slides

• monotonicity
Monotonicity
Let K =(A, ⊕, ⊗, 0, 1) be a semiring, and ≤ a partial ordering over A.
We say K is monotonic if for all a, b, c ∈ A
(a ≤ b) ⇒ (a ⊗ c ≤ b ⊗ c) (a ≤ b) ⇒ (c ⊗ a ≤ c ⊗ b)
• optimal substructure in dynamic programming
A: b ⊗ c
A: b’ ⊗ c
b ⊗ c
• [lemma] idempotent => monotone
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Liang Huang
our focus, totally-ordered semirings, are monotone
10
free lunch!
Dynamic Programming