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