. Elimination Graph v2 v3 v4 v5 v6 Gv1
- 23. Elimination Graph v2 v3 v4 v5 v6 Gv1
- 24. Elimination Graph v2 v3 v4 v5 v6 Gv1
- 25. Elimination Graph v2 v3 v4 v5 v6 Gv1
- 26. Elimination Graph v2 v3 v4 v5 v6 Gv1
- 27. Elimination Graph v3 v4 v5 v6 Gv2
- 28. Elimination Graph v3 v4 v5 v6 Gv2
- 29. Elimination Graph v3 v4 v5 v6 Gv2
- 30. Elimination Graph v3 v4 v5 v6 Gv2
- 31. Elimination Graph v3 v4 v5 v6 Gv2
- 32. Elimination Graph v4 v5 v6 Gv3
- 33. Elimination Graph v4 v5 v6 Gv3
- 36. The filled graph G+ v1 v2 v3 v4 v5 v6 G+ σ1 The implicit ordering σ1 of the vertices causes a fill of 4. G+ σ1 = (V, E ∪ F) is the filled graph, where |F| is the size of the fill.
- 37. There’s a better ordering v1 v1v2 v2v3 v3 v4 v4v5 v5v6 v6 G+ σ2 G+ σ1
- 38. Chordal Completion G+ σ is monotone transitive Eliminate vertices from G+ σ according to the same order σ used to obtain G+ σ from G G+ = (V, E ∪ F) is chordal. G+ is the chordal completion of G v1 v1v2 v2v3 v3 v4 v4v5 v5v6 v6 G+ σ2 G+ σ1
- 39. Overview Introduction • What is Fill-In ? • Perfect Elimination and Chordal Graphs • The Minimum Fill-In Problem Heuristics • Minimum Degree • Minimum Deficiency • Nested Dissection • Hybrid Algorithm Approximation Algorithm
- 40. Problem Formulation Minimum Fill-In Problem Instance: Graph G = (V, E) Task: Find a set of edges F of smallest size such that G = (V, E ∪ F) is chordal. also known as Minimum chordal graph completion Problem
- 41. Minimum Fill-In is NP-hard SAT ↓ 3 - SAT ↓ Simple Max Cut ↓ Simple Optimal Linear Arrangement ⇒ Optimal Linear Arrangement ↓ Minimum Fill-In
- 42. Minimum Fill-In vs. Minimal Fill-In G minimal chordal completion minimum chordal completion • minimal chordal completion (inclusion-minimal set of edges) can be computed in poly- nomial time • can be far from minimum
- 43. Overview Introduction • What is Fill-In ? • Perfect Elimination and Chordal Graphs • The Minimum Fill-In Problem Heuristics • Minimum Degree • Minimum Deficiency • Nested Dissection • Hybrid Algorithm Approximation Algorithm
- 44. Minimum Degree Algorithm repeatedly finds a vertex of minimum degree and eliminates it Enhancements • Mass Elimination using Supernodes [George, Liu 1980] • Multiple Elimination [Liu 1985] • External Degree [Liu 1985] • Approximate Degree [Amestoy, Davis, Duff 1996] fast algorithm, runs in O(nm+ ), where m+ = |E ∪ F|
- 45. Minimum Degree Algorithm 1 2 3 4 5 6 G
- 46. 1. Iteration - MD 1 2 3 4 5 6 G
- 47. 2. Iteration - MD 2 3 4 5 6 G
- 48. 3. Iteration - MD 2 4 5 6 G
- 49. 4. Iteration - MD 2 5 6 G
- 50. 5. Iteration - MD 5 6 G
- 51. 6. Iteration - MD 6 G
- 52. Advantages - Disadvantages Minimum Degree Algorithm Advantages Disadvantages fast algorithm does not recognize chordal graphs does not in general produce a minimal ordering triangulation produced is arbitrarily greater than a minimum triangulation
- 53. Example Let p ≤ q Kq v a1 a2 . . . apa3
- 54. Example Let p ≤ q Kq v a1 a3 . . . apa2
- 55. Example Let p ≤ q Kq v a1 a3 . . . apa2
- 56. Example Let p ≤ q Kq v a1 a3 . . . apa2 deg(v) = p deg(ai) = q + 1 deg(kj) = q − 1 + p
- 57. Example Kq v a1 a3 . . . apa2
- 58. Example Kq v a1 a3 . . . apa2
- 62. A better way Kq v a1 a3 . . . apa2
- 63. A better way Kq v a1 a3 . . . apa2
- 64. A better way Kq v a1 a3 . . . apa2
- 65. A better way Kq v a1 a3 . . . apa2 Turn N(a1) into a clique adding q edges.
- 66. A better way Kq+1 v a3 . . . apa2
- 67. A better way Kq+1 v a3 . . . apa2
- 68. A better way Kq+1 v a3 . . . ap
- 69. A better way Kq+1 v a3 . . . ap
- 73. Performance Guarantee dotsdots total fill of p(p − 1)/2, O(p2 ) Kq+p a1 a3 apa2 total fill of q Kq+1 v
- 74. Minimum Degree Algorithm repeatedly finds a vertex of minimum degree and eliminates it Enhancements • Mass Elimination using Supernodes [George, Liu 1980] • Multiple Elimination [Liu 1985] • External Degree [Liu 1985] • Approximate Degree [Amestoy, Davis, Duff 1996] fast algorithm
- 75. Quotientgraph and Mass Elimination 1. Quotientgraph - Supernodes and Enodes, Reachable Set 2. Mass Elimination - u, v ∈ V are indistinguishable in G if NG(u)∪{u} = NG(v)∪{v}
- 76. Minimum Degree Algorithm repeatedly finds a vertex of minimum degree and eliminates it Enhancements • Mass Elimination using Supernodes [George, Liu 1980] • Multiple Elimination [Liu 1985] • External Degree [Liu 1985] • Approximate Degree [Amestoy, Davis, Duff 1996] fast algorithm
- 77. Multiple Elimination Determine the minimum degree Choose the vertices that have the minimum degree Mark their neighbors as unknown degree
- 78. 1. Iteration Multiple Elimination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
- 79. 1. Iteration Multiple Elimination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
- 80. 1. Iteration Multiple Elimination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
- 81. 1. Iteration Multiple Elimination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
- 82. 1. Iteration Multiple Elimination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
- 83. 1. Iteration Multiple Elimination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
- 84. 1. Iteration Multiple Elimination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
- 85. 1. Iteration Multiple Elimination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
- 86. 2. Iteration Multiple Elimination 2 3 5 6 7 8 9 10 11 12 14 15
- 87. 2. Iteration Multiple Elimination 2 3 5 6 7 8 9 10 11 12 14 15
- 88. 2. Iteration Multiple Elimination 2 3 5 6 7 8 9 10 11 12 14 15
- 89. 2. Iteration Multiple Elimination 2 3 5 6 7 8 9 10 11 12 14 15
- 90. 2. Iteration Multiple Elimination 2 3 5 6 7 8 9 10 11 12 14 15
- 91. 2. Iteration Multiple Elimination 2 3 5 6 7 8 9 10 11 12 14 15
- 92. 3. Iteration Multiple Elimination 3 5 6 7 10 11 12 14
- 93. 4. Iteration Multiple Elimination 5 6 7 10 11 12
- 94. 4. Iteration Multiple Elimination 6 7 10 11 12
- 95. 4. Iteration Multiple Elimination 6 7 10 11 12
- 96. 4. Iteration Multiple Elimination 6 7 10 11 12
- 97. 4. Iteration Mass Elimination
- 98. Minimum Degree Algorithm repeatedly finds a vertex of minimum degree and eliminates it Enhancements • Mass Elimination using Supernodes [George, Liu 1980] • Multiple Elimination [Liu 1985] • External Degree [Liu 1985] • Approximate Degree [Amestoy, Davis, Duff 1996] fast algorithm
- 99. External and Approximate Degree For a set of vertices X ⊂ V , N(X) = v∈X N(v) − X external degree of X is |N(X)|, that is the number of adjacent vertices that belong to other supernodes. NOT the number of adjacent supernodes approximate degree computes an upper bound on the degrees inexpensively instead of computing the exact degrees
- 100. Enhancements Runtime Algorithm Runtime MD O(nm+ ) MD QuotientGraph O(n2 m) MMD O(n2 m) AMD O(nm)
- 101. Overview Introduction • What is Fill-In ? • Perfect Elimination and Chordal Graphs • The Minimum Fill-In Problem Heuristics • Minimum Degree • Minimum Deficiency • Nested Dissection • Hybrid Algorithm Approximation Algorithm
- 102. Minimum Deficiency The deficiency of v is the number of edges needed to turn the neighborhood of a vertex v into a clique The deficiency corresponds to the number of fill edges introduced by eliminating v Try to minimize the overall amount of fill by, at each step, choosing to eliminate a vertex with minimum deficiency
- 103. Minimum Deficiency Algorithm v1 v2 v3 v4 v5 v6
- 104. Minimum Deficiency Algorithm v1 v2 v3 v4 v5 v6 1 3 1 1 3 1
- 105. Minimum Deficinecy Algorithm v2 v3 v4 v5 v6 2 1 0 2 1
- 106. Minimum Deficiency Algorithm v2 v3 v5 v6 1 1 1 1
- 107. Minimum Deficiency Algorithm v3 v5 v6 0 0 0
- 108. Minimum Deficiency Algorithm v5 v6 0 0
- 110. Advantages - Disadvantages Minimum Deficiency Algorithm Advantages Disadvantages good quality ordering implementation too expensive recognizes chordal graphs not always a minimal triangulation 1 1 1 1 1 11 33
- 111. Overview Introduction • What is Fill-In ? • Perfect Elimination and Chordal Graphs • The Minimum Fill-In Problem Heuristics • Minimum Degree • Minimum Deficiency • Nested Dissection • Hybrid Algorithm Approximation Algorithm
- 112. Nested Dissection [George 1973] Nested Dissection examines the graph as a whole before ordering • orders the vertices of the graph backwards • begins by deciding which vertices should be eliminated last works by selecting a balanced separator • a set of vertices, that when removed from the graph, partitions it into connected components vertices in the separator are placed last in the elimination order recursively orders each of the connected components
- 118. Performance Bound At most O(n log n) fill for graphs with small separators (O( √ n))[Lipton, Rose, Tarja 1979] • planar graphs • graphs with bounded genus • graphs with bounded degree Approximation algorithm [Agraval, Klein, Ravi 1990] with performance bound O( √ d log4 n) • for bounded degree graphs Theoretical result [Gilbert 1989]: for any graph there exisits a ND algorithm whose fill is within O(d log n) of minimum, where d is the maximum degree.
- 119. Advantages - Disadvantages Nested Dissection Advantages Disadvantages good ordering for limited practical use solving Ax = b in parallel finding a minimum balanced separator is NP-hard no fast algorithms for finding approximate separators
- 120. Questions left What properties should the vertex separator have? What algorithms should be used to find the vertex separator? When should the nested recursion be halted? How should the separator vertices be ordered? In Practice When connected component small enough use Minimum Degree Algorithm
- 121. Overview Introduction • What is Fill-In ? • Perfect Elimination and Chordal Graphs • The Minimum Fill-In Problem Heuristics • Minimum Degree • Minimum Deficiency • Nested Dissection • Hybrid Algorithm Approximation Algorithm
- 122. Hybrid Algorithms take advantage of of the best characteristics of Minimum Degree and Nested Dissection using a few levels of separators to control the fill introduced by minimum degree orders Hendrickson and Rothberg[1996] is the current champion of ordering algorithms - BEND (Bruce and Ed’s Nested Dissection) algorithm has neither known worst case fill nor work analysis
- 123. BEND - Combining Nested Dissection and Minimum Degree compress the graph Nested Dissection finds a few levels of separators until connected components have size at most n/32 vertices • using a multilevel algorithm to find vertex separators, allowing some imbalance invoke Minimum Degree on the subgraphs finally apply Minimum Degree to reorder all separator vertices
- 124. Difference between Nested Dissection and BEND Nested Dissection BEND select a balanced separator use multilevel to find separator allowing some imbalance recursively order each subgraph recursively order each subgraph until small enough until n/32 vertices in separator are reorder separator vertices placed last in any order using Minimum Degree
- 125. Overview Introduction • What is Fill-In ? • Perfect Elimination and Chordal Graphs • The Minimum Fill-In Problem Heuristics • Minimum Degree • Minimum Deficiency • Nested Dissection • Hybrid Algorithm Approximation Algorithm
- 126. Approximation Algorithm Approximation Ratio is O(8 · OPT) G = (V, E) has minimum fill size k identify in G a kernel set of vertices A ⊆ V , |A| ≤ 4k • using ideas of the partition algorithm • triangulate G by adding only edges between vertices of A no prior knowledge of k
- 127. Summary Introduction • Fill-In • Perfect Elimination and Chordal Graphs • The Minimum Fill-In Problem Heuristics • Minimum Degree • Minimum Deficiency • Nested Dissection • Hybrid Algorithm Approximation Algorithm