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Ordering • idempotent A semiring (A, ⊕, ⊗, 0, 1) is idempotent if for all a in A, a ⊕ a = a. • comparison (a ≤ b) ⇔ (a ⊕ b = a) defines a partial ordering. • examples: boolean, viterbi, tropical, real, ... ({0, 1}, ∨, ∧, 0, 1) (R + ∪{+∞}, min, +, +∞, 0) ([0, 1], max, ⊗, 0, 1) (R ∪{+∞}, min, +, +∞, 0) Liang Huang 9 Dynamic Programming
Ordering • idempotent A semiring (A, ⊕, ⊗, 0, 1) is idempotent if for all a in A, a ⊕ a = a. • comparison (a ≤ b) ⇔ (a ⊕ b = a) defines a partial ordering. • examples: boolean, viterbi, tropical, real, ... ({0, 1}, ∨, ∧, 0, 1) (R + ∪{+∞}, min, +, +∞, 0) ([0, 1], max, ⊗, 0, 1) (R ∪{+∞}, min, +, +∞, 0) • total-order for optimization problems A semiring is totally-ordered if ⊕ defines a total ordering. • examples: all of the above 9 Liang Huang 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 and 10: Motivations for Semirings • in a
Page 11 and 12: Definitions A monoid is a triple (A
Page 13 and 14: Definitions A monoid is a triple (A
Page 15 and 16: Examples Semiring Set ⊕ ⊗ 0 1 i
Page 17: Ordering • idempotent 9 A semirin
Page 21 and 22: Monotonicity • monotonicity 10 Li
Page 23 and 24: • monotonicity Monotonicity Let K
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
Page 61 and 62: What if both work? monotonic optimi
Page 65 and 66: Background: CFG and Parsing Liang H
Page 67 and 68: 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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