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DP on Graphs • optimization problems on graphs => generic shortest-path problem • weighted directed graph G=(V, E) with a function w that assigns each edge a weight from a semiring • compute the best weight of the target vertex t • generic update along edge (u, v) w(u, v) d(v) ⊕ = d(u) ⊗ w(u, v) • how to avoid cyclic updates? • only update when d(u) is fixed d(v) ← d(v) ⊕ (d(u) ⊗ w(u, v)) Liang Huang 12 Dynamic Programming
Viterbi Algorithm for DAGs 1. topological sort 2. visit each vertex v in sorted order and do updates • for each incoming edge (u, v) in E • use d(u) to update d(v): • key observation: d(u) is fixed to optimal at this time w(u, v) d(v) ⊕ = d(u) ⊗ w(u, v) • time complexity: O( V + E ) Liang Huang 13 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 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: Two Dimensional Survey traversing o
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 63 and 64: Two Dimensional Survey traversing o
Page 65 and 66: Background: CFG and Parsing Liang H
Page 67 and 68: Hypergraphs and Deduction tails : a
Page 69 and 70: Hypergraphs and Deduction tails : a
Page 71 and 72: 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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