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Definitions A monoid is a triple (A, ⊗, 1) where 1. ⊗ is a closed associative binary operator on the set A, 2. 1istheidentity element for ⊗, i.e., for all a ∈ A, a ⊗ 1=1 ⊗ a = a. A monoid is commutative if ⊗ is commutative. Liang Huang 7 Dynamic Programming
Definitions A monoid is a triple (A, ⊗, 1) where 1. ⊗ is a closed associative binary operator on the set A, 2. 1istheidentity element for ⊗, i.e., for all a ∈ A, a ⊗ 1=1 ⊗ a = a. A monoid is commutative if ⊗ is commutative. ([0, 1], +, 0) ([0, 1], , 1) ([0, 1], max, 0) Liang Huang 7 Dynamic Programming
Page 1 and 2: Dynamic Programming Algorithms in S
Page 3 and 4: Dynamic Programming Liang Huang 3 D
Page 5 and 6: Dynamic Programming • Dynamic Pro
Page 7 and 8: Two Dimensional Survey traversing o
Page 9: Motivations for Semirings • in a
Page 13 and 14: Definitions A monoid is a triple (A
Page 15 and 16: Examples Semiring Set ⊕ ⊗ 0 1 i
Page 17 and 18: Ordering • idempotent 9 A semirin
Page 19 and 20: Ordering • idempotent A semiring
Page 21 and 22: Monotonicity • monotonicity 10 Li
Page 23 and 24: • monotonicity Monotonicity Let K
Page 29 and 30: Two Dimensional Survey traversing o
Page 31 and 32: Viterbi Algorithm for DAGs 1. topol
Page 33 and 34: Examples Liang Huang 15 Dynamic Pro
Page 35 and 36: Examples • [Number of Paths in a
Page 41 and 42: Example: Word Alignment Liang Huang
Page 43 and 44: Dijkstra Algorithm • Dijkstra doe
Page 51 and 52: Dijkstra Algorithm • keep a cut (
Page 53 and 54: Viterbi vs. Dijkstra • structural
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What if both work? monotonic optimi
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Two Dimensional Survey traversing o
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Background: CFG and Parsing Liang H
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Hypergraphs and Deduction tails : a
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Hypergraphs and Deduction tails : a
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Weight Functions and Semirings d(u)
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Viterbi Algorithm for DAGs 1. topol
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Example: CKY Parsing • parsing wi
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Forward Variant for DAHs 1. topolog
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Example: Treebank Parsers • State
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Dijkstra Algorithm • keep a cut (
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Knuth (1977) Algorithm • keep a c
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Knuth (1977) Algorithm • keep a c
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Example: A* Parsing • Use A* sear
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Same Picture Again monotonic optimi
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Conclusion • surveyed two general
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